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The Innovation Diffusion Process in a Heterogeneous Population: A Micromodeling ApproachAuthor(s): Rabikar Chatterjee and Jehoshua EliashbergSource: Management Science, Vol. 36, No. 9 (Sep., 1990), pp. 1057-1079Published by: INFORMSStable URL: http://www.jstor.org/stable/2632356 .

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MANAGEMENT SCIENCE Vol. 36, No. 9, September 1990

Printed in U.S.A.

THE INNOVATION DIFFUSION PROCESS IN A HETEROGENEOUS POPULATION:

A MICROMODELING APPROACH*

RABIKAR CHATTERJEE AND JEHOSHUA ELIASHBERG School of Business Administration, University of Michigan, Ann Arbor, Michigan 48109-1234

The Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6371

A model of the innovation diffusion process is developed using a micromodeling approach that explicitly considers the determinants of adoption at the individual level in a decision analytic framework, and incorporates heterogeneity in the population with respect to initial perceptions, preference characteristics, and responsiveness to information. The micromodelling approach provides a behavioral basis for explaining adoption at the disaggregate level and the consequent pattern of diffusion at the aggregate level. The analytical implications of the model are compared and contrasted with the traditional, aggregate-level, diffusion models. An advantage of our approach is its micro-theory driven flexibility in accommodating various patterns of diffusion. Examples are provided of conditions under which the model yields diffusion patterns identical to those of some well-known aggregate models. A pilot study is reported, outlining procedures for data collection and estimation of the individual-level parameters, and providing a preliminary test of the predictive performance of the model. Measurement of the individual parameters prior to product launch enables potential applications of the model for segmentation of the target population in terms of their expected adoption times. (MARKETING-NEW PRODUCTS; MARKETING-DIFFUSION OF INNOVATIONS; MARKETING-MEASUREMENT; DECISION ANALYSIS-APPLICATIONS)

1. Introduction

The pattern of sales growth of an innovation-a new product or service-and the factors underlying the diffusion process have been an important subject of study in marketing, from a theoretical and behavioral perspective as well as from a quantitative modeling viewpoint. The diffusion modeling tradition in marketing (Bass 1969) owes its conceptual foundation to mathematical models of contagion in such applications as the diffusion of news and rumors (Bartholomew 1967). The modeling approach takes an aggregate perspective and formulates a differential equation (or a set of equations) to specify the flow(s) between mutually exclusive and collectively exhaustive subgroups (e.g., adopters and nonadopters in a two-state model). This modeling paradigm has produced a rich stream of literature. For detailed reviews, see Mahajan and Peterson (1985) for descriptive models; Kalish and Sen (1986) for normative models incorporating marketing mix variables in a monopoly setting; Eliashberg and Chatterjee (1985) and Dolan, Jeuland, and Muller (1986) for normative models extended to competitive settings; and Eliashberg and Chatterjee (1986) for models explicitly incorporating stochastic con- siderations.

The diffusion of an innovation in a population involves the adoption of the innovation by individuals in the relevant population. Adoption, in turn, involves a deliberate choice decision on the part of the individual, especially in the case of high involvement products (such as consumer durables). Heterogeneity in the population suggests systematic differences in adoption times across individuals. Aggregate diffusion models that conceptualize the innovation diffusion process as analogous to the spread of diseases or news (where no deliberate decision is involved) limit their consideration of the determinants

* Accepted by John R. Hauser, former Departmental Editor; received June 30, 1988. This paper has been with the authors 4 months for 2 revisions.

1057 0025- 1909/90/3609/ 1057$0 1.25

Copyright ? 1990, The Institute of Management Sciences



1058 RABIKAR CHATTERJEE AND JEHOSHUA ELIASHBERG

of adoption at the individual level. Gatignon and Robertson ( 1986) state that "the behavioral assumptions underlying [aggregate level] consumer diffusion models are typically simple and do not provide a behavioral explanation for the rate or pattern of diffusion" (p. 38). As has been pointed out (e.g., by Tanny and Derzko 1988), the mathematical form of the Bass model requires the assumption that the potential adopter population is homogeneous.' The assumption of homogeneity implies that, at any point in the process, all individuals who are yet to adopt have the same probability of adopting in a given time period, so that differences in individual adoption times are purely stochastic.

In this article we develop an analytical model of the innovation diffusion process for a high involvement durable or service that explicitly considers the determinants of adoption at the individual level and allows for heterogeneity with respect to these determinants across the population. The micromodeling approach adopted here thus postulates individual adoption times as an explicit function of the characteristics of potential adopters. The resulting aggregate model consequently incorporates a microlevel behavioral basis to describe the innovation diffusion process in a heterogeneous population. The approach employs the following general framework. An individual's perception of the innovation determines his evaluation of the innovation which in turn determines his adoption decision.2 Perceptions of the innovation are initially uncertain: these perceptions change over time as the potential adopter receives additional information about the innovation. An individual's timing of adoption is thus determined by the dynamics of perceptions, given his preference structure. Aggregation across individuals yields the penetration curve; the distribution of individual adoption times determines the rate and pattern of adoption.

Models of adoption of innovations using the micromodeling approach originally ap- peared in the economics literature (Hiebert 1974; Stoneman 1981; Feder and O'Mara 1982; Jensen 1982). Hiebert characterizes the effect of risk attitude and learning under uncertainty on the individual level adoption decision. In contrast, the other three models employ more specific assumptions, including Bayesian updating of uncertain perceptions, to examine the pattern of diffusion. Stoneman examines intra-firm diffusion of new technology, while Feder and O'Mara and Jensen consider diffusion in a heterogeneous population. Heterogeneity is limited to initial perceptions of the innovation, and the adopting units are assumed to be risk neutral.

Dynamic brand choice models based on a similar conceptualization have been introduced to the marketing literature (Meyer and Sathi 1985; Roberts and Urban 1988). These models allow the consumer to revise his/her evaluation of the brand(s) upon receipt of new information. This in turn influences the probability of brand choice. Roberts and Urban's formulation is of particular relevance to our work. Their framework invokes von Neumann-Morgenstem utility theory to model preference under uncertainty and Baye;,ian updating of beliefs about the value of the brand. A multinomial logit model then links preference to brand choice. Their model is applied to forecast sales of a new automobile.

We adopt a decision analytic conceptualization of the dynamics of individual behavior developed in economics and introduced to marketing in an integrated framework by Roberts and Urban. Like them, we invoke a von Neumann-Morgenstern utility function and Bayesian updating of perceptions. Our focus, however, is on the analytical derivation of the diffusion curve for a major innovation (rather than a new brand in an existing product category), explicitly incorporating the impact of consumer heterogeneity. Further,

1 Lekvall and Wahlbin ( 1973), recognizing this fact, interpreted the Bass model parameters as coefficients of external and internal influence, rather than as coefficients of innovation and imitation as proposed by Bass (1969), who suggested a dichotomous (innovator/imitator) population.

2 Conceptually, the model applies to innovations for consumer or industrial markets. However, for purposes of exposition, we refer to potential adopting units as consumers. We also employ the masculine gender while referring to consumers, purely for ease of presentation.



INNOVATION DIFFUSION IN A HETEROGENEOUS POPULATION 1059

we model diffusion in a given population so that once a consumer adopts the innovation he drops out of the subpopulation of potential adopters, which is consistent with the basic phenomenon of diffusion in a social system (Rogers 1983). In contrast, Roberts and Urban assume an exogenous flow of potential buyers, since they focus on the probability of choosing the new brand given purchase in the product category. Oren and Schwartz (1988) also use a Bayesian decision analytic framework to model the dynamics of market share of a new product (e.g., a new medical technology) assuming an exogenous flow of potential adopters. Their analytical model formally incorporates heterogeneity with respect to the potential adopters' degree of risk aversion. An extension to a quality-heterogeneous population is briefly discussed.

Our analytical focus is similar to the work in economics by Feder and O'Mara (1982) and Jensen (1982), as well as Oren and Schwartz (1988). However, our approach provides a richer conceptualization of the process. First, we allow heterogeneity with respect to: (i) initial consumer perceptions of the innovation's performance, including confidence in initial beliefs; (ii) consumers' preference structure, captured by the price/performance tradeoff and degree of risk aversion; and (iii) the perceived reliability of information. Second, we explicitly model the dynamics of adoption in a stochastic process framework to allow for variability of information. We seek to derive a parsimonious representation of the process based on the combined effects of the various individual-level and innovation- specific determinants of adoption behavior.

We develop the model in ?2. Next, we discuss some of its key analytical properties and implications, contrasting our micromodeling approach with the more traditional aggregate-level diffusion models in the marketing science literature (?3). We demonstrate that our approach yields a flexible diffusion model that provides a microlevel behavioral basis for explaining a variety of diffusion patterns, and illustrate special cases where combinations of the distribution of population characteristics and the pattern of information reproduce some well known aggregate diffusion curves, including the Bass model.

A major advantage of the micromodeling approach is that the individual-level variables postulated to determine adoption timing can be measured via a consumer survey prior to launch. In ?4, we report a pilot study that outlines procedures for data collection and estimation of the individual level parameters and provides a preliminary test of the predictive ability of the model. While the major focus of-our exposition in analytical, the possibility of prelaunch measurement of individual characteristics (based on established marketing science procedures) points to potential applications of the model for segmentation of the potential adopter population. Conceptually, the individual measures may be used to forecast the aggregate penetration curve, given estimates of the nature and extent of information about the innovation that will be generated over time. However, it should be noted that our objective is not to present a comprehensive methodology for prelaunch forecasting; such a methodology would entail the consideration of issues beyond the scope of this paper. We briefly discuss some of these issues in our conclusion (?5).

2. The Model

2.1. Modeling Approach and Assumptions

Potential adopters are assumed to evaluate the innovation along two attributes, performance and price. Prior to adoption, a consumer knows the price but is uncertain about performance. The potential adopter's (uncertain) perception of the innovation's performance is revised over time as he receives information about it. The micromodel predicts individual-level adoption timing. These predictions are then aggregated over the potential adopter population to yield the penetration curve.

Consumers' Preference Structure. Given uncertainty in the potential adopter's assessment, an important factor in the adoption decision is his attitude toward risk. The




importance of perceived risk in the context of high involvement decision making under uncertainty has triggered extensive research in consumer behavior since the pioneering work of Bauer (1960) (see also Bettman 1973 and Ross 1974). We assume that for high involvement innovative products all potential adopters are risk averse, although the degree may vary across the population. We capture the potential adopter's risk aversion via the following uniattribute utility function in a von Neumann-Morgenstern (1947) framework:

u,(xi) = 1 - exp(-c_ci), (1)

where xZi denotes the potential adopter's uncertain perception of performance after receiving i "units" of information about the innovation and c (>0) is the coefficient of risk aversion (Pratt 1964).3 The utility function in (1) is scaled so that ux(O) = 0. The exponential form of the utility function implies constant absolute risk aversion with respect to performance (Keeney and Raiffa 1976). The assumption of constant risk aversion has received empirical support in the decision analysis literature (Howard 1971) as well in marketing applications (Hauser and Urban 1977, 1979; Currim and Sarin 1983, 1984). Roberts and Urban (1988) also adopt an exponential uniattribute utility function, where the single attribute is a composite measure of value the consumer places on the brand.

Given the price, p, of the innovation, the potential adopter's utility for the innovation may be represented by the following widely accepted additive utility function (Keeney and Raiffa 1976):

U(xZi, p) = k,u,(xZi) + kpup(p), (2)

where k, and kp are the scaling constants (which may be interpreted as importance weights) associated with the uniattribute utility functions for price and performance respectively. Assuming that a consumer's utility for price is linear in its argument, the two-attribute utility function (2) may be written, without loss of generality, as

u(ki, p) = u,(xi) + kup(p) = 1 - exp(-cxi) - kp, (3)

where k is a measure of the relative importance of price. Note that u(0, 0) = 0, so that, with the attributes scaled relative to status quo, the utility associated with status quo (i.e., not adopting) is zero. Under these assumptions, the adoption decision rule for a utility maximizing consumer is'

Et[ ux (.i)f] > kp. (4

The above rule suggests an alternative interpretation of k, consistent with the micro- economic model of choice based on maximization of expected utility subject to a budget constraint: k may be interpreted as the shadow price of the budget constraint (Hauser and Urban 1982). The parameter k may vary across the potential adopter population.

The consumer's uncertain perception of performance may be characterized by a probability distribution, with the mean indicating his expectation of performance and the dispersion measuring the degree of perceptual uncertainty. More specifically, if we assume that xi is normally distributed, with mean mi and variance si, then

E[ux(xi)] = E[1 - exp(-cxi)] = 1 - exp(-cmi + c2si/2). (5)

From (4) and (5), we obtain the condition for adoption. An individual adopts the innovation when (and if) the following condition is met:

3Throughout this paper, we use the tilde sign over the notation to indicate a random variable. Technically, von Neumann-Morgenstem theory posits utility as a function of asset levels, rather than deviations

(gains or losses) from base levels, as postulated here, and as commonly used in marketing. Under the assumption of constant risk aversion, it can be easily shown that the adoption decision rule (4) holds even if utility is expressed as functions of asset levels rather than deviations from status quo.




mi > cs2/2 - (1I/c) In (1 - kp). (6)

This condition has an intuitive interpretation: the consumer adopts the innovation as soon as his expectation of its performance (mi) exceeds the sum of the "risk hurdle" (csb'2) and the "price hurdle"5 [-( 1/c) In (1 - kp)]. (6) corresponds to Roberts and Urban's brand choice condition (equation (6), p. 171 ), with the addition of the "price hurdle" resulting from our explicit inclusion of price as a separate attribute in the utility function.

Updating Perceptions: Information Integration. Note that the condition for adoption (6) is based on current perceptions. To forecast an individual's adoption timing, we must predict the dynamics of perceptions, given initial perception about performance (at the time of product launch) and the nature of the information flow. We conceptualize the potential adopter's information integration process as follows. Over time, a stream of information about the innovation's performance is received (for example, by word- of-mouth from adopters). The potential adopter behaves as if the level of performance conveyed by a unit of information is sampled from a normal distribution with mean A, the true performance of the innovation relative to status quo, which is unknown to the potential adopter, and variance, 2, which is an inverse measure of the perceived (i.e., subjective) reliability of the information.6

Under these assumptions, we can apply Bayes' rule (Winkler and Hays 1975, p. 507) to update the potential adopter's perception of the innovation, xi, as one unit of new information is received, where xi is distributed N(m,, si). The formulas for updating the parameters mi and s2 are:

m_X1/s2_I + Z1/Oa2 2 1 Mi= ____+______and = 1/san + i/ 2' (7) 1/S~ I + 1/oU2 1S I+/2

where zi is the level of performance conveyed by the ith unit of information. With increasing information over time, the expected performance mi tends toward the true performance ,u, while the degree of perceptual uncertainty s2 decreases.

The Bayesian modeling approach to conceptualizing the information integration process has been employed in marketing science (Gatignon 1984; Oren and Schwartz 1988; Roberts and Urban 1988) and consumer behavior (Scott and Yalch 1980). The empirical evidence on the descriptive validity of the Bayesian model is mixed. Viscusi (1985), for example, found that behavior patterns were consistent with the principal predictions of a Bayesian learning process. However, past laboratory studies (Slovic and Lichtenstein 1971) indicate systematic deviations in people's revised judgments from the Bayesian norm-in particular, subjects are conservative in updating their uncertain beliefs, i.e., they revise their judgments in the correct direction but to an insufficient degree. More recent literature (Winkler and Murphy 1973; Navon 1978) suggests such systematic deviations are a result of subjects' real world experience of the nature of information, which is perceived as being less than perfectly reliable and possibly conditionally dependent (i.e., some of the information is already contained in previous information). The issue is that the Bayesian learning model, modified to incorporate subjective perceptions about the reliability of the data generating process, can be employed as a reasonable paramorphic, representation of the information integration process, Specifically, in our model, the "effective" data generating process has a variance that captures the potential adopter's perception about the reliability of the information; the smaller the .subjective variance

5 Note that the expression for the price hurdle is positive. Our formulation requires kp < 1; a hypothetical (and nonpotential) consumer with kp > 1 will never adopt the innovation even if its true performance is arbitrarily large.

6 Operationally, any convenient basis for defining a unit of information may be employed. For example, the first piece of word-of-mouth information may be defined as one unit.




2 (implying higher perceived reliability), the larger the extent of revision in perceptions about performance due to a given piece of information. We also allow the population of potential adopters to be heterogeneous in terms of initial perceptions (mo and s2) and perceived reliability of information (a2).

2.2. Predicting Adoption Time for an Individual Consumer

We can now combine the condition for adoption (6), based on the consumer's preference structure, and the information integration model that predicts the dynamics of perceptions to forecast a particular consumer's time of adoption, given the nature of information flow. Let us assume that the consumer receives a stream of information { zi; i = 1, 2,. . . }, where zi is the level of performance conveyed by the ith unit of information.

For ease of exposition, let us define the following:

yi = [csi2/2 - (ll c) In (I1 - kp) - mi] [ a/s (8 )

=yo = [cs2/2 - (1/c) ln (I - kp) - mo][o 2/s2]; and (9)

0 = -(1/c) In (I - kp). (10)

From (6), it is clear that the condition for adoption may be stated in terms of yi as

Yi <0. (11)

That is, the consumer adopts the innovation when (and if) yi drops below zero. From (7), (8), and ( 10), we obtain

Yi = Yi-i - [zi-$ (12)

Given an information stream { zi }, (12) may be applied recursively, and the condition of adoption can be restated as

(z (i - 0) > a, (13) i=1

where i* represents the "critical" amount of information needed by the consumer for adoption.7

The above mathematical representation of the adoption process has an intuitive interpretation. The composite variable yi (see (8)) is a product of two expressions. The first expression represents the difference between the sum of the risk and price hurdles and expected performance. As long as this difference is positive, the consumer does not adopt. The second expression is an inverse measure of the consumer's responsiveness to information. The consumer adopts when (and if) yi drops below zero, that is, when the expected performance exceeds the sum of the risk and price hurdles. Therefore, yi may be interpreted as a measure of how "far" the consumer is from adoption.

The dynamics of Yi provide a parsimonious representation of the adoption process. Let us examine the impact of the ith unit of information, given by ( 12). The information is favorable if the performance level conveyed (zi) is greater than the price hurdle (,B). Favorable information (zi - A > 0) brings the consumer closer to adoption (Yi < Yi- I), while unfavorable information has the opposite effect.

Note that the condition for adoption given by ( 13) is based on a given information stream { zi }. However, our objective is to predict the timing of adoption prior to product

7Our equation ( 12), like Roberts and Urban's brand choice model (equation ( 17), p. 174), is a one-period- ahead Bayesian forecast. Whereas their focus is on brand choice, our is on time to adoption, based on the "critical" amount of information. Our subsequent development uses ( 12) in a stochastic framework to capture information variability. Roberts and Urban's probabilistic choice model, on the other hand, assumes an exogenous random error term that yields their logit formulation.




launch. The values of { zi } are therefore unknown when the prediction is made. We assume that the stream of information generated about the innovation's performance will be unbiased (i.e., E[zi] = ,t, the true performance) but variable, in that the level of performance conveyed by different pieces of information will be distributed about the mean. More formally, we model the zi's as independent draws from a normal distribution with mean A and variance 6 2. The variance 6 2 captures the variability in the information about performance. The distribution N(,u, 62) thus represents the modeler's (or model user's) assessment of the true distribution underlying the zi's. Note the distinction between the distribution N(,u, 6 2) of the information stream and the "effective" data generating process, distributed N(,t, a2). The variance U2 reflects the reliability of information as perceived by the potential adopter, and determines the extent to which information influences his perception.

The dynamics of the consumer's adoption process, parsimoniously represented by yi, can now be conceptualized as a stochastic process. Rearranging (12) and incorporating the distribution of 2i, we obtain

A -

A-1 = -(i- ) -N(-(,u -

0), 62) (14)

with yo = a defined by (9). The stochastic process represented by ( 14) and (9) corresponds to a general random walk (Cox and Miller 1965), which tracks the uncertain prediction (by the modeler) of the path followed by a consumer toward possible adoption. The uncertainty arises from the stochasticity in the information generation process.

In order to formulate our diffusion model in continuous time (similar to the aggregate level models, with which we compare our model in the next section), we need to express the dynamics of the adoption process over time rather than in terms of (discrete) units of information. With this in mind, we first consider the process in terms of a continuous information stream, instead of discrete units. In a continuous framework, i becomes a continuous variable representing the cumulative amount of information. We use the notation y( i) in place of yi to denote the continuous process. The stochastic process y( i) can be conceptualized as a Wiener process with drift parameter - [u- ] and variance parameter 6 2 starting at y(0) = a. Analogous to the discrete case, the consumer adopts the innovation when (and if) y( i) drops below zero for the first time. Figure 1 graphically illustrates the dynamics of the adoption process represented by the dynamics of y(i) for two hypothetical consumers. Consumer 1, for whom It > 3 (i.e., the true performance exceeds his price hurdle) drifts toward adoption, while Consumer 2 (It < 3) drifts away from adoption. This figure also illustrates the intuitive interpretation of y(i) as a measure of how "far" the consumer is from adoption.

Consumers with a-values less than zero adopt the innovation immediately after product launch. For a consumer characterized by the pair of parameters a (?0) and 3, the "critical" (cumulative) amount of information required for adoption is a random variable, denoted by F* I a, 3, having the inverse Gaussian distribution with probability density function (Cox and Miller 1965, pp. 220-222)

g(ila, d) = ^ exp [ (a 62i ) )i] (15)

For a consumer with a 2 0 and : < ,u (for example, Consumer 1 in Figure 1), the mean and variance of the distribution are given by:

E[i*Ia, f] a Var a, ]= (16)

In case a 2 0 and 3 2 u (illustrated by Consumer 2 in Figure 1)' ( 15) is not a proper density function, i.e., fj0 g(i)di < 1, and hence the moments of the distribution are not




y(i)

a,c

\Consumer 2

2(1 > A)

a2 A \aConConsumer 1

(i < 0

FIGURE 1. Stochastic Process Representation of the Adoption Process for Two Hypothetical Consumers.

defined. In substantive terms, this means that the probability of eventual adoption (as i so ) is less than 1. Indeed, a consumer with a > 0 and : > , tends to drift away from adoption, and the nonzero probability of eventual adoption is a consequence of the variability of information.

Table 1 summarizes the implications for adoption behavior for the three "types" of consumers in the potential adopter population. Type I consumers (a < 0) will adopt the innovation as soon as it is launched. Type II consumers (a > 0,5 < u) will eventually adopt the innovation (on receiving the required amount of information), while for Type III consumers (a ? 0, 3 ? ,) eventual adoption is uncertain.

So far, our discussion has focussed on the adoption process in terms of cumulative information rather than time. Let n(t) denote the rate of information at time t, where t - 0 is the point of product launch.8 Therefore, the cumulative amount of information received by the consumer by time t is

i(t)= J'n(T)dt. (17)

The critical time to adoption is distributed according to the density function n(t)g(i(t) I a, p). The probability of adoption by time t for the consumer with a > 0 is (Cox and Miller 1965)

TABLE 1

Summary of Adoption Behavior for Different Consumer "Types"

Quantity of Information Needed for Adoption

Consumer Characteristics Probability of Mean Variance "Type" (a, X) Eventual Adoption E(i* I a, ) Var (i* I a, ,B) Remarks

a < 0 1 0 0 These consumers will adopt the innovation as soon as it is launched.

II. c2 ,,Ba> 05<0 1 , a a

III. a> 2 O a 2 x [ 2 i*Il, [a does not have finite moments

8 The model focuses on information generated subsequent to product launch. Prelaunch information will

affect initial perceptions of potential adopters; this effect is captured by the av values.




( - ) 2[ 6 ]1 ( a

where 4( ) is the standard normal (cumulative) distribution function.

2.3. Aggregation: The Diffusion Curve

The diffusion curve is obtained by aggregating the predicted individual adoption behavior over the population. Analytically, if we assume that the population is sufficiently large so that a and ,B may be treated as continuous and the joint distribution of a and ,B across the population can be represented by the density functionf(., *), the cumulative penetration curve is

A(t) = Fa(O) + J fP(t)Ia, t3)f(a, f)ddad3 where (19)

FY(O) = J ff(a, f)dadf (20) o 00

is the fraction of the population that adopts the innovation immediately when it is launched (the Type I customers in Table 1). The rate of penetration is

A(t) = = n(t) f f g(i(t) Ia, j3))f(a, j3)dad3. (21)

Equations (19) -(21) represent a rigorous analytical statement of our diffusion model, based on stochastic individual adoption times. In the next section, we derive a deterministically approximated aggregate model that is more amenable to analytical manipulation. Operationalization of the model would require a prelaunch survey of a repre- sentative sample of potential adopters to measure the individual level characteristics determining the parameters a and ,B, as well as an assessment of the rate and nature of information over time. Procedures for estimating the individual characteristics are illustrated in ?4.

3. Analytical Implications of the Model

In this section, we discuss some key properties and analytical implications of the model. We compare and contrast our micromodeling approach with the aggregate-level diffusion models, e.g. Bass (1969), to illustrate the benefits of the micromodeling approach in terms of conceptualizing and interpreting the diffusion process.

Aggregate-level models are parsimonious, but their underlying theory, drawn from mathematical models of contagion, is simple since they do not consider the determinants of the adoption decision by individuals. In contrast, our micromodeling approach focuses on these determinants at the disaggregate level by considering initial perceptions, key determinants of preference (risk attitude and price sensitivity), and responsiveness to information about the innovation. The decision analytic framework provides a paramorphic description of the adoption process at the individual level. The perspective that consumers behave as though they follow some normative decision rule underlies much of preference and choice modeling in marketing (Corstjens and Gautschi 1983; Hagerty and Aaker 1984; Hauser and Urban 1977, 1979; Green and Srinivasan 1978; Shocker and Srinivasan 1979).

3.1. A Parsimonious Measure of Innovativeness

In Table 1, we identified three types of consumers: Type I, who will adopt the innovation immediately after launch, Type II, who will eventually adopt the innovation, given the




required amount of information, and Type III, who tend to drift away from adoption with additional information. These three consumer types form natural segments. The segment of interest in terms of predicting the dynamics of adoption over time comprises the Type II consumers. For a high-involvement product or service for which performance information will be available largely after launch (when the performance can be observed), and the target market has been carefully selected (given the price), we would expect this segment to be large relative to the other two.

For Type II consumers, the expected "critical" amount of information required for adoption is al/(Au - ,B) (see ( 16)). For ease of exposition, let us define

a T=Iy -d- (22)

,U-f

The y-value can be interpreted as a composite measure of innovativeness as defined by Rogers (1983): "Innovativeness [is] the degree to which an individual or other unit of adoption is relatively earlier in adopting new ideas than other members of a social system" (p. 245). Mathematically, the composite y captures the combined effect of the various consumer and innovation characteristics that influence the relative expected timing of adoption, under our modeling assumptions. We can "decompose" 'y to verify that the various consumer characteristics influence the expected timing of adoption (holding other characteristics constant) as suggested by behavioral diffusion theory and intuition: thus, lower risk aversion, higher initial expectation of performance, greater confidence in initial beliefs (given favorable initial perception), a lower price hurdle, or greater perceived reliability of information implies a lower y-value and thus earlier expected adoption.

Note that y is also a function of the innovation's performance (,u) and price (p). For a particular individual, a higher performance or lower price lowers 'y and thus implies earlier expected adoption. Thus, y is an innovation specific measure of consumer innovativeness. Given heterogeneity in the population with respect to the price/performance tradeoff, the impact of , and p on the relative expected adoption time varies across consumers. Thus, the ordering of consumers in terms of relative expected adoption times is influenced by the values of , and p. For example, a lower price level decreases 'y for all consumers, but the extent of decrease is greater for more price sensitive consumers.

The micromodeling approach potentially provides a formal basis for segmenting the target population, at two levels. First, the three consumer types define natural segments. Second, Type II consumers, the delayed adopters forming the most important segment, can be further segmented on the basis of their y-values that predict the expected amount of information required by each of them to adopt. A prelaunch consumer survey can provide the necessary individual level measures to estimate y for each respondent, and also collect background information (demographics, socioeconomic characteristics, media habits, etc.) to serve as actionable segment descriptors. An illustrative example is provided in the next section.

3.2. The Pattern of Diffusion

We now examine the implications of our model for the aggregate pattern of diffusion. The rate of diffusion and the shape of the diffusion curve are of theoretical and managerial interest. The shape of the diffusion curve can be characterized by convexity or concavity properties, and, in particular, by the number and location(s) of the point(s) of inflection. In the traditional S-shaped diffusion curve, the single inflection point indicates that sales, having peaked, will decline thereafter. Strategically, this may suggest when to introduce an improved version of the product to extend its life and sustain sales and profits (see, for example, Wind 1982, pp. 57-65). In situations where the diffusion curve exhibits




multiple inflection points, indicating fluctuations in sales over time, managers may consider strategies (for example, promotions) to smooth the temporal pattern of sales. From a theoretical (modeling) perspective, the model's ability to accommodate different patterns of diffusion is an important issue (Mahajan and Wind 1986).

A Deterministically Approximated Diffusion Model. For analytical purposes, the diffusion model given by ( 19)- (21) is cumbersome. Our aim here is to find a reasonable approximation of this model that is more amenable to analytical manipulation. Our approach involves the following deterministic approximation: instead of considering the probability distribution of the critical amount of information "* I a, ,B, our aggregation across individuals is based on the mean of the distribution of * I a, ,B. This is equivalent to ignoring the variability of information (i.e., 6 = 0) and treating individual adoption times as deterministic. Support for this approximation is provided by numerical simulation based on ( 19)-(2 1), which shows that in situations where the population is "sufficiently" heterogeneous (in terms of a and 3) and the fraction of Type II customers is large relative to Type III (i.e., situations we would expect in practice, with a well-defined target pop- ulaiion), the cumulative penetration and penetration rate curves are relatively insensitive to the value of 6, as long as the ratio 1/,u is small. See Figure 2 for an illustration of the impact of 6 on the diffusion curve. For the specific parameters selected for this illustration, the deterministically approximated curves (( = 0) are almost identical to the curves for 6 = 25 (( /,u = 0.25), while deviations are small for (/,u less than 1. These results suggest that aggregation based on the assumption of deterministic individual adoption times may

(a) CUMULATIVE PENETRATION (b) PENETRATION RATE

1. o - o ~~~~~~~~~~~~. 30o

A 0.5 A( 0l

2.2

TIM TIM

Curve S 84t

1 500 5.0

2 100 1.0

3 25&0 0.25&0

FIGURE 2. The Impact of Information Variability on the Diffusion Curve. *

* Based on an illustrative simulation with the following parametric values: ( 1) av, ,B have joint bivariate normal distribution with At = 1000, ,B = 75, i. = 500, a, = 25, P = 0.4; (2) , = 100; (3) n(t) = 5 + lOA(t).




serve as a reasonable approximation. Clearly, the smaller the variation in information about performance, the better the approximation.

The implication of the deterministic approximation is that Type I consumers adopt immediately (as before), Type II consumers adopt on receiving the critical amount of information

a i*la, = (23)

and Type III consumers do not adopt at all. (Note, from Table 1, that the probability of eventual adoption for Type III consumers tends to zero as 6 -* 0.)

Let {I, {I1 and {III denote the fraction of the population belonging to the Type I, Type II and Type III consumer segments, respectively. Further, let the distribution of y across Type II consumers be represented by the density function f*() and the cumulative distribution function F,( ). The cumulative penetration is then

A(t) = {I + I11Fy(i(t)) (24)

and the penetration rate is

A(t) = {jjn(t)fy(i(t)), t > 0. (25)

Equations (24) and (25) define our diffusion model based on the deterministic approximation. Equation (25) shows that the rate of penetration at any point in time is determined by the interaction between the rate of information and the concentration of consumers who are "ready" to adopt, captured by the density f(i(t)). Our micromodeling approach enables us to identify these customers. (In the stochastic model, these customers are identified probabilistically.) Equation (25) also implies that the rate of diffusion at a given level of penetration will be greater when (a) the rate of information is higher and (b) the population is more homogeneous in terms of -y.

Shape of the Diffusion Curve. As Mahajan and Wind (1986) point out, the Bass model and some other popular aggregate diffusion models are structurally limited in terms of flexibility. They suggest further work in the area of "basic diffusion models . that are flexible and can accommodate various diffusion patterns and are based on clearly explicated behavioral assumptions" (p. 23). We believe our micromodeling approach meets these criteria.

The determinants of the shape of the diffusion curve are the distribution of y and the rate of information. The cumulative penetration in the Type II consumer segment as a function of cumulative information is the distribution of y in the segment, since y predicts the cumulative amount of information required for adoption. A time-varying information rate modifies the shape of the penetration curve when time replaces cumulative information as the argument by "stretching" (low information rate) or "compressing" (high information rate) the curve.

For purposes of illustrating the implications of the model for the shape and, in particular, the number and location(s) of inflection point(s), we assume that - has a unimodal distribution (so that the cumulative distribution is S-shaped with a single inflection point, though not necessarily symmetric) and consider three patterns of information flows over time-constant, monotonically decreasing, and monotonically increasing. We define t* as the time by which all Type II consumers with y less than the modal value of the distribution have adopted.

Property. If y has a unimodal distribution over Type II customers then the implications of the pattern of information over time are as follows:




Points of Inflection Shape of Cumulative

Rate of Information Number Location Penetration Curve

(a) Constant 1 t* Same as c.d.f. of y

(b) Monotonically decreasing 0, 1, or more 0 < t < t* Concave for 0 c t < t* if no (if any) inflection point; always

concave for t* < t < co

(c) Monotonically increasing 1 or more (but t* < t < oo Always convex for 0 c t c t* odd number)

PROOF. See Appendix.

3.3. Relationship to Aggregate Models

Given the flexibility of our model, an interesting question is: Under what conditions does the model reproduce the diffusion patterns of some well-known aggregate level models? We consider four popular models: Bass (1969), Fourt and Woodlock (1960), Mansfield ( 1961), and the Gompertz curve (Martino 1975). First, under the scenario of a constant information rate, n(t) = n, we investigate the conditions (specifically, the segment sizes {I and {I1 and the distribution of y in Segment II) under which the cumulative penetration curve implied by our model (24) is identical to each of these aggregate models, and identify the relationship between the parameters of the aggregate models and those of our model. The results of this investigation are presented in Table 2. For example, we can conclude that for a population comprising Type II consumers only, with y having a logistic distribution across the population with scale parameter a and shape parameter b, and a constant information rate n, the implied cumulative penetration curve is identical to the Bass model. The parameter p of the Bass model is directly proportional to the scale parameter a and information rate n but inversely related to the shape parameter b. The parameter q of the Bass model is increasing in all three micromodel parameters a, b, and n.

Another interesting comparison between the aggregate level models and the micromodeling approach is illustrated by contrasting the differential equation specifying the Bass model (see Table 2) with our penetration rate equation (25). The first term of the Bass model [p + qA ] represents the "pressure" from external and internal influences to adopt, and is conceptually similar to the information rate n(t) in our model. The Bass model, however, treats the population as homogeneous and thus all those yet to adopt are equally likely to adopt: hence the term [1 - A (t)] in contrast to the density fl( i( t)) in our model, which captures only those consumers who are "ready" to adopt.

The diffusion patterns implied by various aggregate level models can be reproduced by our model for patterns of information other than the constant information rate case considered above. In particular, the correspondence between the "pressure to adopt" term in the Bass model and the information rate term in our model suggests the following information rate:

n(t) = n1 + n2A(t), (26)

i.e., the information rate consists of a fixed component (from external sources, e.g., advertising) and a word-of-mouth component increasing linearly in the cumulative penetration. Under this time varying information rate, our model exactly reproduces the Bass model if y has the exponential distribution:

F,(y Ia) = - exp(-ay), (27)

with i/' = 0, {1 = 1, and the following relationships exist between the parameters: p = an,




TABLE 2

Coniditions Under Which Micromodeling Approach Reproduces Aggregate Dffilsion Models Under Assumption of Constant Information Rate

Our Model: A(t) = V1 + 4it'Fy (i(t)); i(t) = nt

Distribution (c.d.f.) of y Segment Sizes and Differential Equation for across Segment II Relationships among

Penetration Rate Cumulative Penetration (0 c y < co) Parameters

Model: Bass (1969)

dA/dt = (p + qA)(I - A) A(tIp, q) = 1 - e(p+&)t Logistic: p 0, Ian/ 1 A(O) = 0 + (q/p)e-(p+q)t p=all+b

F,(y I a, b) = I _

b e q = abn/(l + b)

Model: Fourt and Woodlock (1960)

dA/dt = p(l - A) A(tIp) = 1 -e-P' Exponential: -=0,

ll = 1 A(O) = 0 F( I a) = 1-e p =an

Model: Mansfield (1961)

dA/dt = qA(1 - A) A(tj q AO) I Logistic: 4,1 1 /(I + b), 4'11=b/(1 A(O) = AO, 0 < Ao < I ( - q = an

1 + Ieqt FCya b ay AO = 1/(1 + b) 110 / 1~~~~ + be-ay

Model: Gompertz (Martino 1975)

dA/dt = qA ln(l/A) A(tIq, Ao) = (I/AO)-e Q Truncated extreme value: t'1 = Fo, 11 = 1- Fo A(O) = AO, 0 < Ao < 1 I nb) / nb

- FOlk - F O0] F [I _ [-(y-a"lb-]

where Fo = C eal

and q = an2. Note that the Fourt and Woodlock model in Table 2 is a special case (with n2 = q = 0). Similarly, the Mansfield model is reproduced, with n, = p = 0 and, addi- tionally, j1 = Ao, {I1 = 1 - Ao, 0 < Ao < 1. More generally, for any aggregate level model that posits the penetration rate as a product of a "pressure to adopt" term and the fraction of the population yet to adopt [1 - A (t) ], our model will reproduce the diffusion curve if ( 1 ) the information rate is proportional to the pressure to adopt and (2) the distribution of y is exponential, given by ( 27 ). For example, Easingwood et al.'s ( 1983 ) NUI diffusion model is reproduced if the information rate n (t) = n, + n2 [A (t)]d, where d corresponds to Easingwood et al.'s nonuniform influence factor. Other conditions are {I = 0, i11 = 1, p = an1, and q = an2 (i.e., same as for the Bass model).

The above discussion illustrates special cases of our general model that provide diffusion patterns corresponding to the Bass and other aggregate models. It should be noted that other combinations of y-distributions and information patterns can also lead to the same diffusion patterns. The key aspect is that the micromodeling approach provides a rigorous theoretical basis for explaining a wide variety of diffusion patterns under various scenarios.

4. A Pilot Study

This section describes a pilot study conducted in an experimental setting. This study outlines procedures for surveying subjects and estimating the individual level and




information specific parameters of the model from the data collected. The results provide a preliminary test of the predictive performance of the model at the individual level (adoption times) and at the aggregate level (the number of new adopters in each time period).

4.1. Description of the Study

Sixty-five Wharton students participated in the study, which used an interactive com- puter program that provides career counseling as the innovation. Subjects responded to a paper-and-pencil questionnaire. The session, including oral briefing and debriefing, lasted about one hour. The product, its price, and an explanation of the performance rating scale were described, then data were collected to estimate the parameters of the utility function for each respondent. Next, respondents recorded their initial perceptions of performance and also indicated whether they would adopt the innovation at the indicated price, based on the product description.

Respondents then received three pieces of information, with each piece presented as a subjective evaluation of the product by an "adopter" (that is, the information simulated word-of-mouth from previous adopters). After each piece, respondents recorded their perceptions and their adoption intentions. Each questionnaire section focusing on dynamics of perceptions (and adoption intentions) was administered separately.

4.2. Measurement and Estimation9

Preference Parameters. For each respondent, the parameters (c and k) of the utility function (3) are estimated as follows. Estimation of the risk aversion parameter, c, is based on a procedure proposed by Eliashberg and Hauser ( 1985 ). The respondent received a set of 10 binary choice problems, to choose categorically the preferred alternative. The two alternatives in each choice problem are simple lotteries, with probability 7rj of obtaining performance level xj and ( 1 - 7rj) of obtaining some base performance level x0 for alternative j (j = 1, 2). The values of 7r1, 7r2, x1 and x2 are varied across the choice problems and are selected so that

7r2exp(-c*x2) - 7 exp(-c*xl)= =2 - 1, (28)

where c* is a positive constant (equal for all choice problems) and r2 > 7r,. Under the assumed form of the utility function, the estimate of the risk aversion parameter is given by (Eliashberg and Hauser 1985, Proposition 4)

c = -c*/ [ln (I -N/ 10)], (29)

where N is the number of times the first alternative is chosen over the second (over the ten choice problems).

To estimate the price importance parameter k, two distinct sets of questions were used. The first set asked for the maximum price respondents would be willing to pay for a product with a specified performance level (varied over four questions). In the second set (comprising two questions) the respondent compared two alternatives, with performance levels xl and x2 and prices Pi and P2 respectively. Three of the values are specified, and the respondent provides the missing attribute level (performance or price) such that he is indifferent between the two alternatives.

The individual-level data collected are used to estimate k for each respondent as follows: An OLS estimate of k, say k( 1), is obtained, based on the responses to the first set of questions. Similarly, an OLS estimate k( 2) is obtained from the responses to the second set. Next, we derive estimates of the variances for the error terms in the linear models

9 For a more detailed description of the measurement and estimation procedures used in the pilot study, see Chatterjee ( 1986).




employed to obtain k( 1) and k(2), pooling across respondents. Using estimates of the two error variances, a single weighted least squares estimate k is obtained for each individual.

Perception Parameters. Our model assumes that each respondent's uncertain perceptions of performance may be represented by a normal distribution, with mean mi and variance s2, where i indicates the number of pieces of information received.

There is a vast body of literature on encoding, or quantifying, subjective probabilities (for a recent review, see Wallsten and Budescu 1984). The literature suggests that an encoding method using odds (relative likelihoods) may be operationally more appropriate in marketing studies, when responses are self-elicited. Woodruff ( 1972) has suggested an approach based on this method, which has been used in marketing studies to quantify uncertain perceptions (Pras and Summers 1978; Gatignon 1984). We used the Woodruff scale (with minor modifications) in our study. Respondents recorded their uncertain perceptions of product performance after the product description but before receiving any of the pieces of simulated word-of-mouth information (i = 0) and again after receiving the ith piece of information (i = 1, 2, 3).

The scale is used as follows: the respondent (i) marks the performance level he considers most likely on the 0-100 scale, (ii) marks the highest and lowest performance levels expected, (iii) assigns 100 likelihood points to the most likely performance level, and (iv) assigns likelihood points to the lowest and highest performance levels (relative to the 100 points assigned to the most likely level) and then to all round numbered ratings in the range between the lowest and highest levels identified in (ii). For each respondent, the parameters mi and s2 are then estimated by

21 21

M= Lixl/ Li,, and (30) 1=1 1-1

21 21

s i Lil(x,- / m i )I Li, (31) 1=1 1=1

where Li, is the relative likelihood score for the Ith point of 21-point performance scale (which runs from 0 to 100 in steps of 5) after i pieces of information and x, is the performance rating corresponding to the lth point of the scale [x, = 5 (1 - 1 )].

Information Parameters and Perceived Source Reliability. The characteristics of the ith piece of information (i = 1, 2, 3) can be summarized by two parameters: ni, the number of units of information and zi, the product performance conveyed by the ith piece of information. (Note that i here subscripts a piece of information in terms of its order of receipt, rather than denoting the cumulative units of information received by the respondent. Thus we need to consider the number of units of information contained in each piece, ni.) The impact of ni units of information on the perceptions of respondent j, based on our Bayesian learning model (see equation (7)), is given by

1 _s2J- 1/S(2 = nil)n/aJ, and (32)

M i/SJ - M(_1)j/S(2_1)J = nizi/l . (33)

The left-hand sides of (32) and (33) are known (from the estimates given by (30) and (3 1)). Let aij= /12-ls _)j5 bij =mij/s -m _)jsI_)j ,and rj= / a?. Equations (32) and (33) can then be written compactly as (including an error term):

a = nir, + ei1, (34)

b1j = nizir, + t. (35)

Note that ni and rj cannot be estimated independently. In substantive terms, the unit




of information quantity must be defined. We arbitrarily fix n1 = 1 (that is, the first piece of information is assumed to correspond to one unit of information). Then ni and the product nizi are estimated by

ni= aij/ a1j, i = 2, 3; and (36) j j

nizi = E bij/ a1j, i = 1, 2, 3. (37) I i

Given these estimates, the perceived source reliability rj is estimated for respondent j as follows: An OLS estimate, f ( 1), is obtained from (34), which provides three individual level data points (for i = 1, 2, 3). A similar OLS estimate Aj(2) is obtained from (35), again providing three data points. Next, we estimate the variance of the error terms in (34) and (35) by pooling across respondents. Using the estimates of the two error variances, a single weighted least squares estimate rj is obtained utilizing all six individual- level data points.

Estimates of the composite individual level parameters, a and f, are obtained for each respondent from (9) and ( 10), using the estimates for c, k, nmo, s2 and a2.

4.3. Predictive Performance of the Model

Our basis for evaluating the model is a comparison of the predicted adoption timing with that implied by the respondents' stated adoption intentions. While intentions may not capture behavior perfectly, tracking actual behavior was beyond the scope of this pilot study. Predictive tests of choice behavior models in experimental settings commonly employ intentions as surrogates for behavior (Andreasan and Belk 1980; Ryan and Bon- field 1975; Wilton and Pessemier 1981).

Our individual level adoption model is founded on two basic premises-maximization of expected utility and Bayesian updating of uncertain perceptions. The first premise implies (under the assumptions of the model) that an individual adopts as soon as condition (6), i.e., mi > cs2/2 - (1/c) ln (1 - kp), is met. This condition predicts the adoption decision given current perceptions (mi, s2). The second premise (Bayesian learning) gives the model its long-term forecasting capability by predicting the dynamics of perceptions, given initial perceptions and the nature of information. Incorporating the Bayesian learning model (with ni units of information in the ith piece) in (6) yields the adoption condition

z ni(zi - A) > a, (38) i=l1

corresponding to condition ( 13) derived in ?2. (Note that i* in (38) denotes the number of pieces of information needed by the respondent for adoption, with ni units of information in the ith piece, whereas i* in ( 1 3 ) represents the cumulative units of information required for adoption.) Our pilot study permits tests of both conditions, (6) and (38). This provides a two-step evaluation of the model, by considering first the adoption decision component (based on utility maximization) and then the complete adoption timing model (the decision model plus the Bayesian learning model). For ease of discussion, we refer to conditions (6) and (38) as UMAX and UMAX + BAYES, respectively.

We first compare the predictions of UMAX and UMAX + BAYES with the stated timing of adoption at the aggregate level. We aggregate by counting the number of respondents "adopting"in each period (based on intentions or predictions by the two conditions) . Table 3 presents the results. Respondents must belong to one of five categories based on adoption timing: immediate adopters (i = 0), adopters at i = 1, 2, or 3, and




TABLE 3

Stated and Predicted Frequencies of Respondents Categorized by Timing ofAdoption

Number of cases: 49 Adoption after j Pieces of Information No Adoption

j= 0 1 2 3 after 3 Pieces Category: (1) (2) (3) (4) (5)

Stated 12 8 3 2 24 Predicted by:

(a) UMAX 13 5 2 2 27 (b) UMAX + BAYES 13 4 3 1 28

Chi-square statistic*: (a) UMAX 2 460 (p = 0.48) (b) UMAX + BAYES 4.898 (p = 0.18)

* The statistic was computed after collapsing categories (3) and (4). The p-values reported are thus the chi- square upper tail probabilities for 3 degrees of freedom.

nonadopters after the third piece of information. The p-values for the chi-square goodness- of-fit statistic (0.48 and 0.18 for the adoption decision and adoptive timing models, respectively) indicate that the evidence (stated adoption intentions) is not inconsistent with the models predictions at a reasonable level of significance.10

At the individual level, UMAX and UMAX + BAYES correctly predict adoption timing for 48.98% and 42.86% of the cases, respectively. As in the aggregate case, there are no competing theory-based models for comparative assessment of the predictive accuracy of the model. However, some random models may be considered as benchmarks. A naive model would assign each respondent a 1 / 5 probability of belonging to any of the five categories, and would correctly predict adoption timing in 20 percent of the cases, on average. The predictive performance of both conditions is significantly better than that of this naive model (p < 0.001 for both models).

More stringently, we may employ a less naive model based on a random assignment of individuals according to the aggregate-level predictions (i.e., with probabilities proportional to the category frequencies). This test examines whether individual-level predictions provide significantly more information than the aggregate-level prediction of the same model. On average, assignment on this random basis correctly predicts individual adoption timing in 36.28% of the cases, for both UMAX and UMAX + BAYES. For UMAX, the probability of obtaining from the "null" model a percentage of correct predictions of individual adoption timing greater than that actually obtained (48.98% ) is 0.024. Thus, the predictive performance at the individual level is significantly better (at the 0.025 level) than the "null" model. For UMAX + BAYES, the corresponding p-value is 0.179. The individual level predictive performance exceeds the "null" model (42.86% correct predictions vs. 36.28% on average for the "null" model), but the level of significance (0.20) is modest.

4.4. Discussion

The pilot study outlines procedures for data collection and parameter estimation. In evaluating the model, we examined the basic adoption decision condition based on utility maximization (UMAX) and the complete model incorporating the Bayesian learning

'0 16 cases were dropped from the analysis because their risk parameter c could not be estimated from equation (29). Specifically, N = 10 for 3 respondents and N = 0 for 13 respondepns, corresponding to c^ = oo and c^ = 0 respectively.




model (UMAX + BAYES). Both models appear to predict the timing of adoption at the aggregate level reasonably well. At the individual level, UMAX significantly outper- forms the "null" model. The empirical evidence is statistically less conclusive for UMAX + BAYES.

Note that we test the model and the measurement procedure, since both model mis- specification and measurement errors are at issue here. Further, the model's predictions are generated entirely from the measures of individual characteristics obtained via the survey, not on the basis of parameters estimated from fitting the model to given obser- vations of the dependent variable. The "dependent" variable in our study (adoption timing) is used only for evaluating the predictions obtained.

A limitation of the pilot study is that adoption "timing" is based on stated intentions rather than actual behavior. Further, the study asks the adoption intention question after each piece of information, increasing errors and possible bias due to demand and testing effects. These errors have no impact on model calibration; however, they will adversely affect the model performance evaluation measures. Viewed in this perspective, we believe the results of our pilot study encourage larger scale applications to validate the model. Note that in our pilot study, the information stream was "controlled" by the experimenter. In a full scale field application, the nature and extent of information over time must be either assumed or predicted in order to generate a long-term forecast.

A measurement issue relating to risk attitude is that 16 respondents had to be omitted because of "extreme" values (0 or oo) obtained for their risk coefficients (see Footnote 10). An increase in the number of binary questions (from 10 to possibly 20) would increase the sensitivity of the scale, while a higher value of c* would increase its range (see (28) and (29)), thereby reducing the number of cases with indeterminate risk coefficients. It is possible that some subjects may not be risk averse; operationally, for forecasting purposes, such cases can be handled separately using a modified model with a linear (risk neutral) utility function. Our study is the first empirical application of the Eliashberg and Hauser (1985) approach to estimating consumers' risk coefficients; we believe that this approach holds considerable promise in consumer survey settings.

Gamma-values for Type II consumers. For illustrative purposes, we further analyzed -y-values for Type II consumers. 30 subjects (out of 49) fell in this segment (of the rest, 7 were Type I and 12 Type III). The rank order correlation between the y-values and the stated timing of adoption for these 30 subjects was modest in terms of statistical significance (Spearman correlation coefficient = 0.21; p-value = 0.13 for one-tailed test). These y-values were next correlated with 14 background measures (media habits, cred- ibility of information sources) collected during the study. Interestingly, all four items on the importance of advertising (external information source) had significant negative correlations with y, while other correlations were insignificant. Specifically, subjects with low y (i.e., those expected to adopt earlier) tended to agree more strongly with the following statements (where responses were on 5-point "strongly disagree-strongly agree" scales):

Spearman Statement Correlation (p-value)

1. Commercials on TV provide useful information about the product they advertise -0.311 (0.094)

2. Magazine advertisements provide useful information about the products they advertise -0.537 (0.003)

3. I usually buy an expensive product on the basis of ads and information from such sources as Consumer Reports -0.432 (0.017)

4. Ads can provide a lot of useful information about products -0.430 (0.018)




These results are consistent with the literature on innovativeness suggesting that early adopters rely more on external sources of information such as advertising (Rogers 1983). While this small-scale analysis is illustrative, we find these results encouraging in the context of our claim that y-values can potentially provide a meaningful basis for segmentation.

An analysis of the sample distribution of the y-values suggested an exponential pattern. The exponential distribution, F,( y I a) = 1 - exp(-ay), with a = 0.0156 fits reasonably (chi-square statistic for goodness-of-fit = 5.66, with 4 d.f.; p-value = 0.23). As discussed in ?3.3, if the rate of information increases linearly in the cumulative adopters (word- of-mouth effect), the exponential distribution of y implies that the (deterministically approximated) diffusion curve will follow the logistic (S-shaped) pattern.

5. Conclusions

In this paper, we have taken an individual level decision analytic approach to examine the pattern of diffusion of an innovation. The approach considers the dynamics of consumers' perceptions and their impact on expected utility and thereby on the timing of adoption. An important focus of our endeavor has been the explicit and rigorous consideration of heterogeneity in the population along various dimensions that influence adoption at the individual level, in a parsimonious stochastic process framework. Spe- cifically, the individual-level determinants of adoption timing considered in our model are initial perceptions of the performance of the innovation (both expectation of performance and the degree of perceptual uncertainty), key determinants of preference (degree of risk aversion and price sensitivity), and responsiveness to information about the innovation. These determinants can be measured at the individual level via a consumer survey prior to launch, as illustrated by the pilot study. The diffusion curve is obtained by aggregating individual-level behavior (in terms of adoption timing) across the potential adopter population.

The individual-level characteristics, combined with the true performance and price of the innovation, can be parsimoniously described by two composite parameters, a and a

(defined by (9) and (10)). The parameter a may be interpreted as a measure of how "far" the consumer is from adoption prior to product launch, while : measures the price hurdle and is conceptually equivalent to the consumer's reservation price for the product under full information, measured in the same units as performance. (If performance is evaluated in dollar terms, 3 is the reservation price in dollars under full information.) Consumers with a below zero will adopt when the innovation becomes available; consumers with positive a will eventually adopt if their price hurdle a is less than the innovation's true performance level ,u; on the other hand, consumers with a greater than ,u will drift away from adoption as they receive more information. These three consumer types form natural segments. For a high involvement innovation and with a properly defined target market, Type II consumers (with a 2 0,5 < tu) constitute the major segment of interest. These consumers will adopt once they receive the "critical" amount of information predicted by the ratio al/(Au - 3), denoted by -y. The y-value can be interpreted as a composite measure of innovativeness as defined by Rogers ( 1983 ). This composite variable is a theoretically appealing basis for segmenting Type II consumers. Such a segmentation may be used by the manager to target the firm's marketing efforts, particularly advertising.

The micromodeling framework model is flexible and, given its conceptualization of the adoption process, provides a behavioral basis for explaining a variety of diffusion patterns, as demonstrated for some popular aggregate level diffusion models. In essence, the pattern of diffusion is determined by the heterogeneity of the population with respect to y and the characteristics of the information generated about the innovation. The




dynamics of the rate of penetration over time, for example, are governed by the interaction between the rate of information and the concentration of consumers who are "ready" to adopt at that point in time, captured by the density function. This is in contrast to aggregate diffusion models that assume that the population is homogeneous and therefore consider the entire fraction of the population that has not yet adopted.

Our modeling focus has been on a rigorous consideration of the individual characteristics determining adoption. While the innovation's price and performance are included in the model, they are assumed constant over time. Dynamic pricing and product performance can conceptually be accommodated if discrete changes are considered by em- ploying a sequential approach in which the model is applied to successive time periods between changes in price and/or performance (see Chatterjee 1986 for an illustrative example). A normative extension of our work, deriving implications for marketing policies, is left for future research. We have considered a monopolistic setting, which may be reasonable when limited time horizons are considered, particularly if the innovation enjoys patent protection. However, extending the model to consider competing brands in an innovative product category is a promising direction for future research.

While the primary focus of our exposition has been analytical, we believe that the illustrative pilot study encourages a large scale empirical application. A full scale field application would entail (1) identification of the potential adopter population, (2) sample selection, (3) estimation of the individual-level parameters, (4) assessment of the levels of performance conveyed by the information stream and (5) an estimate of the rate of information over time. The level of performance (,u) can be based on the firm's "expert" assessment of the innovation and / or consumer response to prototypes (depending on the innovation). The variation in consumers' assessments may be used to estimate the variability in information (62). If this variability is small, the deterministic approximation (that ignores 3) may be appropriate. In the context of modeling the diffusion of information, the basic premise of contagion models (considering information from external sources as well as word-of-mouth) may be conceptually relevant as a basis for specifying the information rate. Such a field application would permit an empirical assessment of the model and the measurement procedures as a tool for segmentation prior to product launch, as well as a potential component of the new product forecasting process.11

"We thank John R. Hauser, the Associate Editor, and three anonymous reviewers for their valuable comments.

Appendix. Proof of Property

(a) Constant Information Rate. With the information rate constant over time, we have from (24), A(t) = I' + /,,F',(nt), t > 0. Since /,, 6,, and n are constants, the result follows.

(b) Monotonically Decreasing Information Rate. Differentiating (25) with respect to t, we have

-A(t) = 6,,f'(i(t))[Nt)] + _,Y(i(t))h(t). (A.1I)

Now n(t) < 0 and, for t > t*, f; (i(t)) ? 0. Hence, from (A. 1), A(t) < 0 for t > t*; therefore, the penetration curve must be concave for t* < t < oo.

At an inflection point (say, at t = T), A( T) = 0; from (A. I),

f;(i(T)) = -f_(i(T)) [n( T))2 (A.2)

In the absence of any specific functional forms assumed for n(t) and fg(*), it is not possible to derive the specific number of points of inflection in the range 0 < t c t*; in general, there may be any number of such points. In particular, if

n(t) < -((i(( [n(t)]2 for 0 < t < t*, fi(i(t))




(c) MonotonicallyIncreasingInformation Rate. Since n(t)> 0 and f(i(t)) 2 0 and 0 <t c t*, A(t)> 0 for 0 < t c t*. Therefore, the penetration curve must be convex in this range.

In the absence of any specific functional forms for n (t) and f,( * ), it is not possible to derive the number of points of inflection in the range t * < t < oo. However, if the right tail of the distribution of -y tapers smoothly, i.e.,f,(i(t)) -O 0 as i(t) -- oo, A(t) asymptotically approaches 6, + ijj. Thus, as t -o oc, the penetration curve must be concave. Since the first segment of the penetration curve is convex, there must exist an odd number of inflection points in the range t * < t < oo. Q.E.D.

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