Identifying Critical Mass in the Global Cellular Telephony Market

Identifying Critical Mass in the Global Cellular Telephony Market

Michał Grajek ESMT European School of Management and Technology, Schlossplatz 1, 10178 Berlin, Germany, [email protected]

Tobias Kretschmer

Institute for Strategy, Technology and Organization, LMU Munich, Schackstr. 4/III, 80539 Munich, Germany, ifo Institute and Centre for Economic Performance, [email protected]

June 2012

Abstract:

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.

Keywords:

Critical Mass, Network Effects, Technology Diffusion, Cellular Telephony

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.


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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; Lee et al. 2003; Golder and

Tellis 1997), a “catastrophe” (Cabral 1990, 2006), a “punctuated equilibrium” (Loch and

Huberman 1999) or “critical mass” (Mahler and Rogers 1999; Cool et al. 1997; Markus 1987;

Evans and Schmalensee 2010). 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 (Loch and Huberman 1999; Grajek and Kretschmer 2009). 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, 2006; Rohlfs 1974; Kretschmer 2008;

Granovetter 1978; Markus 1987), and research identifying multiple stable equilibria separated by

an unstable one (Evans and Schmalensee 2010; Economides and Himmelberg 1995; 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


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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 (Ohashi

2003; Gowrisankaran and Stavins 2004; Dubé et al. 2010). 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


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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 cent) 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 effects1 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 (Ohashi 2003; Dranove and Gandal 2003; Jenkins et al. 2004; Dubé et al. 2010;

Cantillon and Yin 2008) 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 (Saloner and Shepard 1995; Gowrisankaran and Stavins 2004) 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.


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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 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 (Golder and Tellis 1997; Agarwal and Bayus

2002; 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

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.


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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).

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. 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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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 txvu , , where v

is the individual preference parameter, tx is lagged network size at time t , and the perception

lag is a non-negative number. Further, we assume v to be distributed according to a

CDF vF , and txvu , 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 tx .

Including lagged network size tx in the willingness-to-pay function captures network

effects in demand for the good, and the perception lag 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.

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 , consumers are rational as their subscription decisions resemble those by forward-looking consumers.


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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:

(1) tt pxvu , .

There is one market price, pt. If (1) is non-negative, the consumer will join, otherwise not. The

consumer indifferent between joining or staying out at time t ( *tv ) is given by the following

equation:

(2) ttt pxvu ,* .

All consumers with *tvv will join. Define

(3) ** 1 tt vFvH ,

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:

(4) *tt vHx .

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:

(5) tt xx .

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


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failed products. Note that the steady-state equilibrium coincides with the static fulfilled-

expectations equilibrium in the literature (Rohlfs 1974, Katz and Shapiro 1985).

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 vF and the willingness-

to-pay function txvu , 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 by equation (4) (Cabral 1990). Because H maps the

change in network size from time t to t, it is convenient to think of it as of a function of

lagged network size tx . We calculate the derivatives of H with respect to lagged network

size tx and price p in Appendix A.1. The slope of H increases in the strength of installed

base effects measured by

t

tt

x

xvu ),( *

, as shown in Lemma 1.

Figure 1 illustrates the diffusion dynamics. In the top panel we show H as a function of

lagged network size tx . Given the steady-state condition (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

Figure 1. The dynamics of the model let us discriminate among these multiple steady states.

Suppose market price is *p in Figure 1. According to state equation (4), network size will evolve

as indicated in the top panel. If it starts at some size x < x’, it will eventually reach 0x ; if x > x’,


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it will end up in x’’. If x = x’, it will stay there, but any arbitrarily small shock will lead to an

equilibrium at 0x or x’’. Therefore, 0x and x’’ are stable steady states, whereas x’ is unstable.

--------------------------------------------- INSERT FIGURE 1 ABOUT HERE ---------------------------------------------

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 pD in the lower panel of Figure 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 pD 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.

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 pD . 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 tp be a continuous

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.


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and decreasing function of time, and let ht pp 0 (as in Figure 1) and otpx be the unique

steady-state network size given 0tp . As price falls, network size initially follows the low-

adoption steady state. Eventually price reaches lp and network size jumps to the high-adoption

steady state and progresses along it. Thus, the low-adoption steady-state network size at price lp

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

hp 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 hp and lp . 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 lp , the suppliers need to grow the installed base to critical mass point (x’ in Figure

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.


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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.

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.

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

7 This data has 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


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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 one-minute 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

------------------------------------------- INSERT TABLE 1 ABOUT HERE -------------------------------------------

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

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.


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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.

We specify consumer v’s willingness-to-pay function as follows:

(6) 21,1,1,, tititi dXcXvXvu ,

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 1, tiX 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

11 In the Appendix we show how the model simplifies to the Bass diffusion equation. 12 Note that in the empirical model, δ is determined by data frequency. Consequently, we replace δ with 1 meaning “one period” from now.


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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:

(7) ai,t = a0i + a1(GDP/POP)i,t

(8) bi,t = bPOPi,t.

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 willingness-to-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 differently-sized

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 still have a much smaller effect on

the subscriptions (in absolute terms) in Greece than in the US in the model. This restriction does

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


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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 equation (4) becomes

(9) 211,,10, )/( tttitiiti bdXbcXbpPOPGDPbabaX .

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

(10) tititititiiti XXpPOPGDPX ,2

1,21,1,,10, )/( ,

where ti, 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 equation (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 . Unless we have

population size is defined here in absolute terms and not in relative terms as in the theoretical model in section 3. 14 We estimated such a model and the installed base parameters turned out statistically insignificant.


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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 instruments depends on the extent to which the cost structure of 2G operators

is correlated across countries. The existence of a global input market for the telecommunications

industry suggests that cost structures will be significantly correlated.


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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 equation (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.

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.


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Consequently the IV 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.



19

We can use the parameters reported in Table 3 to identify combinations of installed base ( 1tX )

and prices ( tp ) that give an upward-sloping long-run demand curve – points at which critical

mass occurs. Figure 2 gives steady-state demand functions based on the parameters in Table 3.

------------------------------------------- INSERT FIGURE 2 ABOUT HERE -------------------------------------------

Figure 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 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.


20

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 Figure

3, the long-run 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 could imply that some countries do not display critical mass while

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. 18 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.


21

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 Figure 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

19 Our results are robust to including all three interaction terms simultaneously. 20 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.


22

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.

------------------------------------------- INSERT FIGURE 3 ABOUT HERE -------------------------------------------

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%

(Rogers, 2003; Golder and Tellis, 1997) 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.

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.


23

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 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 Figure 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 upward-sloping 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


24

more pronounced in panel c of Figure 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 Figure 5 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

(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.


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


25

Our counterfactuals indicate that in otherwise identical markets (i.e., with the same structural

parameters and price sensitivity), 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

Figures 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 Figure 1.23 However, the simulated demand functions in Figures 2 to 4

approximate the more general function in Figure 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.

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


26

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

23 Figure 1 implies a long tail in the distribution of types that captures consumers with very high willingness to pay


27

modified to accommodate between-network competition at the firm rather than market level

(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.

even if no one else subscribes (the innovators).


28

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31

Figure 1. Stable vs. unstable equilibria

x’’ x’

p

xt

x

p*

1

pl

ph

0% x0 100%

H(v*(xt-δ,p*))

xt-δ


32

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


33

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


34

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

distribution parameters


35

Table 1. Descriptive statistics

Variable Variable Name Mean Std. Dev. Min Max Cellular penetration Xi,t .676 .317 .009 1.484 Cellular penetration squared X2

i,t .557 .424 .00008 2.202 GDP per capita (GDP/POP)i,t 21.469 13.916 .870 77.589 Population (in millions) POPi,t 80.090 209.500 3.813 1329.48 Average price/minute pi,t .200 .089 .019 .870 Others’ average price pj,t .217 .085 .070 .770 Above median GDP per capita WEALTH .504 .500 0 1 First to adopt 2G PIONEER .231 .422 0 1 Average # of competitors ≥ 3 COMPETITION .694 .461 0 1


36

Table 2. Baseline estimation results

(1) (2) (3) Estimation: OLS IV IV FE

GDP per capita 0.001*** 0.002*** 0.023*** (0.000) (0.001) (0.005) Average price -0.259*** -0.623*** -1.845*** (0.076) (0.151) (0.222) Lagged penetration 1.091*** 1.042*** 0.770*** (0.068) (0.076) (0.203) Lagged penetration squared -0.147** -0.118* -0.370** (0.065) (0.065) (0.151) Constant 0.049*** 0.129*** 0.246a (0.016) (0.035) (0.255) Autocorrelationb 0.055** 0.223*** 1.20 (0.026) (0.072) N 1134 1112 1057 * p<0.1, ** p<0.05, *** p<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).


37

Table 3. Identified structural parameters from the baseline estimation results

(1) (2) (3) Estimation: OLS IV IV FE

a0 0.188*** 0.207*** 0.133a (0.056) (0.030) (0.138) a1 0.006*** 0.003*** 0.012*** (0.001) (0.000) (0.003) b 0.259*** 0.623*** 1.845*** (0.076) (0.151) (0.222) c 4.220*** 1.674*** 0.417*** (1.184) (0.435) (0.144) d -0.570** -0.190* -0.201** (0.221) (0.104) (0.097) Xmax 0.310*** 0.180 -0.311 (0.107) (0.231) (0.397) * p<0.1, ** p<0.05, *** p<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.


38

Table 4. Estimation results using varying data frequency

(1) (2) (3) Frequency: Quarterly Semiannual Annual

GDP per capita 0.023*** 0.003** 0.003** (0.005) (0.001) (0.001) Average price -1.845*** -0.312*** -0.534*** (0.222) (0.097) (0.160) Lagged penetration 0.770*** 0.893*** 0.766*** (0.203) (0.073) (0.102) Lagged penetration squared -0.370** -0.044 -0.027 (0.151) (0.055) (0.085) Constanta 0.246 0.136 0.288 (0.255) (0.048) (0.082) Autocorrelationb 1.20 1.56 -0.86 N 1057 479 191 * p<0.1, ** p<0.05, *** p<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.


39

Table 5. Varying network effects: Estimation results with interaction terms

(1) (2) (3) INDICATOR: WEALTH PIONEER COMPETITION

GDP per capita 0.021*** 0.022*** 0.022*** (0.003) (0.004) (0.004) Average price -1.761*** -1.848*** -1.838*** (0.228) (0.220) (0.223) Lagged penetration 0.523** 0.617** 0.758** (0.221) (0.242) (0.370) Lagged penetration squared -0.101 -0.275 -0.356 (0.151) (0.177) (0.311) Lagged penetration

*INDICATOR 0.465 0.731** -0.002 (0.322) (0.341) (0.420) Lagged penetration squared *INDICATOR -0.448* -0.453* 0.012 (0.229) (0.261) (0.337) Constant (INDICATOR=1)a 0.084 0.026 0.266 (0.218) (0.202) (0.244) Constant (INDICATOR=0)a 0.429 0.331 0.266 (0.165) (0.258) (0.247) Autocorrelationb 1.02 1.11 1.11 N 1057 1057 1057 * p<0.1, ** p<0.05, *** p<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.


40

Appendix

A.1. Derivatives of the function H(.) with respect to xt – δ and p

For simplicity, we slightly abuse the notation in this section by treating price as a constant

parameter p. Recall that vt* is an implicit function of xt – δ and p is defined by

(A.1) u(vt*, xt – δ) = p.

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

(A.2)

t

t

t

tt

t x

v

v

vFvH

x

*

*

*)(*)(

The first term on the RHS of (A.2) is just the density of v at vt*. To calculate the second term,

note that the total derivative of u(vt*, xt – δ) with respect to xt – δ must stay constant in order to

satisfy equation (A.1). This holds for

(A.3)

t

t

t

tt

t

tt

x

v

v

xvu

x

xvu *

*

*,()*,(

Solving (A.3) for

t

t

x

v * and substituting that into (A.2) yields

(A.4)

t

tt

t

tttt

t x

xvu

v

xvuvfvH

x

)*,(

*

*,(*)(*)(

1

,

where f is the density function of v.

Examination of (A.4) gives the following lemma.


41

Lemma 1: Whenever the solution to equation (2) exists and is unique so that vt* is well

defined, the extent of network externalities measured by

t

tt

x

xvu ),( *

determines the

slope of the function H in the xt-δ domain, 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

vt*. The first component of this product is non-negative (density function), the second is 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

(A.5) p

v

v

vFvH

pt

t

tt

*

*

*)(*)( .

Then we note that from (A.1) we have

(A.6) 1*

*

)*,(

p

v

v

xvu t

t

tt ,

and substitute to get

(A.7)

1

*

)*,(*)(*)(

t

tttt v

xvuvfvH

p .


42

Lemma 2 follows directly from examination of (A.7).

Lemma 2: Whenever the solution to equation (2) exists and is unique so that vt* 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’<p’’.

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

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 equation (4) (illustrated by the arrows in the upper panel of Figure

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,


43

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’ is a critical mass point, that is, there exists a price p* for which H(.) crosses the 45-degree

line from below at x’. Therefore, there must exist a neighborhood of x’, ',' xx , such that

'' 1 xxx and H(.) has a positive slope over ',' xx . Note that the density of types that

corresponds to ',' xx and p* must be strictly positive for H(.) to have a positive slope over ',' xx

(Lemma 1). An increase in network effects in the neighborhood ',' xx thus increases the slope of

H(.) over the entire neighborhood, shifting the critical mass point x’ to the left.

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

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:

(A.8) 2111, ttt dxcxvxvu ,

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 equation (4) becomes


44

(A.9) 211 tttt bdxbcxbpabx .

The structural parameters of this model can be recovered from the coefficients of the following

estimation equation:

(A.9) ttttt xxpx 2

1211 ,

where t denotes the error term. Equation (A.9) simplifies to the original Bass model if 0 .

To see this, rearrange the terms to obtain:

(A.10) ttttt xxxx 2

12111 )1( .

The left-hand side of (A.10) corresponds to sales at t and the right-hand 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 equation (9)

Given the willingness-to-pay function (6), an indifferent consumer in market i and time t, v*i,t, is

given by

(A.11)

21,1,

*,, titititi dXcXvp , hence

(A.12) 21,1,,

*, titititi dXcXpv .

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

(A.13) 21,,1,,,,,,, tititititititititi dXbcXbpbbax .

Equation (9) results from substituting (7) and (8) for the parameters ai,t and bi,t in (A.13).


45

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 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

(A.14) 22110 1)/( iii

ii X

β

γX

β

γPOPGDP

β

α

β

αp

.

And the level at which the maximum price is achieved is 2

1max 2

1

γ

γX

. 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) Others’ average price 0.559*** (0.053) Lagged penetration -0.015 (0.061) Lagged penetration squared -0.005 (0.041) Constant 0.057** (0.024) N 1112 * p<0.1, ** p<0.05, *** p<0.01 Robust standard errors in parentheses.

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Identifying Critical Mass in the Global Cellular Telephony Market