Vol. 00, No. 0, Xxxxx 2007, pp. 1–23issn 0732-2399 �eissn 1526-548X �07 �0000 �0001

informs ®

doi 10.1287/mksc.1060.0224©2007 INFORMS

New-Product Diffusion withInfluentials and Imitators

Christophe Van den Bulte, Yogesh V. JoshiThe Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania 19104

{vdbulte@wharton.upenn.edu, yogesh@wharton.upenn.edu}

We model the diffusion of innovations in markets with two segments: influentials who are more in touchwith new developments and who affect another segment of imitators whose own adoptions do not affect

the influentials. This two-segment structure with asymmetric influence is consistent with several theories insociology and diffusion research, as well as many “viral” or “network” marketing strategies. We have four mainresults. (1) Diffusion in a mixture of influentials and imitators can exhibit a dip or “chasm” between the early andlater parts of the diffusion curve. (2) The proportion of adoptions stemming from influentials need not decreasemonotonically, but may first decrease and then increase. (3) Erroneously specifying a mixed-influence modelto a mixture process where influentials act independently from each other can generate systematic changes inthe parameter values reported in earlier research. (4) Empirical analysis of 33 different data series indicatesthat the two-segment model fits better than the standard mixed-influence, the Gamma/Shifted Gompertz, andthe Weibull-Gamma models, especially in cases where a two-segment structure is likely to exist. Also, the two-segment model fits about as well as the Karmeshu-Goswami mixed-influence model, in which the coefficientsof innovation and imitation vary across potential adopters in a continuous fashion.

Key words : asymmetric influence; diffusion of innovations; innovation; market segments; social contagion;social structure

History :

1. IntroductionUnder pressure to increase their marketing ROIthrough more astute targeting of resources, marketersare rediscovering the importance of social contagion.Recent “viral” and “network” marketing strategiesoften share two key assumptions: (1) some customersare more in touch with new developments than oth-ers, and (2) some (often, the same) customers’ adop-tions and opinions have a disproportionate influenceon others’ adoptions (e.g., Gladwell 2000, Moore 1995,Rosen 2000, Slywotzky and Shapiro 1993). Targetingthose influential prospects who are more in touchwith new developments and converting them intocustomers, the logic goes, allows marketers to bene-fit from a social multiplier effect on their marketingefforts. The two assumptions are quite reasonable, asthey are consistent with several theories and a largebody of empirical research (e.g., Katz and Lazars-feld 1955, Rogers 2003, Weimann 1994), and the socialmultiplier logic cannot be faulted either (e.g., Caseet al. 1993, Valente et al. 2003). However, marketingscience provides little or no additional theoretical ordescriptive insight into how new products diffuse insuch markets. The reason is that the great majorityof marketing diffusion models assume homogeneityrather than heterogeneity in the tendency to be in

tune with new developments and the tendency toinfluence (or be influenced by) others. We addressthis gap between theory and emerging practice onthe one hand, and marketing diffusion models on theother. Specifically, we model the aggregate-level dif-fusion path of a new product when the set of ulti-mate adopters is not homogeneous, but consists oftwo segments: influentials who are more in touch withnew developments and who affect another segmentof imitators whose own adoptions do not affect theinfluentials. We allow for the presence or absence ofcontagion among influentials and among imitators.Many diffusion models incorporate the dual drivers

of independent decision making affected by beingin touch with new developments, and of imitationdriven by others’ prior adoptions, but they do sounder the assumption that all potential adopters areex ante equally affected by both factors. Taga andIsii (1959) in statistics; Mansfield (1961), Pyatt (1964),and Williams (1972) in economics; Coleman (1964) insociology; and Bass (1969) and Massy et al. (1970) inmarketing, all advanced a model specifying the rateat which actors who have not adopted yet do so attime t as h�t� = p + qF �t�, where F �t� is the propor-tion of ultimate adopters that have already adopted,parameter q captures social contagion, and parame-ter p captures the time-invariant tendency to adopt

1

Van den Bulte and Joshi: New-Product Diffusion with Influentials and Imitators2 Marketing Science 00(0), pp. 1–23, © 2007 INFORMS

early affected by consumer characteristics, the innova-tion’s appeal, and efforts of change agents.1 Becausethe proportion that adopts at time t can be written asf �t�= dF �t�/dt = h�t��1− F �t�], one obtains:

f �t�= dF �t�/dt = �p+ qF�t���1− F �t�� (1)

The solution of this differential equation can be writ-ten as:

F �t�= �1− e−g−�p+q�t�/�1+ �q/p�e−g−�p+q�t�� (2)

where g acts as a location parameter fixing the curveon the time axis (e.g., Mansfield 1961). When t = 0 cor-responds to the actual launch time such that F �0�= 0,then g = 0 and Equation (2) reduces to the solutionpopular in marketing.The rate is influenced by the intrinsic tendency to

adopt both �p� and social contagion (q� at all timesexcept at t = 0 when qF �0� = 0. To reflect this dualinfluence, Mahajan and Peterson (1985) refer to themodel as the mixed-influence model. Because therate contains no contagion pressure at t = 0, thoseadopting at that time are sometimes referred to asinnovators and contrasted against all others adopt-ing later, who are called imitators (e.g., Bass 1969).However, this terminology can only be used ex post,and the model does not represent a diffusion pro-cess in an ex ante mixture of two segments, thefirst adopting independently at rate p and the sec-ond adopting because of social contagion at rate qF �t�(Bemmaor 1994, Jeuland 1981, Lekvall and Wahlbin1973, Manfredi et al. 1998, Steffens and Murthy 1992,Tanny and Derzko 1988).The objective of this study is to mathematically for-

malize prior theoretical arguments and research find-ings on social structure and diffusion, and to usethis formalization to generate more refined theoret-ical insights on new product diffusion in a popula-tion of influentials and imitators. This is importantbecause marketing practitioners increasingly deploystrategies assuming such a market structure, andbecause marketing researchers increasingly incorpo-rate social structure into their diffusion investiga-tions (e.g., Bronnenberg and Mela 2004, Frenzenand Nakamoto 1993, Garber et al. 2004, Godes andMayzlin 2004, Putsis et al. 1997, Van den Bulte andLilien 2001).Our results offer formalized insights into some cur-

rent substantive and methodological research ques-tions. First, diffusion in a mixture of influentials andimitators can exhibit a dip between the early and

1 Following the convention in marketing, we refer to the rate atwhich nonadopters turn into adopters as the hazard rate anddenote it as h�t�, even though the models we discuss are determin-istic rather than probabilistic.

later parts of the diffusion curve. In contrast to whatMoore (1991) claims, our model shows that it need notalways be necessary for firms to change their prod-uct to gain traction among later adopters and for theadoption curve to swing up again.2 Like Steffens andMurthy (1992) and Karmeshu and Goswami (2001),but unlike Goldenberg et al. (2002), we obtain thisresult from a closed-form solution, and unlike thoseprior analyses, we show that a dip can occur evenwhen influentials act independently from each other.Second, the proportion of adoptions stemming frominfluentials need not decrease monotonically, but mayfirst decrease and then increase. The managementimplication is that while it may make sense to shiftthe focus of one’s marketing efforts from influen-tials to imitators shortly after launch, as shown byMahajan and Muller (1998) using a two-period model,one may want to revert one’s focus back to influen-tials later in the process. Third, erroneously specifyinga mixed-influence model to a two-segment processcan generate the systematic changes in the parame-ter values over time reported in several studies (e.g.,Van den Bulte and Lilien 1997, Venkatesan et al. 2004).This analytical result is a specific formalization ofVan den Bulte and Lilien’s (1997) more general butqualitative argument that unaccounted heterogene-ity in p or q can generate changes in these param-eters’ estimates as one extends the data window.Our result also complements Bemmaor and Lee’s(2002) simulation analysis because we consider het-erogeneity in a process with genuine contagion ratherthan in a Gamma/Shifted Gompertz process withoutcontagion.We also perform an empirical analysis and assess

the descriptive performance of the two-segmentmodel compared to that of the mixed-influence modeland of three diffusion models incorporating hetero-geneity in the form of a continuous rather than adiscrete mixture. Given the difficulty of unambigu-ously identifying causal processes from aggregate dif-fusion data (Bemmaor 1994, Hernes 1976, Lekvall andWahlbin 1973, Lilien et al. 1981, Van den Bulte andStremersch 2004), the objective of this empirical anal-ysis is not to conclusively demonstrate the validityof any model. Rather, it is to assess whether thedifferences between the discrete mixture and othermodels are sufficiently important to lead to differ-ences in descriptive performance when applied todata of interest to marketing researchers. The two-segment model fits better than the mixed influ-ence, Gamma/Shifted Gompertz (Bemmaor 1994),

2 Changing the product might consist of augmenting the core prod-uct with complementary services and products to provide a “wholeproduct,” or consist of offering simpler and more user-friendly ver-sions of the core product.

Van den Bulte and Joshi: New-Product Diffusion with Influentials and ImitatorsMarketing Science 00(0), pp. 1–23, © 2007 INFORMS 3

and Weibull-Gamma models (Hardie et al. 1998,Massy et al. 1970, Narayanan 1992), especially incases where a two-segment structure is likely (oreven known) to exist, and fits about as well as arecently advanced mixed-influence model where pand q vary across potential adopters in a continuousfashion (Karmeshu and Goswami 2001).We proceed by first outlining our model setting,

and within that context, discuss five theories andframeworks that suggest the existence of ex ante influ-entials and imitators. Next, we develop a macro-levelmodel of innovation diffusion in such a setting. Sub-sequently, we discuss how this model relates to thefamiliar mixed-influence model and to prior workon two-segment models. Finally, we report on thedescriptive performance of the influential-imitatormodel compared to that of the mixed-influence andcontinuous-mixture models.

2. Theories Motivating aTwo-Segment Structure ofInfluentials and Imitators

The situation we model is the following. The set ofeventual adopters has a constant size M and consistsof two a priori different types of actors, influentialsand imitators. We use the subscripts 1 and 2 to denoteeach type, and the subscript m to denote the entiremixture population of adopters. We use � to denotethe proportion of Type 1 actors in the population ofeventual adopters (0 ≤ � ≤ 1), and F �t� to denote thecumulative penetration. Finally, w denotes the relativeimportance that imitators attach to influentials’ ver-sus other imitators’ behavior (0≤w ≤ 1). Each type’sadoption behavior is then captured by the followinghazard functions:

h1�t�= p1+ q1F1�t�� (3)

h2�t�= p2+ q2�wF1�t�+ �1−w�F2�t�� (4)

Note the asymmetry in the influence process: Type 1may influence Type 2, but the reverse is not true.Because, ex ante, anyone of Type 1 may influence any-one of Type 2, we label the former influentials and thelatter imitators. When p2 = 0, contagion from influen-tials to imitators �wq2 > 0� is critical for the diffusionprocess among the latter to get started. Obviously,when � = 1 or � = 0, everyone falls into a single seg-ment and the situation reduces to the mixed-influencemodel (MIM). When 0 < � < 1 but w = 0, the modelreduces to two disconnected MIMs and, with furtherrestrictions, to a model with two disconnected logisticor exponential functions (e.g., Moe and Fader 2001,Perrin 1994). Also, when imitators put equal weighton all prior adoptions regardless of origin, then wehave h2�t�= p2+q2Fm�t�, which implies w= � (see §3).

The distinction between influentials and imitatorsis based on what drives their adoption behavior, noton whether they adopt early or late. Hence, the dis-tinction is different from that of innovators versusimitators in Bass (1969) and innovators versus earlyadopters versus early majority versus late majorityversus laggards in Rogers (2003). Conceptually, causaldrivers and time of adoption need not map one-to-one. Empirically, while those adopting early may actindependently of others, and those adopting late maybe subject to contagion, this is not always so: Manyearly adoptions may be driven by contagion, and thebulk of the late adoptions may stem from people notsubject to social contagion (e.g., Becker 1970, Colemanet al. 1966).Several theories and conceptual models suggest

such a two-segment structure, although there is somedisagreement on whether q1 and p2 may be larger thanzero. We first describe sociological arguments focus-ing on social character, social status, and social norms.We then, turn to the two-step flow hypothesis, whichfocuses on interest in new developments, and finallyto the chasm idea, which focuses on enthusiasm forinnovations versus risk aversion.

2.1. Social CharacterIn his classic treatise on the changing nature of mod-ern society, Riesman (1950) distinguished three typesof social character: autonomous, inner directed, andother directed. The first two have in common the pres-ence of clear-cut internalized goals, but differ as towhether these are consciously chosen (autonomous)or inculcated during youth by elders (inner directed).Other-directed actors, in contrast, use their peersas their source of direction. The typology is in es-sence about conformity stemming from the need forapproval and direction from others. Riesman workedon a broad social and cultural canvas and his typol-ogy is best used to refer to patterns of behavior foundin a variety of specific contexts rather than to typesof persons or personalities. However, his conceptshave direct relevance for consumer behavior (e.g.,Riesman 1950, Schor 1998). Some actors in some situa-tions will exhibit autonomous or inner-directed adop-tion behavior independent from their peers (henceq1 = 0), while others will exhibit other-directed behav-ior driven by social contagion from peers. Riesmandid not narrowly specify who these peers are, andallowed them to be all of society (therefore w = �being possible).

2.2. Status Competition and MaintenancePeople buy and use products not only for functionalpurposes, but also to construct a social identity, and toconfirm the existence, and support the reproduction,of social status differences (Bourdieu 1984). A long-held idea in diffusion theory is that people seek to

Van den Bulte and Joshi: New-Product Diffusion with Influentials and Imitators4 Marketing Science 00(0), pp. 1–23, © 2007 INFORMS

emulate the consumption behavior of their superiorsand aspiration groups (e.g., Simmel 1971), and alsoquickly pick up innovations adopted by others of sim-ilar status if they fear that such adoptions might undothe present status ordering (Burt 1987). In short, actorstend to imitate the adoptions of those of higher andsimilar social status.Assuming one can divide the population into a

high-status and a low-status group, status consider-ations suggest that both groups may exhibit conta-gion. Higher-status actors may imitate each other outof fear of falling behind �q1 ≥ 0�, and lower-statusactors imitate to catch up. Whose adoptions the imi-tators act upon is not clear a priori. If they care onlyabout adoptions by the high-status influentials, thenw → 1. However, most authors follow Simmel andposit a finer-grained hierarchy with multiple strata(approximated imperfectly by a dichotomy) and a cas-cading pattern where all prior adoptions contributeequally to social contagion �w = ��. Finally, to theextent that status is maintained by adhering to socialnorms enforced among one’s direct peers of similarposition, imitators should care mostly about fellowimitators �w→ 0�.2.3. Middle-Status ConformityLike theories of status competition and maintenance,middle-status conformity theory is about one’s properplace in society. The main claim is that the relationshipbetween status and conformity to norms—and hencesusceptibility to social contagion—is an inverted U(e.g., Homans 1961, Philips and Zuckerman 2001). Be-cause high-status actors feel confident in their socialacceptance, they feel comfortable in deviating fromconventional behavior and adopt appealing innova-tions independently from others. Low-status actorsfeel free to deviate from accepted practice and adoptinnovations independently as well, because they feelthat this cannot hurt their already low status. Middle-status actors, in contrast, feel insecure and striveto demonstrate their legitimacy by engaging in newpractices only after they have been socially validated.Therefore, middle-status conformity theory is consis-tent with the presence of two kinds of actors, oneadopting as a function of the innovation’s appeal irre-spective of others’ actions �q1 = 0�, and one adoptingas a function of the legitimation stemming from prioradoptions.The theory does not specify whose adoptions are

being imitated �w�. Adoptions by high-status actorsmight legitimate the innovation in the eyes of themiddle-status actors disproportionately, in which casethe relation of w to � is unclear because the lattercaptures both high and low status. Conversely, imita-tors may care only about social acceptability amongtheir middle-status peers, and hence care only about

the latter’s adoptions �w = 0�. Finally, applications ofneoinstitutional theory to innovation adoption tend toposit that the legitimacy of an innovation is affectedby the overall penetration rate �w= ��.Note that higher status is often associated with

higher economic resources, and hence a higher abilityto adopt innovations. This leads to the interesting pre-diction that only the adoptions at an intermediate stage ofthe overall diffusion process �made by middle-status actors�exhibit contagion (e.g., Cancian 1979), because the ear-liest adoptions will come from high-status actors andthe latest from low-status actors, none of which aresubject to contagion.

2.4. Two-Step FlowThe two-step flow hypothesis, originally proposedto explain unexpectedly weak mass media effects inpresidential elections, posits that “ideas often flowfrom radio and print to the opinion leaders and fromthem to the less active sections of the population”(Lazarsfeld et al. 1944, p. 151, emphasis in origi-nal). Therefore, in its original and starkest version,the two-step flow hypothesis posits two groups, onebeing affected only by mass media �q1 = 0� and theother being affected only by social contagion �p2 = 0�.What distinguishes the two groups is the level ofinterest in the subject matter and alertness to newdevelopments rather than exposure to mass commu-nications (Lazarsfeld et al. 1944). Later studies inmarketing have corroborated a strong relationshipbetween opinion leadership and product interest andinvolvement (e.g., Coulter et al. 2002, Myers andRobertson 1972). Note that the two-step flow hypoth-esis does not require that an opinion leader in onesphere (politics, fashion, computer games, etc.) alsobe a leader in another sphere, and several studiesindeed document only moderate to little overlap inleadership across product categories (e.g., Katz andLazarsfeld 1955, Merton 1949, Myers and Robertson1972, Silk 1966). Therefore, the relative size of the seg-ments ��� may vary across innovations. While earlystudies focused on information flows from opinionleaders to less active members of the population,subsequent research has documented extensive infor-mation exchange among opinion leaders and (e.g.,Coulter et al. 2002, Katz and Lazarsfeld 1955) consis-tent with q1 > 0.The two-step flow hypothesis emphasizes the flow

of information. The contagion mechanism is oneof information transfer increasing awareness of theproduct’s existence and decreasing its perceived risk,not of normative legitimation or status competition.Of the five theories we consider, this is perhaps themost flexible. For low-risk innovations, for instance,the fraction of imitators in need of guidance can bequite small, and � quite large. Who is being imitated

Van den Bulte and Joshi: New-Product Diffusion with Influentials and ImitatorsMarketing Science 00(0), pp. 1–23, © 2007 INFORMS 5

Table 1 Theoretical Frameworks Suggesting an Influential-Imitator Mixture

Framework Influentials Imitators Reason to imitate Who gets imitateda

Social character Autonomous and Other-directed Looking for approval and —Not specified, possiblyinner-directed; q1 = 0 direction all adopters �w = ��

Status competition High status; q1 ≥ 0 Low status Gaining or maintaining status —All adopters �w = ��

and maintenance —Only influentials �w = 1�—Only imitators �w = 0�

Middle-status High and low Middle status Conforming to social norms —All adopters �w = ��

conformity status; q1 = 0 —Only influentials withhigh status

—Only imitators �w = 0�

Two-step flow Active and involved Not active or involved Transferring information —All adopters �w = ��

(opinion leaders); q1 ≥ 0 —Only influentials �w = 1�—Only imitators (w = 0)

Technology chasm Technology enthusiasts Mainstream customers Reducing risk —All adopters �w = ��

and visionaries; q1 ≥ 0 —Only imitators �w = 0�

aParameter w denotes how much the social contagion affecting the imitators stems from the influentials �w� rather than fellow imitators �1−w�.Parameter � is the fraction of ultimate adopters belonging to Segment 1 (influentials).

is not clearly specified, and w may range from zero toone. The original two-step flow idea emphasizes thatmass media influence on the less-active segment oper-ates through opinion leaders who are the only ones totake an active interest in information available in themedia. It does so without constraining the social influ-ence exerted on the less-active segment to come onlyfrom opinion leaders, and allows for a cascading orrolling pattern through the population where all prioradoptions contribute to social contagion (e.g., Katz1957, Merton 1949). This suggests w ≈ �. However, itis quite possible that opinion leaders are more influen-tial, suggesting that—in the extreme case—they maybe the only ones being imitated �w= 1�. Conversely,it is also quite possible that imitators consider fellowimitators to be more representative, and hence valu-able as information sources, suggesting low valuesof w.

2.5. High-Technology Adoption ChasmIn Moore’s (1991) chasm framework for technologyproducts, the so-called early market consists of “tech-nology enthusiasts” and “visionaries” who are quickto appreciate the nature and benefits of the inno-vation, whereas the mainstream market consists ofmore risk-averse decision makers and firms who fearbeing stuck with a technology that is not user friendly,poorly supported, or at risk of losing a standardswar. Whereas the mainstream market can be repre-sented as responding only to the size of the installedbase, i.e., prior adoptions (Mahajan and Muller 1998),Moore is unclear about the process among technologyenthusiasts and visionaries. Whereas his textual dis-cussions suggest that they act independently �q1 = 0�,his stylized graph of the bell-shaped adoption curvewith a chasm is mathematically inconsistent with a

constant-hazard process in the early stages of diffu-sion, and requires q1 > 0. Note that for the chasm tobe truly problematic, p2→ 0 is required.Moore does not clearly specify whose adoptions are

being imitated �w�. On the one hand, one might arguethat the legitimacy of a new technology is affectedby the penetration rate in the overall population, i.e.,the total installed base regardless of who adopted�w = ��. On the other hand, Moore emphasizes thatproduct and service offerings appealing to technol-ogy enthusiasts and visionaries need not appeal to themainstream market, which implies that mainstreamcustomers discount adoptions by technology enthusi-asts and visionaries and care only about adoptions byother mainstream customers �w= 0�.3

2.6. ConclusionAt least five different theoretical frameworks suggestmodeling innovation diffusion using a two-segmentstructure consisting of influentials and imitators(Table 1). Two theories suggest that influentialsadopt independently, implying q1 = 0, but the otherthree suggest that influentials may exhibit contagionamongst themselves.4 While one might intuitively

3 Moore himself is far from clear on the issue when discussingthe relationship between “visionaries” in the early market and“pragmatists,” i.e., the early adopters among the members of themainstream market. At one point, he admonishes the reader to“do whatever it takes to make [visionaries] satisfied customers sothat they can serve as good references for the pragmatists,” buton the very next page he writes that “pragmatists think vision-aries are dangerous. As a result, visionaries, with their highlyinnovative projects do not make good references for pragmatists”(Moore 1995, pp. 18–19).4 Independent decision making among influentials is also consis-tent with Midgley and Dowling (1978), who define innovativenessas “the degree to which an individual makes innovation decisions

Van den Bulte and Joshi: New-Product Diffusion with Influentials and Imitators6 Marketing Science 00(0), pp. 1–23, © 2007 INFORMS

expect p1 > p2 and none of the theories rules this out,this inequality is suggested only by adherents of thechasm framework. Also, several studies have docu-mented that the majority of earliest adopters neednot always be opinion leaders with disproportionateinfluence (Weimann 1994), implying �p1 < �1 − ��p2and leaving p1 < p2 as a possibility. Similarly, whileone might intuitively expect q1 < q2 and none of thetheories rules this out, this inequality is required onlyby the two theories implying q1 = 0 and q2 > 0, andseveral studies have documented that opinion lead-ers with disproportionate influence may also greatlyinfluence one another (e.g., Weimann 1994). All theo-ries allow for the initial impetus among imitators tostem from influentials, and so allow for p2 = 0.The theories vary in their causal mechanisms and,

consequently, in what kind of actors belongs to eachsegment and who the imitators imitate (w�. The the-ories also suggest that the relative size of the seg-ments (�� can vary from innovation to innovation.It may be quite low for very nonmainstream prod-ucts that only a very small pocket of “bleeding-edge” customers find attractive, but that in spite ofthe latter’s enthusiasm take a long time to diffuse,resulting in an adoption curve with a long left tail.Conversely, for products with low functional or finan-cial risk and with little implications for social status,like marginally novel drugs or CDs and movies withalready famous performers, most adopters may feellittle need for information or legitimation from peers.This implies a high �, a low q1, and an exponential-like diffusion process (e.g., Moe and Fader 2001, Vanden Bulte and Lilien 2001).

3. Two-Segment Mixture ModelsWe seek closed-form solutions in the time domainfor an innovation’s diffusion path when the set ofeventual adopters, which has a constant size M , con-sists of two a priori different types of actors adopt-ing according to Equations (3) and (4). The overallcumulative penetration is simply the average of bothtypes’ cumulative penetration weighted by their con-stant population weights (e.g., Cox 1959):

Fm�t�= �F1�t�+ �1− ��F2�t� (5)

Similarly, the fraction of the population adopting attime t is:

fm�t�= �f1�t�+ �1− ��f2�t� (6)

independently of the communicated experience of others” (p. 235).Therefore, our distinction between independent influentials (withq1 = 0) and pure imitators with p2 = 0 is the same as their dichotomybetween “innate innovators” and “innate noninnovators.”

In contrast, the population hazard function is not anaverage of the two hazards weighted by each seg-ment’s constant population weights, but is given by:

hm�t� = fm�t�/�1− Fm�t��

= ��f1�t�+ �1− ��f2�t��/�1− Fm�t��

= ��t�h1�t�+ �1−��t��h2�t�� (7)

where fi�t�= hi�t��1− Fi�t�� and ��t� is the proportionof actors not having adopted yet at time t that belongto Type 1:

��t�= �1− F1�t�

1− Fm�t� (8)

Finally, the proportion of adoptions made by actorsof Type 1 taking place at time t is:

��t�= �f1�t�/fm�t� (9)

3.1. Asymmetric Influence Model (AIM) withq1 > 0

Having defined the key functions, and having madethe behavioral assumptions in the hazard functions(Equations (3) and (4)), we now develop the asym-metric influence mixture model (AIM). The processamong the influentials is the well-known mixed-influence model. When F1�0�= 0, the cumulative pen-etration function and instantaneous adoption functionfor influentials are:

F1�t�= �1− e−�p1+q1�t�/(1+ q1

p1e−�p1+q1�t

)� (10)

f1�t�=(p1

(1+ q1

p1

)2e−�p1+q1�t

)/(1+ q1

p1e−�p1+q1�t

)2 (11)

The diffusion path among imitators, in contrast, doesnot follow any standard diffusion model, because itis driven by the prior adoptions of both influentialsand other imitators. As shown in Appendix A1, whenF2�0�= 0, the cumulative penetration function for imi-tators in the AIM is:

F2�t� = 1+ �p2q1+ q2�p1w− q1�1−w���

·(q1q2�1−w�H1+e�p2+q2�t

(p1+q1e

−�p1+q1�t

p1+q1

)wq2/q1

·�p2q1+q2�q1�1−w��1−H2�−p1w��

)−1� (12)

where

H1 = 2F1

(1�

wq2q1

�1+ wq2q1

− p2+ q2p1+ q1

�p1

p1+ q1e−�p1+q1�t

)�

H2 = 2F1

(1�

wq2q1

�1+ wq2q1

− p2+ q2p1+ q1

�p1

p1+ q1

)�

Van den Bulte and Joshi: New-Product Diffusion with Influentials and ImitatorsMarketing Science 00(0), pp. 1–23, © 2007 INFORMS 7

Figure 1 Adoption Functions for Four IIM Diffusion Processes

10 20 30 40 50 60 70

0.02

0.04

0.06

0.08

10 20 30 40 50 60 70

0.02

0.04

0.06

0.08

10 20 30 40 50 60 70

Time

Time

0.02

0.04

0.06

0.08A

dopt

ions

Ado

ptio

ns

Ado

ptio

nsA

dopt

ions

Case (a)

Case (c) Case (d)

Case (b)

10 20 30 40 50 60 70

Time

Time

0.02

0.04

0.06

0.08TotalInfluentialsImitators

and 2F1�1� b� c�k� is the Gaussian hypergeometricfunction:

2F1�1� b� c�k�=∑n=0

�b+n� �c�

�b� �c+n�kn (13)

This hypergeometric series is convergent for arbitraryb, c if �k�< 1; and for k=±1 if c > 1+ b. This impliesthat the closed-form solution in Equation (12) is welldefined as long as q1 > 0.5 Once F1�t� and F2�t� areknown, one can obtain the instantaneous adoptionfunction f2�t� by substituting Equations (10) and (12)into:

f2�t�= q2�wF1�t�+ �1−w�F2�t���1− F2�t�� (14)

With solutions for F1�t�, f1�t�, F2�t�, and f2�t� available,one can enter those into Equations (5) through (9) toobtain closed-form solutions for the population-levelfunctions.6

5 While the Gaussian hypergeometric functions 2F1�1� b� c�k� can besimplified to incomplete beta functions, we do not perform thissimplification because it requires the overly restrictive conditionthat p1w> q1�1−w+ p2/q2�.6 Even though our closed-form solution for F2�t� in the AIM looksquite different from the solution presented by Steffens and Murthy(1992), theirs is actually nested in ours. After imposing the con-straints p2 = 0 and w = �, reparameterizing the Steffens-Murthysolution in terms of m, �, p1, q1, and q2, correcting for an (mostlikely typographic) error in their solution, and performing addi-tional derivations, one can show that our closed-form solution for

In Figure 1, we plot the function fm�t� and its twocomponents �f1�t� and �1 − ��f2�t� for four sets ofparameter values chosen to illustrate various types ofdiffusion behavior possible in this model when p2 = 0and interconnection between segments is crucial:Case (a): p1 = 0 05; q1 = 0 1; q2 = 0 2; � = 0 15;

w= 0 20;Case (b): p1 = 0 01; q1 = 0 5; q2 = 0 2; � = 0 15;

w= 0 01;Case (c): p1 = 0 05; q1 = 0 5; q2 = 0 2; � = 0 30;

w= 0 30;Case (d): p1 = 0 01; q1 = 0 1; q2 = 0 2; � = 0 15;

w= 0 001.Diffusion process (a) exhibits a bell-shaped adop-

tion curve fm�t� that is unimodal and close to sym-metric around its peak. This is the pattern commonlyassociated with the mixed-influence model. Diffusionprocess (b) is bimodal and exhibits a marked dipbecause adoptions by influentials are already wellpast their peak by the time the imitators start adopt-ing in numbers (the delay being caused by the loww value). This is the much-debated “chasm” pattern.Diffusion processes (c) and (d), finally, are again uni-modal but exhibit a clear skew to the right or left,which the mixed-influence cannot account for very

F2�t� in the AIM, and hence Fm�t�, is identical to theirs. One differ-ence, though, is that their solution requires q1 > q2� (or q1 > q2w�for a series expansion term in their solution to converge, whereasthe solution in Equation (12) only requires q1 > 0.

Van den Bulte and Joshi: New-Product Diffusion with Influentials and Imitators8 Marketing Science 00(0), pp. 1–23, © 2007 INFORMS

well (e.g., Bemmaor and Lee 2002).7 Note that in allfour cases, f1�t� reaches zero before f2�t� does, sothe commonly expected association between being animitator and being a late adopter holds. Also notethat, as one would intuit, low values of w cause thediffusion among imitators to be delayed and f2�t� toshift to the right. We now turn to the case whereq1 = 0, and study it in more detail using the functionshm�t�, ��t�, and ��t�.

3.2. Asymmetric Influence Model (AIM) withq1 = 0 and Pure-Type Mixture Model (PTM)

When influentials adopt independently and q1 = 0, theprocess among the independents is the well-knownconstant-hazard exponential process. When F1�0�= 0,we have:

F1�t�= 1− e−p1t� (15)

f1�t�= p1e−p1t (16)

As shown in Appendix A2, when q1 = 0 and F1�0� =F2�0�= 0, the cumulative penetration function for imi-tators in the AIM is:

F2�t� = 1+ exp(−p2t− q2t−

q2p1we−p1t

)

·(q2p1�1−w�

(q2p1w

)−�p2+q2�/p1

·(

(p2+ q2p1

�q2p1we−p1t

)−

(p2+ q2p1

�q2p1w

))

− exp(− q2p1w

))−1� (17)

where �!�k� is the “upper” incomplete gamma func-tion:

�!�k�=∫

kv!−1e−v dv

The instantaneous adoption function f2�t� is obtainedby substituting Equations (15) and (17) into (14). Withsolutions for F1�t�, f1�t�, F2�t�, and f2�t� available, onecan enter those into Equations (5) through (9) toobtain closed-form solutions for the population-levelfunctions.A case of special interest is that of a pure-type mix-

ture (PTM) of pure independents with q1 = 0 and pureimitators with p2 = 0. In Figure 2, we plot the functionsfm�t�, hm�t�, ��t�, and ��t� for three sets of parameter

7 All four patterns for the total number of adoptions shown inFigure 1 have been documented in prior research. Pattern (a) isprobably the most commonly reported in the marketing literature.Steffens and Murthy (1992) and Karmeshu and Goswami (2001)report data series exhibiting the bimodal pattern (b). Dixon (1980)reports the presence of long right tails, i.e., pattern (c), in many ofthe data he analyzed. Van den Bulte and Lilien (1997) report severaldata series exhibiting long left tails, i.e., pattern (d).

values chosen to illustrate various types of diffusionbehavior possible in this model:8

Case (a): p1 = 0 15, q2 = 0 50, �= 0 25, w= 0 25;Case (b): p1 = 0 25, q2 = 0 40, �= 0 15, w= 0 01;Case (c): p1 = 0 15, q2 = 0 65, �= 0 60, w= 0 05.Diffusion process (a) exhibits the common uni-

modal, symmetric-around-the-peak adoption curvefm�t� well captured by the mixed-influence model.More interesting is that the hazard function is notmonotonic as in the mixed-influence model. Rather,it is roughly bell shaped and seems to convergeto a value in-between the minimum and the maxi-mum. Here is why. The very earliest adopters con-sist of independents, and the population hazardequals �p1 = 0 0375 at first. As more and more imi-tators adopt with hazard q2Fm�t�, the population haz-ard increases. Once q2Fm�t� > p1, which can happenquickly when q2 is markedly larger than p1, the setof imitators not having adopted yet will start deplet-ing faster than the set of independents not havingadopted yet. As a result, the laggards still remain-ing to adopt consist increasingly of independents—asindicated by the function ��t� reaching a minimumaround t = 5 and then increasing to 1—and the pop-ulation hazard converges back to an asymptote ofp1 = 0 15. This pattern of relative speed of depletionalso explains the nonmonotonic pattern in ��t�, theproportion of adoptions taking place at time t stem-ming from independents. Note that in this diffusionprocess, independents make up the bulk not only ofthe early adopters, but also of the very late adopters.Importantly, the point at which ��t� starts increas-ing and independents start gaining rather than losingimportance �t = 7 3� occurs when the process is stillfar from complete and the remaining market poten-tial is still quite sizable (37% because Fm�t� = 0 63 att = 7 3).Diffusion process (b) differs in several respects from

process (a). First, the adoption curve f �t� does nothave a smooth bell shape, but exhibits a clear dipearly on. This is easily explained. The independentsadopt rapidly because p1 = 0 25 is rather high. How-ever, imitators’ reaction to those independent adop-tions is very muted because they imitate mostlyfellow imitators (w = 0 01). As a result, the adop-tions by independents show an exponential declinethat is not immediately compensated by the imitators’slowly developing adoptions, resulting in an early

8 Of the three shapes of adoption curve in Figure 2, pattern (a) isprobably the most commonly reported in the diffusion literature.The other two shapes have not been documented as extensively, butdo occur in previously analyzed data. For instance, the sales curveof several music CDs studied by Moe and Fader (2001) exhibit pat-tern (b) or (c), and the classic Medical Innovation data analyzed byColeman et al. (1966) also exhibit pattern (c).

Van den Bulte and Joshi: New-Product Diffusion with Influentials and ImitatorsMarketing Science 00(0), pp. 1–23, © 2007 INFORMS 9

Figure 2 Plots of Functions Characterizing Three PTM Diffusion Processes

0 5 10 15 20 25

0.00

0.05

0.10

0.15

t

f (t)

h(t)

0 5 10 15 20 25

0.0

0.1

0.2

0.3

0.4

0.5

t

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

t

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

t

0 5 10 15 20 25

0.00

0.05

0.10

0.15

t

0 5 10 15 20 25

0.0

0.1

0.2

0.3

0.4

0.5

t

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

t

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

t

0 5 10 15 20 25

0.00

0.05

0.10

0.15

t

0 5 10 15 20 25

0.0

0.1

0.2

0.3

0.4

0.5

t

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

t

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

t

(a)

p1 = 0.15, q2 = 0.50, θ = 0.25, w = 0.25 p1 = 0.25, q2 = 0.40, θ = 0.15, w = 0.01 p1 = 0.15, q2 = 0.65, θ = 0.60, w = 0.05

(b) (c)

π(t)

φ(t)

f (t)

h(t)

π(t)

φ(t)

f (t)

h(t)

π(t)

φ(t)

dip in the population curve. Note that independentsaccount for the bulk of the adoptions only early inthe diffusion process, as ��t� declines steeply to closeto zero. Therefore, while the adoption curve does notfit the standard model, we do have the commonly

expected association between being an imitator andbeing a late adopter.Diffusion process (c) looks mostly like an ex-

ponential-like process commonly observed for fast-moving consumer goods, CDs and films, but with a

Van den Bulte and Joshi: New-Product Diffusion with Influentials and Imitators10 Marketing Science 00(0), pp. 1–23, © 2007 INFORMS

marked boost after the early periods. What is happen-ing is that most adopters are independents ��= 60%�,so the majority of adoptions follow an exponentialdecline. However, there is also a sizable segment ofimitators that are very sensitive to social contagion(q2 = 0 65), but mostly from fellow imitators ratherthan independents �w = 0 05�. As a result, the imita-tors are slow to adopt at first, but once the snowballstarts rolling, tend to adopt in a very short time. Thisis reflected in the shape of ��t�: The proportion ofadoptions accounted for by independents tends to beclose to 100%, except for a relatively narrow time win-dow during which it first declines and then increasesagain. The contrast between process (a) and (c) isinformative: They have similar p1 and q2 values, andthe composition of both adopters ��t� and remainingnonadopters ��t� tend to evolve similarly, as do theirrespective population hazard functions h�t�. How-ever, because of the different segment sizes � and con-tagion weights w in the two processes, the resultingadoption curves are quite different.

3.3. Some Special Cases of Theoretical InterestOur review of prior theories and frameworks indi-cates that three cases of the social influence struc-ture captured by w are of special theoretical interest.The first is where imitators imitate only influentials�w= 1�, such that h2�t� = p2 + q2F1�t�.9 The second iswhere imitators imitate only other imitators �w = 0�,such that h2�t�= p2+ q2F2�t�. The third is where imita-tors mix randomly with both independents and imi-tators, such that w = � and h2�t�= p2+ q2Fm�t�. In thefirst and third case, F2�t� and f2�t� are easily derivedby imposing w = 1 and w = �, respectively, in Equa-tions (12), (14), and (17). The second case poses anissue when p2 = 0 and the process among imitators isonly a function of prior adoptions by other imitators:The process is then simply the well-known logisticprocess, which does not allow for F2�0�= 0.10A fourth case of special interest is less obvious:

When all independents adopt instantaneously withp1 → and pure imitators �p2 = 0� have a veryspecific influence weight w = ��1 + q2 − ��/q2 > �,then the PTM reduces to the MIM (see TechnicalAppendix A).11

9 This model, with the additional constraints q1 = p2 = 0, was alsodeveloped independently from us by Beck (2005).10 Note that when p2 =w = 0 or p2 = � = 0, the process among imi-tators cannot get started within the model. As is well known, theclosed-form solution for the logistic requires that F2�0� > 0. Hence,while the cases with p2 =w= 0 or p2 = �= 0 are conceptually nestedwithin the AIM, their closed-form solutions are not because theymake different assumptions about the initial conditions.11 All technical appendices are available online at the Marketing Sci-ence website.

4. Relation to Prior Diffusion Models4.1. Mixed-Influence Model vs. Pure-Type Mixture

ModelAs the closed-form solutions and the plots in Fig-ure 2 indicate, the mixed-influence model (MIM) doesnot capture diffusion processes in a discrete mixtureof pure independents and pure imitators (PTM). Twoexceptions to this are the case where p1 = 0 or � = 0and both models collapse to the logistic model, andthe case where q2 = 0 or � = 1 and both models col-lapse to the exponential model. A third, less obvious,exception is when p1→ and w= ��1+q2−��/q2 > �,and the PTM also reduces to the MIM.Our analysis allows one to assess the widely

accepted notion (e.g., Mahajan et al. 1993) that rewrit-ing the standard differential equation for the mixedinfluence model (Equation (1)) into

f �t�= p�1− F �t��+ qF �t��1− F �t�� (18)

allows one to interpret the term p�1 − F �t�� as thenumber of adoptions made by people adopting withhazard p; and the term qF �t��1 − F �t�� as the num-ber of adoptions made by people adopting with haz-ard qF �t�. While the manipulation of the equation isevidently correct, the interpretation is not. The mainreason is that, in each term, the fraction of actors nothaving adopted yet, 1− F �t�, refers to the total popu-lation, rather than to the fractions in each of the seg-ments, 1− F1�t� and 1− F2�t�. In addition, the sizes ofeach segment are ignored. The correct expression fora mixture is:

fm�t� = �f1�t�+ �1− ��f2�t�

= �h1�t��1− F1�t��+ �1− ��h2�t��1− F2�t��

= �p1�1− F1�t��+ �1− ��q2

· �wF1�t�+ �1−w�F2�t���1− F2�t�� (19)

When imitators randomly mix with independents andimitators and are equally affected by both, then w= �and the equation simplifies to:

fm�t�= �p1�1− F1�t��+ �1− ��q2Fm�t��1− F2�t�� (20)

Even if p = �p1, q = �1 − ��q2, and one omits them-subscript from the population-level fm�t� and Fm�t�,the mixture Equation (20) is different from the mixed-influence Equation (19).Within a homogeneous population with mixed in-

fluence, one can only interpret the relative size of thetwo terms p�1− F �t�� and qF �t��1− F �t�� as reflectingthe relative influence of time-invariant elements �p�versus social contagion �qF �t�� on the adoptions attime t, keeping in mind that each and every adop-tion is influenced by both p and qF �t� for any t > 0.

Van den Bulte and Joshi: New-Product Diffusion with Influentials and ImitatorsMarketing Science 00(0), pp. 1–23, © 2007 INFORMS 11

For instance, the ratio p/�p + qF �t�� can be used asa measure of the relative strength of time-invariantelements at time t (Lekvall and Wahlbin 1973), ascan the decomposition presented by Daley (1967) andMahajan et al. (1990), but neither can be interpreted asthe fraction of all adoptions at time t stemming frompure-type actors adopting a priori with hazard p.Another common belief about the mixed-influence

model that is inconsistent with its mathematical struc-ture is that “the importance of innovators will begreater at first but will diminish monotonically withtime,” where innovators are defined as those who“are not influenced in the timing of their initial pur-chase by the number of people who have alreadybought the product” (Bass 1969, p. 217). In a homo-geneous population where everyone behaves accord-ing to the hazard rate p+ qF �t�, the only actors withhazard p are those adopting at t = 0 when F �0� = 0.Anyone adopting afterwards is influenced by prioradoptions. Hence, in the mixed-influence model, theproportion of adoptions occurring at time t that areunaffected by social contagion follows a step functionwith value 1 at t = 0 and value 0 for any t > 0. Con-versely, in a mixture with p1 �, the proportion ofindependents adopting with a constant hazard, i.e.,function ��t�, need not diminish monotonically overtime, as shown in Figure 2.

4.2. Consequence of Imposing a Mixed-InfluenceStructure on a Pure-Type Mixture Process

From comparing Equations (18) and (20), one may getthe impression that a diffusion process in a discretemixture with h1�t� = p1 and h2�t� = q2Fm�t� could beapproximated quite well by a mixed-influence modelwith h�t� = p + qF �t�, even if they are not identical.However, the adoption functions fm�t� and hazardfunctions hm�t� suggest some potentially importantdeviations. More insight comes from rewriting theexpression for fm�t� in Equation (20) into a form sim-ilar to that for f �t� in the mixed-influenced model(following Manfredi et al. 1998):

fm�t� = �p1�1− F1�t��+ �1− ��q2Fm�t��1− F2�t��

=[�p11−F1�t�

1−Fm�t�+�1−��q2Fm�t�

1−F2�t�

1−Fm�t�

]�1−Fm�t��

= �p�t�+ q�t�Fm�t���1− Fm�t�� (21)

where

p�t�= �1− F1�t�

1− Fm�t�p1 =��t�p1 (22)

q�t�= �1− ��1− F2�t�

1− Fm�t�q2 = �1−��t��q2 (23)

Deleting the m subscript from Equation (21) to reflectone’s ignoring that the population consists of a mix-ture results in:

f �t�= �p�t�+ q�t�F �t���1− F �t�� (24)

Therefore, one is able to rewrite the pure-type mix-ture model with w = � into an expression akin tothe mixed-influence model, but with both hazard-rateparameters varying systematically over time. Morespecifically, p�t� changes in exactly the same way as��t�, the proportion of actors not having adopted yetby time t that belong to the segment of indepen-dents. At t = 0, ��t� = � and p�t� = �p1. Because atthe very beginning adoption tends to be more preva-lent among independents than among imitators, thenumber of independents who have not adopted yetgets depleted faster than the number of imitators whohave not. Consequently, ��t� and p�t� decline at first.However, when q2 p1, the relative speed of adop-tion between the two segments quickly reverses andthe set of actors who have not adopted yet tends tobecome increasingly dominated by independents. Asa result, ��t� and p�t� increase over most of the timewindow. The reverse pattern takes place for q�t� =�1−��t��q2. It starts at �1− ��q2, increases for a veryshort period, but starts decreasing very soon. Note,when � ≈ 0 or � ≈ 1, then ��t� will not vary much,and neither will p�t� or q�t�.In short, specifying a mixed-influence model with

h�t� = p + qF �t� when the true data-generating pro-cess is that of a discrete mixture with h1�t� = p1 andh2�t�= q2Fm�t�, where q2 p1 will yield increasing val-ues of p and decreasing values of q (except for thefirst very few periods). This is consistent with thepattern in mixed-influence model estimates describedin prior research. Although Van den Bulte and Lilien(1997) focused their analysis on ill conditioning inthe absence of model misspecification, they recog-nized that unobserved heterogeneity in p and q formsan alternative explanation for the systematic changesthey observed in empirical applications. Our resultsformalize their argument for the case of two segmentswhere one segment has p= 0 and the other has q = 0.4.3. Relation to Other Two-Segment ModelsFigure 3 shows how our models relate to a few othermodels, including two earlier two-segment models.Tanny and Derzko (1988) used a discrete mixture withh1�t�= p1 and h2�t�= p2+q2Fm�t�. Steffens and Murthy(1992) used a discrete mixture with h1�t�= p1+ q1F1�t�and h2�t� = q2Fm�t�. Therefore, as shown in Figure 3,both these models conceptually nest both the mixed-influence model and PTM3 with w = �. The diagramalso shows that, like the mixed-influence model, thepure-type mixture models have both the exponentialand logistic models nested in them, with the excep-tion that PTM1 with w = 1 does not nest the logisticbecause if h2�t� = q2F1�t� and either � = 0 or p1 = 0,then h2�t� is undefined. Note that only the PTMs fea-ture two “pure types,” i.e., independents and imita-tors without any mixed influence.

Van den Bulte and Joshi: New-Product Diffusion with Influentials and Imitators12 Marketing Science 00(0), pp. 1–23, © 2007 INFORMS

Figure 3 Relations Among the AIM and PTM Models, the Steffens-Murthy and Tanny-Derzko Models, and the Mixed-Influence, Exponential, andLogistic Modelsa

h(t) = p h(t) = qF(t)

Mixed influence

h(t) = p + qF(t)

Pure-type mixture 1

h1(t) = p1h2(t) = q2F1(t)

h1(t) = p1h2(t) = q2F2(t)

h1(t) = p1h2(t) = q2Fm(t)

Pure-type mixture 2 Pure-type mixture 3

Tanny-Derzko

h1(t) = p1h2(t) = p2 + q2Fm(t)

Steffens-Murthy

h1(t) = p1 + q1F1(t)h2(t) = q2Fm(t)

Pure-type mixture (free weights)

h1(t) = p1h2(t) = q2[wF1(t) + (1– w)F2(t)]

Asymmetric influence mixture

h1(t) = p1 + q1F1(t)h2(t) = p2 + q2[wF1(t) + (1– w)F2(t)]

Exponential Logistic

Note. aA model receiving an arrow is conceptually nested in the model where the arrow originates. For instance, the general PTM with w = 1 generates PTM1and the PTM1 with q2 = 0 generates the exponential. The link between PTM and MIM is indicated by a broken line as it holds only as p1 →.

As shown in Technical Appendix B, the solution forF2�t� in PTM3 is consistent with Jeuland’s (1981, p. 14)earlier work. The differences are that he did not spec-ify h1�t�, but kept it general, and that his partial solu-tion still contained unknown integrals. In contrast, wespecify the process among independents and solvethe equations using incomplete gamma functions,making parameter estimation and empirical analysispossible.

5. Empirical AnalysisTo what extent does the two-segment asymmetricinfluence model, consistent with several theoreticalframeworks, agree with empirical diffusion patterns?And how well does it do compared to the mixed-influence model and other, more flexible, models? Weprovide insights on those issues through an empiricalanalysis of 33 data series.

5.1. DataOne must use an informative variety of data sets ifone is to draw sound conclusions on model perfor-mance. We therefore analyze four sets of data. Thefirst consists of a single series on the diffusion of

the broad-spectrum antibiotic tetracycline among 125Midwestern physicians over a period of 17 months inthe mid-1950s. This series comes from the classicMed-ical Innovation study (Coleman et al. 1966). It warrantsspecial attention because it is commonly accepted asan instance of diffusion in a mixture of independentsand imitators (e.g., Jeuland 1981, Lekvall and Wahlbin1973, Rogers 2003).The second set of data series consists of 19 music

CDs, a category where a two-segment structure is alsolikely to exist a priori. Some customers are dedicatedfans buying products by their favorite performersalmost unconditionally, while others end up buyingthe CD only after it has become popular and a must-buy (Farrell 1998, Yamada and Kato 2002). Therefore,q1 = 0 and p2 = 0 are quite possible. We use the weeklyU.S. sales data analyzed previously by Moe and Fader(2001).12 Because people are very unlikely to buy twoidentical CDs for themselves or to replace an oldercopy, the sales data are unlikely to be contaminatedby multiple or repeat purchases and can be treated asnew product adoptions. Figure 4 shows the data of

12 The full set consists of 20 data series, but we deleted one that stillhad not reached the time of peak sales.

Van den Bulte and Joshi: New-Product Diffusion with Influentials and ImitatorsMarketing Science 00(0), pp. 1–23, © 2007 INFORMS 13

Figure 4 Weekly Sales (Adoption) Data for Four CDs

AdamAnt

0

2,000

4,000

6,000

8,000

1 6 11 16 21 26 31 36 41 46 51 56

1 6 11 16 21 26 31 36 41 46 51 61 7156 66

Dink

0

1,000

2,000

3,000

4,000

Blind Mellon

0

10,000

20,000

30,000

40,000

1 3 5 7 9

Everclear

0

10,000

20,000

30,000

40,000

1 53 7 9 23211917151311 25 27 29 31 33 35 37 39 41 43 45

333129272523211915 171311

four CDs, each illustrating one typical path: a rathersmooth decline for Blind Mellon, an early dip fol-lowed by a recycle for AdamAnt, a slowly developing“sleeper” pattern for Everclear, and a bell shape forDink.The third set of data consists of five series of high-

technology products, for which a two-segment struc-ture with q1 > 0 is quite possible (e.g., Moore 1991).The first three series consists of adoptions of CTscanners, ultrasound, and mammography equipmentamong hospitals of all sizes (Van den Bulte and Lilien1997). The fourth series consists of the penetrationbetween 1979 and 1993 of CT scanners among hospi-tals with 50 to 99 beds. Controlling for size may beimportant, as larger hospitals have larger budgets andmore highly skilled staff, and these differences maymask genuine contagion processes (e.g., Davies 1979).The fifth series consists of the penetration of personalcomputers among U.S. households. The series coversthe years 1981–1996, but to avoid left-censoring arti-facts we impose 1975 as the actual launch year. Thefirst three series are roughly bell shaped, the lattertwo series show two “bells” separated by a dip or“chasm.”The final set is a miscellaneous mix of eight data

series analyzed previously by Van den Bulte and Lilien(1997) and Bemmaor and Lee (2002) (these studies

also included the tetracycline and three of the high-tech series). There is no compelling a priori reason toexpect a mixture of independents and imitators to beable to better account for those diffusion data thantraditional models, and several innovations need nothave diffused through contagion at all (Griliches 1962,Van den Bulte and Stremersch 2004). The adoptioncurves all have a very pronounced bell shape, withseveral showing skew that the MIM cannot accountfor (Bemmaor and Lee 2002).

5.2. Parameter EstimatesOne of our closed-form solutions involves Gaussianhypergeometric functions, the estimation of which isvery troublesome.13 Fortunately, one can estimate theAIM through direct integration, that is, by comput-ing nonlinear least-squares estimates at the same time

13 Nonlinear regression using the “difference in-closed-form-cdfs”approach (Srinivasan and Mason 1986) in R and Mathematica eitherdid not converge at standard convergence criteria or enabled usto obtain point estimates, but not standard errors. We experiencedthese problems even with simulated data, which rules out modelmisspecification as an explanation for these difficulties. Maximum-likelihood estimation is known to be troublesome as well, evenwhen the parameters of interest enter the function linearly ratherthan nonlinearly as in the AIM (e.g., Fader et al. 2005).

Van den Bulte and Joshi: New-Product Diffusion with Influentials and Imitators14 Marketing Science 00(0), pp. 1–23, © 2007 INFORMS

as one numerically solves the following differentialequation:14

dX�t�/dt

=M��f1�t�+ �1− ��f2�t��+ $�t�

=M

[�f1�t�+�1−��q2

{wF1�t�+�1−w�

X�t�/M−�F1�t�

1−�

}

·{1− X�t�/M − �F1�t�

1− �

}]+ $�t� (25)

where X�t� is the cumulative number of adoptersobserved at time t, f1�t� and F1�t� are the closed-formsolutions to the adoption and penetration functions ofthe MIM, and f2�t� is expressed as in Equation (15),but with �X�t�/M −�F1�t��/�1−�� replacing F2�t�. Thelatter is based on X�t� = MFm�t� (absent error) andFm�t� = �F1�t�+ �1− ��F2�t�. We allow the error term$�t�∼N�0�%2� to exhibit serial correlation up to order2 when the time series contains more than 20 obser-vations or the Durbin-Watson statistic falls outsidethe 1.5–2.5 range. We impose the condition that haz-ard parameters p1, q1, p2, and q2 be nonnegative (≥0)and that 0≤ � ≤ 1. Because hazard rates can be largerthan one in continuous time, we do not impose p1,q1, p2, and q2 ≤ 1. As to w, we impose 0 01%≤w ≤ 1,choosing a very small but positive lower bound so themodel itself ensures the “seeding” of the contagionprocess among imitators even when p2 = 0. Becauseestimation through direct integration fits the cumu-lative adoptions X�t� rather than the periodic adop-tions X�t�−X�t−1�, the R2 values are often extremelyhigh and noninformative (the lowest we obtained was0.992, and several were higher than 0.9995). There-fore, we report the mean absolute percentage error(MAPE) instead, as well as an alternative R2 metricdefined as the squared Pearson correlation betweenthe actual periodic adoptions and the difference inpredicted cumulative adoptions �R2p�.Table 2 reports the results of estimating the AIM

to all 33 data series.15 Values for p1 tend be smallerthan 0.3. There are two exceptions to this: ForeignLanguage, where � is so low that fm�0� = �p1 equalsonly 0.04, and the Beastie Boys CD that exhibited anextreme “blockbuster” pattern, i.e., extremely quicklydeclining sales. Values for q1 show much more vari-ance. This is especially so for CDs. For about half ofthem, q1 equals zero, indicating the absence of wordof mouth among influentials. In six cases, q1 is largerthan one, suggesting very strong word of mouthamong influentials. However, these large estimates

14 This can be done quite conveniently, e.g., using the model proce-dure in SAS or the odesolve package in R.15 We do not report the ceiling parameter values M due to spaceconstraints in the table.

are very imprecise, and only two are significant at95% confidence. Values for p2 are most often zero, andonly four of the 33 estimates are significantly differentfrom zero. Values for q2 also show considerable vari-ance, with several high values recorded for the set ofmiscellaneous innovations. The latter may result fromthe strong left skew in the adoption time series (Bem-maor and Lee 2002). Finally, � is often significantlydifferent from both zero and one, indicating that theAIM does not reduce to the mixed-influence or logis-tic models, and only weakly correlated with w (r =−0 16). That � is often larger than 2.5% or 16%, tra-ditional values used to separate innovators from imi-tators based on time of adoption, is an indication—in addition to the ��t� function—that the dichotomybased on drivers of adoption underlying the modelis conceptually different from that based on time ofadoption. The MAPE and R2p values indicate that themodel tracks the data well. While the MAPE is higherthan 10% for some of the shorter data series, like the15% value for scanners in small hospitals with 50–99 beds, such high MAPE values can be misleadingbecause they tend to result from a few deviationsearly in the process when the base for calculatingthe percentage error is small. Figure 5 shows that themodel can indeed track bimodal patterns rather welleven with a high MAPE.Because combining nonlinear least-squares estima-

tion with direct integration may be new to mar-keting (diffusion) researchers, we briefly report, forthe case of tetracycline, estimates obtained throughdirect integration (DI) with those obtained throughthe popular Srinivasan-Mason (SM) procedure fittingthe difference in closed-form cdfs to the difference incumulative adoptions. The results in Table 3 clearlyshow that both procedures produce very similar esti-mates for the PTM and the MIM. Direct integrationhas somewhat higher serial correlation because it fitsthe cumulative adoptions X�t� rather than the peri-odic adoptions X�t�−X�t − 1�. The difference in thedependent variable also explains why direct integra-tion produces much lower MAPE values even whenthe mean squared error (MSE) values are very sim-ilar. That the DI method leads to lower R2p valuesthan the SM method is not surprising, because thelatter method finds those estimates that minimize thesum of squared errors (SSE), and hence maximizesthe correlation between predicted and observed peri-odic adoptions. The parameter estimates of the AIMand PTM, with the zero value of q1 meaning thatSegment 1 consists of independents, and the highvalue of � meaning that contagion affected only aminority, are consistent with previous analyses usingindividual-level data on adoption times and actualnetwork structure (Coleman et al. 1966, Van den Bulteand Lilien 2003), as is the decomposition of totaladoptions in Figure 6. The graph indicates that by

Van den Bulte and Joshi: New-Product Diffusion with Influentials and ImitatorsMarketing Science 00(0), pp. 1–23, © 2007 INFORMS 15

Table 2 IIM Results for All Data#

N p1 q1 p2 q2 � w AR1 AR2 DW MAPE R2p

Tetracycline 18 0102c 0∗ 0∗ 0998c 081c/c 001%∗ 182 22% 0799AdamAnt 57 0061c 0∗ 0∗ 0369c 063c/c 010/c −014 007 065 06 0986Beastie Boys 97 1256c 0∗ 0∗ 0041c 028c/c 1∗ 020c 006c 067 02 0991Blind Mellon 34 0210c 3291 0∗ 0073c 024c/c 1∗ 040 016 144 05 0964Bob Seger 24 0084c 0∗ 0∗ 1357c 081c/c 001%∗ −002 008 167 13 0814Bonnie Raitt 1 107 0291c 0∗ 0∗ 0040c 041c/c 1∗ 007a −009b 131 02 0984Bonnie Raitt 2 22 0096c 0∗ 0∗ 1538c 074c/c 001%∗ 002 −003 145 19 0823Charles & Eddie 32 0024b 0541c 0050c 0007 026a/c 001% −082 −070 175 07 0971Cocteau Twins 127 0000 14848 0∗ 0051c 010c/c 1∗ 086c 025b 188 02 0950Dink 73 0019c 0162c 0∗ 0011 067a/ 1∗ 036c 037c 167 13 0938Everclear 46 0024 0273a 0∗ 0188c 005c/c 001%∗ 037a 022 188 32 0969Heart 124 0000 1909c 0074c 0∗ 008c/c 001% −018c 004 033 03 0993John Hiatt 24 0274 3282a 0∗ 0192b 016a/c 029/c 037 § 204 11 0683Luscious Jackson 85 0065 4153 0∗ 0028c 010c/c 1∗ 041c 020 143 04 0883Radiohead 73 0041c 0141c 0001 0102c 016/c 001% 043c 010 144 15 0867Richard Marx 113 0122c 0074 0023a 0023 043c/c 001% 021c −008 092 03 0982Robbie Robertson 79 0075c 0054 0∗ 0010 058c/a 1∗ 022b −004 132 06 0888Smoking Popes 40 0089c 0143c 0∗ 0142c 075c/c 001%∗ −022b 001 096 07 0966Supergrass 38 0157 2715 0∗ 0058a 009b/c 066a/ 071c 041a 143 06 0876Tom Cochrane 22 0108c 0∗ 0∗ 1741 097c/ 072 −001 013 072 18 0915Home PC 17 0000 0407c 0∗ 2567c 065c/c 065∗∗ 220 119 0333Mammography 15 0000 1350c 0015b 0602c 038b/b 038∗∗ 289@ 59 0976Scanners (all) 18 0003c 0634c 0∗ 0476 063c/c 001/c 205 197 0927Scanners (50–99) 15 0002 1031a 0000 0821c 060c/c 001%∗ 179 150 0831Ultrasound 15 0022 0309c 0∗ 1113b 058a 000 249 77 0937Hybrid corn 1943 16 0000 0868c 0192 2866 085c/c 001%∗ 088a 027 239 131 0974Hybrid corn 1948 15 0037 0482 0∗ 0861 020 001%∗ 247 126 0744Accel. program 13 0001 0786c 0∗ 2394c 085c 001%∗ 244 269 0842Foreign language 13 0656 0∗ 0∗ 0716c 006/c 000a 281@ 31 0919Comp. schooling 15 0006 0746b 0∗ 0694 069 001/c 182 176 0627Color TV 17 0000a 0361c 0∗ 1272c 078c/c 001%∗ 148@ 40 0391Clothes dryers 17 0000 0508c 0∗ 5593b 061c/c 1∗ 204 35 0819Air conditioners 17 0000 1044a 0000 0511c 028/c 001%∗ 237 95 0706

#N = number of observations (incl. X�0�= 0�; AR1, AR2= first-order and second-order serial correlation, DW= Durbin-Watson statistic, R2p = r 2 of actual

adoptions with difference in predicted cumulative adoptions.∗Boundary constraint; ∗∗ constrained to equal � to aid convergence; § including AR2 results in convergence problems; @ adding AR1 and AR2 does not

improve DW.ap≤ 005, bp≤ 001, cp≤ 0001; for � and w , the entry left of the slash (/) refers to the significance of the test against zero and those to the right refer to

the test against one.

Month 11, when 25% of all physicians still had toadopt, all imitators had already adopted and the “lag-gards” consisted only of independents. This is consis-tent with the original finding by Coleman et al. (1966)

Figure 5 Actual and Predicted Adoptions of CT Scanners in SmallHospitals (50–99 Beds)

0

2

4

6

8

10

12

14

321 4 6 87 9 10 11 12 13 14 15

InfluentialsImitatorsTotalActual

5

using individual-level data that the laggards tendedto be very poorly integrated in the social network,and hence unaffected by social influence. Finally, themixture models generate an estimate of M close tothe entire sample of physicians (N = 125), whereasthe mixed-influence estimates are very close to thenumber of adopters having adopted at the end ofthe observation period (X�t17� = 109). This is consis-tent with our analytical result that imposing a mixed-influence model on a mixture process can generate thekinds of estimation artifacts documented by Van denBulte and Lilien (1997).

5.3. Descriptive Performance Compared toBenchmark Models

To assess the descriptive performance of the two-seg-ment model, we compare it against that of the mixed-influence (MIM), Gamma/Shifted Gompertz (G/SG),

Van den Bulte and Joshi: New-Product Diffusion with Influentials and Imitators16 Marketing Science 00(0), pp. 1–23, © 2007 INFORMS

Table 3 AIM, PTM, and MIM Results for Medical Innovation Tetracycline Data, Using Estimation by Direct Integration (DI)and by the Srinivasan-Mason Procedure (SM)#

M p1 q1 p2 q2 � w AR1 AR2 DW MSE MAPE R2p

HMIM-DI 1270c 0102c 0∗ 0∗ 0998c 081c/c 001%∗ — — 182 210 22% 0799PTM-DI 1270c 0102c — — 0998c 081c/c 001%∗ — — 182 210 22 0799PTM-SM 1312c 0097c 0∗ 0∗ 1059c 081c/c 003c/c — — 169 202 388 0908MIM-DI 1116c 0097a 0155 — — — — 010 014 147 415 26 0717MIM-SM 1113c 0085a 0188 — — — — 032 182 435 431 0784

#AR1, AR2= first-order and second-order serial correlation; DW= Durbin-Watson statistic; for estimation on cumulative data usingdirect integration (DI), R2

p = r 2 of actual adoptions with difference in predicted cumulative adoptions; for estimation on periodic datausing SM-method, R2

p = r 2 of actual and predicted adoptions.∗Boundary constraint.ap ≤ 005, bp ≤ 001, cp ≤ 0001; for � and w , the entry left of the slash (/) refers to the significance of the test against zero and

those to the right refer to the test against one.

Weibull-Gamma (WG), and Karmeshu-Goswami (KG)models. Because all of these benchmark models havea closed-form solution, we estimate them using thestandard Srinivasan-Mason (1986) approach. To avoidhaving comparisons across model specifications beaffected by differences in estimation method anddependent variable, we do not estimate the full AIMusing the DI approach. Instead, we estimate tworestricted versions, one with w= 0 and the other withq1 = 0, that lead to closed-form solutions for Fm�t�that do not involve Gaussian hypergeometric func-tions and that can hence be estimated using the SMapproach.We assess model performance under three error

structures: (1) i.i.d. additive error, (2) additive errorwith AR1 serial correlation, and (3) lognormal multi-plicative error.16 Estimating the models without serialcorrelation provides a more informative assessment ofdescriptive performance because incorporating serialcorrelation into a model might alleviate a poor fitof its mean function to the data (Franses 2002). Still,the question remains to what extent serial correlationalone helps close the gap between two models.We use four measures of descriptive performance:

mean absolute deviation (MAD), mean absolute per-centage error (MAPE), mean square error (MSE), andthe Bayesian Information Criterion (BIC). Note thatonly the latter two penalize models with a largernumber of free parameters.17 To save space and aidinterpretation, we report only the ratio of the baseline

16 For the model with lognormal multiplicative error, we estimateits log-transform, i.e., ln)X�t� − X�t − 1�* = lnM + ln)F �t� − F �t −1�*+ $�t�, where F �t� is the closed-form solution of the cdf underthe model, and $�t� is i.i.d. normal.17 MSE= SSE/�n−k�, where n is the number of observations and kthe number of free parameters. BIC=−2LLc+k ln�n�, where LLc isthe concentrated log-likelihood function. Under the assumption ofnormally distributed errors, the latter is computed from the non-linear regression solution as LLc = 1/2n)ln�n�− 1− ln (SSE)} (e.g.,Davidson and MacKinnon 1993, Seber and Wild 1989). The useof the concentrated rather than true log-likelihood is immaterial

models’ MSE and MAD to that of the two-segmentmodel. This relative measure controls for differencesacross data series in their total variance, with onebeing the neutral value and higher values indicatingsuperior fit of the two-segment model. To save space,we report only the difference in BIC and MAPE, withzero being the neutral value and higher values indi-cating superior fit of the two-segment model.Table 4 reports the performance indicators aver-

aged for each of the four sets of data as well asfor all 33 data series. Technical Appendix C reportsresults for the individual series. The first panel per-tains to models with additive i.i.d. errors. Let us startby focusing on the BIC, where a three-point differenceis large enough to be evidence of superior fit and a10-point difference provides strong to very strong evi-dence of superior fit (Raftery 1995). The two-segmentmodel fits markedly better than the MIM, G/SG, andWG models for tetracycline, music CDs, and high-tech products, but not for the miscellaneous productswhere the presumption of a discrete mixture is notstrong a priori. The two-segment model fits aboutequally as well as the continuous-mixture KG model,except for tetracycline, where it beats it by a sizablemargin. The same pattern exists for the other threeperformance measures: The two-segment model fitsmarkedly better than MIM, G/SG, and WG for datawhere a two-segment structure is a priori likely, butnot elsewhere; and the two-segment model fits aboutequally as well as the Karmeshu-Goswami model inall data sets.Turning our attention to the second panel in

Table 4, we see that allowing for serial correlationin the more poorly specified models tends to some-what narrow the gap with the two-segment model.However, the performance gap for products wherea two-segment structure is a priori likely does not

for our purpose. For instance, for nested models, the likelihoodratio test statistic constructed using the concentrated log-likelihoodremains +2 distributed (Seber and Wild 1989).

Van den Bulte and Joshi: New-Product Diffusion with Influentials and ImitatorsMarketing Science 00(0), pp. 1–23, © 2007 INFORMS 17

Figure 6 Actual and Predicted Number of Adopters in MedicalInnovation (Predictions from SM Estimates of the PTMWithout Serial Correlation)

0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

IndependentsImitators

TotalActual

vanish. For high-technology products, adding serialcorrelation even increases the gap in BIC and MSEvis-à-vis MIM, G/SG, and WG. The results in thethird panel of Table 4 indicate that using a multi-plicative rather than additive error structure does notaffect the main conclusion from the first two panelsvery much: The two-segment model fits about as wellas the continuous-mixture KG model, and markedlybetter than the MIM, G/SG, and WG models for newproducts for which a two-segment structure is a priorilikely.

Table 4 Descriptive Performance of the Two-Segment Model Compared to Mixed-Influence, Gamma/Shifted Gompertz, Weibull-Gamma, andKarmeshu-Goswami Models for Different Error Structures#

BIC difference MAPE difference MSE ratio MAD ratio

MIM G/SG WG KG MIM G/SG WG KG MIM G/SG WG KG MIM G/SG WG KG

1. Additive error without serial correlation (AR0)Tetracycline 1044 1327 1445 899 796 810 788 −011 2.21 2.38 2.55 1.46 1.71 1.72 1.76 1.24Music CDs 5883 4409 3498 117 1873 1792 882 −158 2.66 2.06 2.21 0.92 1.69 1.54 1.47 0.98High-tech 984 382 447 205 5119 675 716 821 2.41 1.57 1.74 1.10 1.92 1.35 1.47 1.05Miscellaneous 073 −216 054 −055 556 113 383 −276 1.21 0.93 1.11 0.90 1.24 1.05 1.11 0.88All 3514 2527 1951 112 2017 1167 712 −034 2.14 1.62 1.76 0.96 1.60 1.37 1.37 0.97

2. Additive error with serial correlation (AR1)Tetracycline 892 819 1006 881 444 −1111 694 114 2.00 1.75 2.13 1.47 1.73 1.44 1.82 1.17Music CDs 3419 2259 2824 765 487 279 930 210 2.12 1.65 1.93 1.11 1.47 1.17 1.54 1.11High-tech 1267 515 783 022 3539 680 682 722 2.97 1.70 2.19 0.98 2.12 1.46 1.67 0.91Miscellaneous 262 −162 279 −159 412 516 133 −502 1.30 0.91 1.25 0.79 1.23 0.99 1.13 0.80All 2245 1275 1625 421 860 363 639 109 1.95 1.42 1.75 1.01 1.48 1.17 1.43 0.99

3. Multiplicative error (log-log model) without serial correlation (AR0)Tetracycline −124 −137 109 −168 321 302 290 208 1.11 1.10 1.16 0.78 1.24 1.22 1.25 1.02Music CDs 5173 3595 1532 404 158 120 063 −009 2.34 1.97 1.54 0.97 1.70 1.54 1.30 0.95High-tech 910 −729 351 −793 854 438 1449 −556 2.25 0.76 1.65 0.55 1.81 1.06 1.83 0.74Miscellaneous 446 −045 782 168 1404 −475 1511 097 1.60 1.04 1.93 1.03 1.37 1.01 1.48 0.96All 3232 1977 1126 178 574 014 654 −044 2.06 1.45 1.64 0.92 1.60 1.31 1.40 0.93

#To save space and aid interpretation, we report only the relative fit performance by comparing the fit of the two-segment discrete mixture model againstthat of the alternative models. For BIC and MAPE, we report the alternative models’ value minus that of the two-segment model. For MSE and MAD, we reportthe alternative models’ value divided by that of the two-segment model. Therefore, for the BIC and MAPE differences, the neutral value is zero; for the MSEand MAD ratios, it is one. For all metrics, higher values indicate superior fit of the two-segment model. For the BIC and MAPE differences, the average valuesreported are arithmetic means. For the MSE and MAD ratios, they are geometric means because this is a better measure of the central tendency of a ratio thanthe arithmetic mean.

6. ConclusionWe have analyzed the diffusion of innovations in mar-kets with two segments: influentials who are more intouch with new developments and who affect anothersegment of imitators whose own adoptions do notaffect the influentials. Such a structure with asym-metric influence is consistent with several theories insociology and diffusion research, including the clas-sic two-step flow hypothesis and Moore’s more recenttechnology adoption framework. Our model allowsdiffusion researchers to operationalize these theorieswithout recourse to micro-level diffusion data and toestimate parameters from real data. There are fourmain results.(1) Diffusion in a mixture of influentials and imita-

tors can exhibit the traditional symmetric-around-the-peak bell shape, asymmetric bell shapes, as well as adip or “chasm” between the early and later parts ofthe diffusion curve. In contrast to Moore’s contention,the model suggests that it need not always be nec-essary to change the product to gain traction amonglater adopters and for the adoption curve to swing upagain. Tetracycline is an example.(2) The proportion of adoptions stemming from

independents need not decrease monotonically; it canalso first decline and then rise again to unity. This

Van den Bulte and Joshi: New-Product Diffusion with Influentials and Imitators18 Marketing Science 00(0), pp. 1–23, © 2007 INFORMS

result disproves a common contention among diffu-sion researchers based on an erroneous mixture inter-pretation of the mixed-influence model (e.g., Bass1969, Mahajan et al. 1993, Rogers 2003).(3) Specifying a mixed-influence model to a mixture

process with pure independents and pure imitatorscan generate systematic changes in the parameter val-ues. As several authors have noted, diffusion withina pure-type mixture of independents and imitatorswith hazards p and qF �t�, respectively, is distinct fromdiffusion in a homogeneous population with mixed-influence, where everyone adopts with hazard p +qF �t�. The closed-form solutions we present not onlyprove this mathematically, but also show that impos-ing a mixed-influence specification on a pure-typemixture process can generate the systematic changesin the parameter values reported by Van den Bulteand Lilien (1997), Bemmaor and Lee (2002), and Vanden Bulte and Stremersch (2004), unless � is close toeither zero or one, or unless p1 → and pure imi-tators (p2 = 0) have a very specific influence weightw= ��1+ q2− ��/q2 > �.(4) Empirical analysis of four sets of data com-

prising a total of 33 different data series (the classicMedical Innovation data, 19 music CDs, five high-techproducts, and eight miscellaneous innovations) indi-cates that the two-segment model fits markedly betterthan the MIM, the G/SG, and the WG models, at leastfor innovations for which a two-segment structure islikely to exist. Hence, the model does better when it istheoretically expected to, and does not when it is nottheoretically expected to. The two-segment model fitsabout equally as well as the mixed-influence modelproposed by Karmeshu and Goswami (2001), wherep and q vary in a continuous fashion. Overall, thefindings on descriptive performance are robust tochanges in the error structure and indicate that thediscrete-mixture model is sufficiently different andthe data sufficiently informative for the model to fitreal data better than other models.The models we presented provide sharper insight

into how social structure can affect macro-level diffu-sion patterns, and should prove useful in five areasof application where influentials and imitators are apriori likely to exist. The first two are high-technologyand health care products, including pharmaceuticals.In these two areas, innovations are often perceived tobe complex or risky, and mainstream imitators refuseto be on the “bleeding edge,” unlike opinion leadersand lead users. The third area is that of entertain-ment and mass culture products like gaming soft-ware, music, books, and movies, where the distinctionbetween aficionados and the casual mainstream audi-ence can loom large.18 Teen marketing is the fourth

18 Explicitly allowing for influentials and imitators may be espe-cially useful for products carried by characters, writers, actors, or

area where the distinction between influentials andimitators may be critical in the new product diffu-sion process. For several years, P&G has been oper-ating Tremor as a mechanism to connect with highlyinvolved and influential teens, encourage adoptionamong them, and through them reach out to thelarger teen population. Categories in which Tremorand similar services have been used include notonly fashion-oriented apparel and entertainment, butalso more mundane fast-moving consumer goods likebeauty aids and food. The fifth area of particularpotential consists of situations where a segment ofenthusiasts has pent-up demand. For instance, whenInternet access providers started operating in Francein 1996, a rather large number of people adoptedtheir services. New adoptions dipped in 1997, onlyto increase again from 1998 onwards. The deviationfrom the standard bell shape was not the low num-ber in 1997, but the high initial number in 1996, whenmany university users who had been accessing theInternet exclusively through the university RENATERnetwork were finally able to start using the Internetat home as well (Fornerino 2003). In the case whereenthusiasts can place advance orders that the market-ing analyst can observe (e.g., Moe and Fader 2002),it may be useful to explicitly allow for a differencebetween the start time of the diffusion process of thetwo segments.

6.1. Implications for PracticeThe first two of our results have clear managerialimplications. Because dips in the adoption curve canstem from the mere presence of influentials and imi-tators, it need not always be necessary for firms tochange their product to gain traction among lateradopters and for the adoption curve to swing upagain. In contrast to what Moore (1991) claims,launching a new version to appeal to prospects whohave not yet adopted need not always be necessary,let alone optimal, to get out of the dip. Of course,when the dip results not from a social chasm between

directors who already have a small following among aficionados,but have not yet broken through to the mainstream. In such cases,one would expect the former to adopt according to an independentprocess and the latter to adopt only through contagion, if at all. Thismight result in a temporary dip. Movies starring Christina Ricciand movies directed by Ang Lee exhibit this pattern. Early in hercareer, Ricci played in several independent movies that won criti-cal acclaim and earned her the label of “Indie Queen.” These earlymovies exhibited the bell curve typical of very successful “sleepers”(The Ice Storm-1997, The Opposite of Sex-1998, Buffalo 66-1998).Then followed a small movie exhibiting a dip (Desert Blue-1998),while her recent movies are more standard Hollywood fare exhibit-ing the standard monotonic, exponential decline (e.g., The ManWho Cried-2001). The same pattern is observed for movies directedby Ang Lee: bell-shaped for The Ice Storm-1997, a temporary dipfor Ride with the Devil-1999, and monotonic decline for his morerecent Hollywood production, The Hulk-2003.

Van den Bulte and Joshi: New-Product Diffusion with Influentials and ImitatorsMarketing Science 00(0), pp. 1–23, © 2007 INFORMS 19

segments (very low w� but from a difference in whatconstitutes an acceptable product offering, changingthe product will be necessary to gain traction in thesecond segment.We have also shown that the proportion of adop-

tions stemming from influentials need not decreasemonotonically; it can also first decline and then riseagain. Hence, while it may make sense for firmsto shift the focus of their marketing efforts fromindependents to imitators shortly after launch asshown by Mahajan and Muller (1998) using a two-period model, they may want to start increasing theirresource allocation to independent decision makersagain later in the process. Managers who confuse thedistinction between influentials and imitators withthat between early and late adopters, and ignore ourresults and others’ empirical evidence that the bulk ofthe late adoptions may stem from people not subjectto social contagion (e.g., Becker 1970, Coleman et al.1966), may end up wasting money by poor targeting.Both of these prescriptive implications assume the

existence of influentials and imitators. Of course,thoughtful managers will want to check these assump-tions against data from their own markets to assess towhat extent they should trust these implications. Stan-dard aggregate-level data and models can be quitemisleading for identifying a causal mechanism affect-ing new product diffusion (e.g., Bemmaor 1994, Vanden Bulte and Stremersch 2004). Managers and mar-ket researchers must realize that disaggregate data arenecessary to gain a better understanding of whetherand how social contagion drives the diffusion of theirproducts (e.g., Burt 1987, Van den Bulte and Lilien2001).Our work also has important implications for how

managers should develop more effective networkmarketing efforts. Several firms in the pharmaceuti-cal industry, longtime leaders in applying marketinganalytics, are now conducting research in which theyask physicians to name the opinion leaders in theirsocial network. Typically, firms use this informationto guide their sales reps to the more central physi-cians. In terms of our model, they are allocating theirresources to make F1�t� grow faster, in the hope thatthis will get F2�t� growing faster as well through thesocial multiplier effect captured by wq2. This makessense, but should be complemented with efforts toincrease the multiplier, especially the weight factor w.Rather than focusing only on identifying and convert-ing influentials, firms should also identify ways toincrease their impact (e.g., Valente et al. 2003).The limited value of aggregate-level data to detect

contagion effects does not mean that nothing can belearned from them. Following the lead of studies likethat of Hahn et al. (1994), firms could analyze the

sales evolution of multiple products and look for sys-tematic differences in parameters like �, w, and q2 thatcan be related to product or market characteristics.This, in turn, may help firms develop a better under-standing of why product sales evolve the way theydo, and might even result in better forecasting mod-els. Such analysis should be useful in all five areasof application identified earlier. From a data availabil-ity point of view, it should be particularly appealingto firms in the book, music, and film industries wholaunch many products each year, and to consultingand research firms with many clients in pharmaceu-ticals or in high-tech industries.

6.2. Additional Implications for Education andResearch

We have shown that some ideas in mathematical dif-fusion modeling that have become part of the stan-dard marketing curriculum through influential papersand books (Bass 1969, Rogers 2003) are wrong andhave misleading marketing implications. We hope ourwork will help redress this situation in both educationand research training.Several of the implications for practice we presented

above have clear research opportunities attached tothem. Another important extension of our work wouldbe to incorporate control variables, including market-ing efforts. This may not only be useful for empir-ical research (e.g., to what extent are dips simplycaused by exogenous demand shocks?), but may alsoenable one to more rigorously study the decision totarget independents versus imitators. Even a simpli-fied three-period model might be helpful in study-ing under what conditions it is profit maximizing tochange one’s targeting from independents to imitatorsand, possibly, to independents again (Esteban-Bravoand Lehmann 2005). Like the models we presented,this extension would allow one to better understandcurrent arguments and findings, to formalize richertheoretical arguments, and perhaps even to opera-tionalize them into estimable models that help bridgethe gap between theory and data.

AcknowledgmentsThe authors benefited from comments by David Bell, AlbertBemmaor, Xavier Drèze, Peter Fader, Donald Lehmann,Gary Lilien, Piero Manfredi, Paul Steffens, Stephen Tanny,Masataka Yamada, the Marketing Science reviewers, asso-ciate editor and editor, and audience members at the 2005INFORMS Marketing Science Conference. They also thankPeter Fader for providing the SoundScan CD sales data.

Appendix

A.1. Solution for F2�t� in AIM with q1 > 0To simplify notation, we omit the time argument from func-tions and write F1 instead of F1�t�, etc. We know that

F1 = �1− e−�p1+q1�t�/(1+ q1

p1e−�p1+q1�t

)

Van den Bulte and Joshi: New-Product Diffusion with Influentials and Imitators20 Marketing Science 00(0), pp. 1–23, © 2007 INFORMS

Because h2 = p2+ q2�wF1+ �1−w�F2� we write:

dF2dt

=(p2+ q2w

1− e−�p1+q1�t

1+ q1e−�p1+q1�t/p1

)

+(q2�1−w�− p2− q2w

1− e−�p1+q1�t

1+ q1e−�p1+q1�

t /p1

)F2

− q2�1−w�F 22 (A.1.1)

This is a Ricatti equation of the general form dy/dx =P�x�+Q�x�y+R�x�y2. Setting y = F2 and x= t� we get

P�x�= p2+ q2w1− e−�p1+q1�t

1+ q1e−�p1+q1�t/p1

�

Q�x�= q2�1−w�− p2− q2w1− e−�p1+q1�t

1+ q1e−�p1+q1�t/p1

�

and R�x�=−q2�1−w�. F2 = 1 is a potential solution for thisRicatti equation. We use the transformation

z= 1F2− 1

⇒ F2 =z+ 1z

For F2 continuous in �0�1�, z is continuous in �−�−1�.Note,

dF2dt

=− 1z2

dz

dt

The equation now becomes:

dz

dt= q2�1−w�

+(q2�1−w�+p2+q2w

1−e−�p1+q1�t

1+q1e−�p1+q1�t/p1

)z (A.1.2)

This is of the form d1/dx+P1�x�1=Q1�x� with 1= z; x= t;

P1�x�=−(q2�1−w�+ p2+ q2w

1− e−�p1+q1�t

1+ q1e−�p1+q1 �t/p1

)�

and Q1�x� = q2�1 − w� The general solution for such anequation is

1=∫u�x�Q1�x�dx+ c

u�x��

where u�x� = exp�∫ P1�x�dx� is the integrating factor.Because∫

P1�x�dx=−p2t− q2t−q2w

q1ln�p1+ q1e

−�p1+q1�t��

we get

u�x�= exp(−p2t− q2t−

q2w

q1ln�p1+ q1e

−�p1+q1�t�

)

Hence,∫u�x�Q1�x�dx

= e−�p2+q2�tq1q2�1−w��p1+ q1e−�p1+q1�t�−wq2/q1H1

q2�p1w− q1�1−w��− p2q1�

where

H1 = 2F1

(1�

wq2q1

�1+ wq2q1

− p2+ q2p1+ q1

�p1

p1+ q1e−�p1+q1�t

)

and 2F1�1� b� c� k� is the Gaussian hypergeometric function,the series representation of which is

∑n=0

�b+n� �c�

�b� �c+n�kn

This series is convergent for arbitrary b, c when �k�< 1; and

when k = ±1 if c > 1 + b. This implies that the series isconvergent as long as q1 > 0.Substituting back, we get

z=e−�p2+q2�tq1q2�1−w��p1+ q1e

−�p1+q1�t�−wq2/q1H1q2�p1w− q1�1−w��− p2q1

+ c

exp(−p2t− q2t−

q2w

q1ln�p1+ q1e

−�p1+q1�t�

)

Transforming z back to F2� we obtain

F2 = 1+exp

(−p2t− q2t−

q2w

q1ln�p1+ q1e

−�p1+q1�t�

)

e−�p2+q2�tq1q2�1−w��p1+ q1e−�p1+q1�t�−wq2/q1H1

q2�p1w− q1�1−w��− p2q1+ c

(A.1.3)Because F2�0�= 0,

c= �p1+ q1�−wq2/q1 �p2q1+ q2�q1�1−w��1−H2�− p1w��

q2�p1w− q1�1−w��− p2q1�

where

H2 = 2F1

(1�

wq2q1

�1+ wq2q1

− p2+ q2p1+ q1

�p1

p1+ q1

)

Simplifying, we obtain as a closed-form expression:

F2�t�= 1+ �p2q1+ q2�p1w− q1�1−w���

·(q1q2�1−w�H1+ e�p2+q2�t

(p1+ q1e

−�p1+q1�t

p1+ q1

)wq2/q1

· �p2q1+ q2�q1�1−w��1−H2�− p1w��

)−1 (A.1.4)

As w → 0, this expression for F2�t� reduces to the closed-form solution for the MIM.

A.2. Solution for F2�t� in AIM with q1 = 0We know that F1 = 1 − e−p1t , and hence f2 = dF2/dt =�p2+ q2�wF1+ �1−w�F2���1− F2� equals:

dF2dt

= p2+ q2w�1− e−p1t�+ �q2�1− 2w+we−p1t�− p2�F2

− q2�1−w�F 22 (A.2.1)

The above equation is a Ricatti equation of the general formdy/dx = P�x�+Q�x�y+R�x�y2. Setting y = F2 and x = t, wehave P�x�= p2+ q2w�1− e−p1t�; Q�x�= q2�1− 2w+we−p1t�−p2; R�x�=−q2�1−w�.

F2 = 1 is a potential solution for this Ricatti equation. Weuse the transformation:

z= 1F2− 1

⇒ F2 =z+ 1z

� anddF2dt

=− 1z2

dz

dt

For F2 continuous in �0�1�, z is continuous in �−�−1�.Substituting in Equation (A.2.1):

dz

dt= q2�1−w�+ z�p2+ q2�1−we−p1t�� (A.2.2)

Equation (A.2.2) is of the form: d1/dx + P1�x�1 =Q1�x�,with 1 = z; x = t; P1�x� = −p2 − q2�1 − we−p1t�; Q1�x� =q2�1−w�. The general solution for this equation is

1=∫u�x�Q1�x�dx+ c

u�x�� (A.2.3)

Van den Bulte and Joshi: New-Product Diffusion with Influentials and ImitatorsMarketing Science 00(0), pp. 1–23, © 2007 INFORMS 21

where u�x� = exp�∫ P1�x�dx� is the integrating factor. Be-cause ∫

P1�x�dx=−p2t− q2t−q2p1we−p1t�

we get

u�x�= exp(−p2t− q2t−

q2p1we−p1t

)�

and hence∫u�x�Q1�x�dx

=∫exp

(−p2t− q2t−

q2p1we−p1t

)q2�1−w�dt

Now let us define

I=q2�1−w�∫exp

(−p2t−q2t−

q2p1we−p1t

)dt (A.2.4)

To solve this integral, we do another transformation:

a= e−p1t ⇒ t =− 1p1ln a ⇒ dt =− 1

p1ada

Equation (A.2.4) then becomes

I =− q2p1�1−w�

∫a�p2+q2�/p1−1 exp

(− q2p1wa

)da�

with the solution

I = q2p1�1−w�

(q2p1w

)−�p2+q2�/p1

(p2+ q2p1

q2p1wa

)� (A.2.5)

where �!�k� is the “upper” incomplete gamma function:

�!�k�=∫

kv!−1e−v dv

Substituting a = e−p1t in Equation (A.2.5), and then I =∫u�x�Q1�x�dx from Equation (A.2.5) back into Equation(A.2.3), we obtain:

z�t� = I + c

u�x�

=

[q2p1�1−w�

(q2p1w

)−�p2+q2�/p1

(p2+ q2p1

�q2p1we−p1t

)+ c

]

�exp�−p2t− q2t− �q2/p1�we−p1t��

(A.2.6)

Transforming z back to F2, we get:

F2�t�=1+exp�−p2t−q2t−�q2/p1�we−p1t�

q2p1�1−w�

(q2p1w

)−�p2+q2�/p1

(p2+q2p1

�q2p1we−p1t

)+c

(A.2.7)As F2�0�= 0, we get

c = −exp(− q2p1w

)

− q2p1�1−w�

(q2p1w

)−�p2+q2�/p1

(p2+ q2p1

�q2p1w

)

Hence:

F2�t� = 1+ exp(−p2t− q2t−

q2p1we−p1t

)

·(q2p1�1−w�

(q2p1w

)−�p2+q2�/p1(

(p2+ q2p1

�q2p1we−p1t

)

−

(p2+ q2p1

�q2p1w

))− exp

(− q2p1w

))−1

(A.2.8)

As w → 0, this expression for F2�t� reduces to the closed-form solution for the MIM.

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Page: 16Sequence number: 1Author: Production EditorDate: 1/29/2007 4:07:59 PM Type: Note

Footnote mark 'b' is not used in the body of the table, yet it appears in the notes.

Page: 17Sequence number: 1Author: Production EditorDate: 1/29/2007 4:08:40 PM Type: Note

Edit correct?

Sequence number: 2Author: Production EditorDate: 1/29/2007 4:08:56 PM Type: Highlight

Moore’s

Sequence number: 3Author: Production EditorDate: 1/29/2007 4:09:26 PM Type: Note

Insert year.

Sequence number: 4Author: Production EditorDate: 1/29/2007 4:09:40 PM Type: Highlight

Moore’s

Sequence number: 5Author: Production EditorDate: 1/29/2007 4:09:49 PM Type: Note

Insert year.

Page: 18Sequence number: 1Author: Production EditorDate: 1/29/2007 4:10:22 PM Type: Note

Edits OK?

Page: 20Sequence number: 1Author: Production EditorDate: 1/29/2007 4:11:04 PM

Type: NoteExtra closing paren removed, OK as set?

Page: 21Sequence number: 1Author: Production EditorDate: 1/29/2007 4:11:46 PM Type: Note

Update available?

Sequence number: 2Author: Production EditorDate: 1/29/2007 4:12:03 PM Type: Note

Pages correct?

Page: 22Sequence number: 1Author: Production EditorDate: 1/29/2007 4:12:20 PM Type: Highlight

Serv.

Sequence number: 2Author: Production EditorDate: 1/29/2007 4:12:30 PM Type: Note

Spell out term.

Sequence number: 3Author: Production EditorDate: 1/29/2007 4:12:41 PM Type: Highlight

Sociol.

Sequence number: 4Author: Production EditorDate: 1/29/2007 4:12:51 PM Type: Note

Spell out term.

Sequence number: 5Author: Production EditorDate: 1/29/2007 4:13:07 PM Type: Note

Confirm correct year.

Sequence number: 6Author: Production EditorDate: 1/29/2007 4:13:18 PM Type: Note

Update available?

Sequence number: 7Author: Production EditorDate: 1/29/2007 4:14:09 PM Type: Note

One author or two? Cited as Karmeshu and Goswami in text.

Sequence number: 8

Author: Production EditorDate: 1/29/2007 4:14:22 PM Type: Highlight

Swed.

Sequence number: 9Author: Production EditorDate: 1/29/2007 4:14:32 PM Type: Note

Spell out term.

Page: 23Sequence number: 1Author: Production EditorDate: 1/29/2007 4:14:55 PM Type: Highlight

Model.

Sequence number: 2Author: Production EditorDate: 1/29/2007 4:15:03 PM Type: Note

Spell out term.

Sequence number: 3Author: Production EditorDate: 1/29/2007 4:15:10 PM Type: Highlight

Statist.

Sequence number: 4Author: Production EditorDate: 1/29/2007 4:15:18 PM Type: Note

Spell out term.

Sequence number: 5Author: Production EditorDate: 1/29/2007 4:15:31 PM Type: Highlight

Medical Innovation

Sequence number: 6Author: Production EditorDate: 1/29/2007 4:15:55 PM Type: Note

In italics in source?

Sequence number: 7Author: Production EditorDate: 1/29/2007 4:16:07 PM Type: Note

Update available?