No. 044/2020/MKT

Diffusion of Innovations:
Modeling, Estimation, and Normative Developments

Dipak C. Jain*
Department of Marketing
China Europe International Business School (CEIBS)Trichy V. Krishnan
Department of Marketing
Great Lakes Institute of Management
Sophie Feng
Department of Marketing
Monash Business School

December 2020––––––––––––––––––––––––––––––––––––––––* Corresponding author:
1. Dipak C. Jain (dipakcjain@ceibs.edu). Address: 699 Hongfeng Road, Pudong, Shanghai, China, 201206

Diffusion of Innovations:
Modeling, Estimation, and Normative Developments

Dipak C. Jain*
Trichy Krishnan**
Sophie Feng****President (European), China Europe International Business School, Shanghai, China**Professor of Marketing, Great Lakes Institute of Management, Chennai, India***Faculty, Monash University, Melbourne, Australia

January 2021

1

ABSTRACT

Understanding the adoption process of a new product and modeling the resulting diffusion
pattern over time has been of great interest to both academicians and practitioners for the
past 50 years. The keen interest in modeling diffusion of innovations would continue to growin the new digital economy also, perhaps more vigorously given the increased speed withwhich information spreads on social media.

We are presenting a research manuscript that has three purposes. The first is to give a treatiseon the interesting model developments that have happened in modeling diffusion of
innovations. Noting that diffusion modeling is one of the strong theory-based quantitative
investigations that has evolved in the marketing literature, we explain how theoreticalmodeling by various researchers has advanced the knowledge on new product sales growth.

Second, given that no theoretical model can claim to be valid unless it is supported strongly
by empirical data, we discuss in detail the development of estimation procedures as they
pertain to offering validation support to the diffusion of innovation models. From theapplication perspective of diffusion models, we also look at the forecasting implications andnormative developments, e.g., optimal marketing mix policies.

Third, acknowledging that there have been comprehensive review papers in the diffusion of
innovations area, including: Mahajan and Muller (1979), Mahajan and Wind (1986), Mahajan,
Muller and Bass (1990), Mahajan, Muller and Bass (1993), Mahajan, Muller and Wind (2000),
and the latest one being Meade and Islam (2006). Our objective is not to just do anotherreview article on diffusion models but rather concentrate on emerging areas such as:
multigeneration diffusion models, social media related product diffusion, and repeatpurchase behavior, to name a few. Our focus is on building strong analytical models withtheoretical insights and in-depth empirical analysis.

Our work is likely to be more useful for the PhD students and researchers interested in new
product adoption and diffusion. We focus our discussion on the following three areas: model
formulations, estimation methods and normative implications.
2

INTRODUCTION

Rogers (1983) conducted several field experiments and found that a new product was notgetting adopted immediately by all but rather gradually, in a systematic way. He also foundthat the adoption process was largely influenced by word of mouth (WOM) spreading fromadopters to potential adopters. Rogers theorized that a new product diffuses through fivegroups of users namely: innovators, early adopters, early majority, late majority and laggards,
in that order. Of the 5 groups, the innovator group, Rogers argued, adopted the product firstby looking at how the product was useful to them and its role in their lifestyle. Then they
make a decision of whether to adopt. Once they decide to adopt the product, the other fourgroups are influenced by WOM and chose to gradually adopt over time.

Frank M Bass (1969) gave a mathematical structure to the Roger’s findings after collapsing
the last four groups into one group and calling it “imitator group”. Thus, Bass took the Roger’s
finding to mean that a small fraction of a market adopted a new product without getting
influenced by WOM and that the rest of the market adopt it after getting enough information
through WOM. He called the two groups innovators and imitators respectively. He used thistheory to develop a mathematical expression for the adoption of a new product by anindividual and derived a diffusion model that described the periodic sales growth of the
product at the market level. He further offered strong support to his model through fitting it
on eleven new consumer durable goods’ sales growth data.

The rather strong theoretical foundation and empirical support helped the Bass (1969) model
to set off an unprecedented research movement in Marketing and elsewhere. This resulted
in the Bass model of the 1969 article, which was published in Management Science, getting
the maximum ever citation count in Management Science. Till today, it holds that honor.

3

1: Diffusion Models

Bass (1969) asked how a given individual at time t would go about deciding to adopt a new
product introduced at time 0. Bass took the Rogers’ main finding on “innovators andimitators” dichotomy in the marketplace and transformed that to mean that the individual attime t would be influenced partly by innovative thinking and partly by WOM i.e. imitating
tendency. He captured the individual’s new product adoption decision in a hazard framework
as follows.
𝑓(𝑡)= 𝑝 + 𝑞𝐹(𝑡)1 − 𝐹(𝑡)(1)where f(t) is the probability that the individual would adopt the new product at time t, and[1-F(t)] is the probability he had not adopted the product until time t. Thus, if we take t is therandom variable of “time to adoption”, f(t) is the pdf (Probability Density Function) and F(t)is the cdf (Cumulative Distribution Function). The left-hand side function is thus the hazard
that an individual would adopt a new product at time t given that she has not adopted that
product until time t. The right-hand side has two components: a) “q F(t)” captures the WOM
influence of previous adopters on the individual, where q is the parameter that captures thestrength of the WOM influence generated by F(t), b) the parameter p captures all those nonWOM forces influencing the individual’s decision. These include advertising, salesforce efforts(especially in a B2B context), brochures and information on the website. Bass referred p andq as the coefficients of innovation and imitation respectively.

Recognizing that f(t) is actually dF(t)/dt, the differential equation (1) can be solved to yield:

𝐹 (𝑡 ) =1 − exp[−(𝑝 + 𝑞 )𝑡]𝑞
1 + 𝑝 exp[−(𝑝 + 𝑞 )𝑡](2)

This is the probability an individual might have adopted by time t. The pdf is obtained bydifferentiating F(t) with respect to t, resulting in:4

(𝒑 + 𝒒)𝟐 −(𝒑+𝒒)𝒕]𝒆𝒑𝒇(𝒕) =𝒒[𝟏 + 𝒆−(𝒑+𝒒)𝒕 ]𝟐
𝒑
[

(3)

Multiplying equation [2] by market potential, m, we get:𝐶𝑆 (𝑡) = 𝑚

1 − exp[−(𝑝 + 𝑞 )𝑡]𝑞
1 + 𝑝 exp[−(𝑝 + 𝑞 )𝑡](4)

In equation [4], m is the market potential i.e. the total number of adopters in the target
market, and CS(t) is the cumulative number of adopters until time t. 1 Interestingly, Mansfield(1961) proposed a model that was same as equation [1] but in deriving a closed form solution,
Mansfield made an assumption that yielded a different model (i.e. a model without p) whichbecame a simple logistic regression equation.
The sales growth pattern of a new product is shown in Figure 1:

In Figure 1, T* is the time to peak sales. To find T*, differentiating f(t) where f(t) is given inequation (3), and setting the result equal to zero will yield:

1

For a durable good like a smart phone or a laptop, the number of adopters would equal the sales of the product because
an adopter would buy only one unit in general. This is true of adopting a service product like signing up for a club, cell phoneservices, Facebook or WhatsApp. Hence, for durable goods and services, we can say that CS(t) represents Cumulative Sales
up to t.

5

𝑇∗ =

1
𝑞
ln( )
𝑝+𝑞 𝑝

𝑆 (𝑇 ∗ ) =(5)𝑚 (𝑝 + 𝑞 )2
4𝑞

Substituting T* in equation (2), we get:𝐹 (𝑇 ∗ ) =(𝑞 − 𝑝)2𝑞

And cumulative sales will be:
𝐶𝑆 (𝑇 ∗ ) = 𝑚𝐹(𝑇 ∗ ) =𝑚(𝑞 − 𝑝) 𝑚𝑝
= (1 − )2𝑞
2𝑞

Given that p is normally smaller than q, cumulative sales can be approximated as,𝑚
𝐶𝑆 (𝑇 ∗ ) =2
Therefore, cumulative peak sales of a new product will be half the total market potential.

Noting that S(t) = m f(t) and CS(t) = m F(t), where m is the market potential, equation [1]becomes:
𝑆 (𝑡) = 𝑚𝑝 + (𝑞 − 𝑝)𝐶𝑆(𝑡) − (𝑞/𝑚)𝐶𝑆 2 (𝑡)(6)Equation [5] was used by Bass (1969) to estimate the model with sales data in elevenconsumer durables, where he found a high R-square indicating an excellent fit of the model)
to the annual sales data 2. RCA, the leading Color TV manufacturer of those times, and WSJ
mentioned about the usefulness of the Bass model for forecasting purposes.

2.1 Model Extensions: Role of Marketing Mix variables
After the rather very successful empirical demonstration of the diffusion model by Bass in1969 and acknowledgement of the same by the industry, there was surprisingly no follow up
work for almost 6 years. Robinson and Lakhani (1975) of Kodak used the Bass model to derive
optimal pricing policies for their new cameras. An interesting thing is that the original Bassmodel (1969) doesn’t have price in their function, and so Robinson and Lakhani (1975)2

Equation [4], which is a better equation to use in estimation, was introduced 15 years later by Srinivasan and
Mason (1986).

6inserted the price in a rather ad-hoc way in the Bass model equation [1] and proceeded toderive the optimal dynamic pricing policy for a new camera. We use the phrase ad-hoc forthe reason that the authors didn’t test the model with empirical data. Later, other researchers
carried out empirical analysis but couldn’t find support for this model 3.

In fact, in the two decades following the publication of the Bass model, only a few research
articles offered empirically supported models while investigating the role of marketing mix
variables in the Bass model. Almost all of the models followed the Robinson and Lakhani(1975) approach in that their main focus was on deriving optimal policies without empiricallyjustifying them with empirical data4.

Bass (1980) proposed that the marginal cost decreasing over time with sales volume would
encourage firms to decrease the price accordingly, and used that principle to infer the roleof price. Horsky and Simon (1983) incorporate advertising in the Bass model by making thecoefficient of innovation (p) a function of advertising. They tested their model empirically aswell as derived the optimal advertising policy.

Horsky (1990) proposed that income-wage distribution would affect the market potential
while Kalish (1985) used the reservation price framework to explain the role of price on
adoption. Although the theoretical frameworks used in these two models are plausible, theempirical exercise carried out are difficult to replicate.

Jain and Rao (1990) focused on empirically testing out if price should be added to the Bassmodel to affect the market potential or the rate of adoption.

3

Although one could propose a model that logically explains a marketing process, providing good empiricalsupport for the model is important to justify the mechanism underlying the model because there could bemany such competing models one could come up with. In such a case, strong empirical support for a model
would provide the researchers and the managers enough confidence in the model and in the mechanism
proposed in the model.
4
Later we will show how optimal policies depend critically on the model one uses to derive those policies
although all models try to explain the same marketing process.

7

Two research articles had a more comprehensive approach to incorporate marketing mix
variables in the Bass diffusion model: Bass, Krishnan and Jain (1994) and Bass, Jain andKrishnan (2000).

Bass, Krishnan and Jan (1994) turned the puzzle upside down and asked: Why does the Bassmodel fit without decision variables? Analyzing the unique structure of the Bass model, theresearchers found that the only explanatory variable, i.e. t in the Bass model (see equation[3]) was somehow capturing also the impact of the decision variables such as price andadvertising over time. This is a classic case of the widely known statistical principle that if
omitted variables are correlated with the included variables the model would provide a good
fit to the data even in the absence of those omitted variables. This is of course good news forforecasting purposes because the forecaster now doesn’t have to worry about the othervariables but guess the values for the three parameters Bass model, namely, {p, q, m} and use
equation [4] and forecast the sales growth into future. However, for controlling the sales
growth process, managers need a modified Bass model that is theoretically sound andempirically proven. It was provided by Bass, Krishnan and Jain (1994). The model is given
below.
𝑓(𝑡)= [𝑝 + 𝑞𝐹(𝑡)] 𝑥(𝑡)1 − 𝐹(𝑡)(7)where x(t) was called marketing efforts at t (or current marketing efforts), which wasformulated as:𝑥(𝑡) = 1 + 𝛽11where Pr(𝑡)𝑑𝑃𝑟(𝑡)
𝑑𝑡

1 𝑑𝑃𝑟(𝑡)1 𝑑 𝑎𝑑𝑣(𝑡)+ 𝛽2
Pr(𝑡) 𝑑𝑡
adv(𝑡)𝑑𝑡
1is the percentage change in price at time t, and adv(𝑡)(8)

𝑑 𝑎𝑑𝑣(𝑡)𝑑𝑡percentagechange in advertising at time t, and the new parameters {𝛽1 , 𝛽2 } are respectively the impactof those price and advertising variables. This was called “Generalized Bass Model” because
when price and adv change at more or less a constant rate over time, these two factors
would be a constant, yielding x(t) to be some constant. In such a case, equation [6] would
become observationally equal to the Bass model [1] 5.

5

In order to accommodate the fact that advertising is likely to have a carry-over effect implying that an
𝑑 𝑎𝑑𝑣(𝑡)occasional drop in advertising might not mean negative influence on sales the variable 𝑑𝑡 in expression [7]8Solving the differential equation [7] with [8], we get:𝐹 (𝑡 ) =1 − exp[−(𝑝 + 𝑞 )𝑋(𝑡)]𝑞
1 + 𝑝 exp[−(𝑝 + 𝑞 )𝑋(𝑡)]

(9)

𝑃𝑟(𝑡)
𝑎𝑑𝑣(𝑡)+ 𝛽2
𝑃𝑟(0)𝑎𝑑𝑣(0)(10)

where,
𝑋(𝑡) = 𝑡 + 𝛽1 𝐿𝑛

GBM has found wide support in the academia thanks to its theoretical appeal on explaining
why the Bass model is so successful without including any decision variables.

Bass, Jain and Krishnan (2000) proposed another interesting framework that stems from thehazard model literature. They claim that the Bass (1969) model represents the basic hazard
process while the marketing variables act on that basic process like in a proportional hazard
model. This is as given below.
ℎ(𝑡 |𝑥(𝑡)) = ℎ(𝑡) exp[𝛽1 Pr(t) + 𝛽2 adv(t)](11)

where h(t) on the righthand side is the base level hazard function as given by Bass model(equation [1]) and h(t|x(t)) on the left hand side is the hazard that includes the impact ofprice and adv. This model has proved to be more versatile than GBM, especially inestimating the role of decision variables on the diffusion.

was devised to have the lower bound of zero. This of course can be modified to suit different conditions ofadvertising expenditure policies.

9A list of models proposed in the literature with empirical support and normative
implications is presented in Table 1.
Table 1
Diffusion of Innovations: Model Extensions
Research
Robinson and Lakhani (1975)
Bass (1980)

Dolan and Jeuland (1981)

Modeling Approach
Experience Curve and PriceNormative Model
Experience Curve andReservation Price –Empirical
AnalysisPrice – Normative Model

Bass and Bultez (1982)

Price – Normative Model andEmpirical Analysis

Horsky and Simon (1983)

Advertising – Normative
Model and Empirical Analysis
Advertising – Normative
Model
Price – Normative Model

Tang and Thomson (1983)
Kalish (1983)

Kalish (1985)

Price and Advertising –Normative Model

Simon and Sebastian (1987)
Horsky (1990)

Advertising – Empirical
Analysis
Income-Wage Distribution –Normative Model andEmpirical Analysis

Jain and Rao (1990)

Price – Empirical Analysis

Jones and Ritz (1991)

Distribution – Empirical
Analysis

Krishnan, Bass and Jain
(1994)
Krishnan, Jain and Bass
(1999)

Price and Advertising –Empirical Analysis
Price and Advertising –Normative Model

Specific Formulation
Price affects the rate ofadoption multiplicatively.
Price acts multiplicatively
on the rate of adoption.
Price has a multiplicative
effect on the adoption
rate.
Price and rate ofadoption have amultiplicative
relationship.
Advertising effects thecoefficient of innovation
Advertising affects the
diffusion coefficients
Price affects the market
potential and the rate ofadoption multiplicatively
Price affects the market
potential, the rate ofadoption is controlled byadvertising.
Advertising effects thecoefficient of imitation
Existence of incomeprice effect on market
potential is hypotheticaland tested.
Price effects market
potential and rate ofadoption
Market potential is afunction of distribution
growth.
Accelerated Hazard
Model Formulation
Advertising and priceaffect rate of adoption

10Bass, Jain and Krishnan(2000)
Danaher, Hardie and Putsis(2001)
Krishnan and Jain (2006)

Price and Advertising –Empirical Analysis
Price – Empirical Analysis
Advertising – Normative
Model

Proportional Hazard
Model Formulation
Proportional Hazard
Framework
Advertising affects rateof adoption

2.2 Modeling other aspects
There are many other aspects that are of interest to academicians and practitioners as wellconcerning a new product. We list a few of them here first and take up for discussion one by
one to explain how researchers handle them.

Micro-modelling Adoption
Although Bass model and its extensions were found to be successful and useful in describing
the sales growth process of a new durable, it is interesting that they don’t describe theadoption of a new product at an individual level. Bass model is actually founded on theindividual level adoption process but by assuming that all individuals are the same, theindividual level differences get washed out by the aggregate model. In other words, theaggregate model ignores the rich information contained in the differences amongindividuals with respect to how they view a given new product. This is a little troubling
because we don’t set to gain insights that could answer questions such as “are there
individuals who are more risk averse than others so a manager could approach them with adifferent strategy?”

In order to address this critical issue Chatterjee and Eliashberg (1990) advanced aninteresting model that goes to the heart of the individual’s decision regarding adopting a
new product. They start with a utility model for an individual that builds on how far the
information gathered by the individual affects his perception of the product’s expected
performance, and augment that utility with a price function. The individual is modeled to
update his utility in a Bayesian manner as more and more information bits come to him, andupon reaching a threshold the individual is said to adopt the product. There is unobserved
stochasticity in the information process, represented by Weiner process, and this creates11the heterogeneity among the individuals. This leads to a nicely structured individual-level
adoption model, which when aggregated gives Bass model under certain conditions. The
authors did a pilot study with Wharton MBA students to demonstrate how their model
could be used and estimated, leading to rich insights that could not be obtained from using
aggregate models.

Diffusion Models with Repeat Purchase
Even though Bass acknowledges that his model is concerned with initial purchases only, theBass Model is frequently applied to fit sales data that counts both initial purchases andrepeat purchases. Such applications are reasonable when repeat purchases are infrequentor negligible, which might be the case for durable products during the early stages of
product sales.

For products that have frequent upgrades or multiple versions, repeat purchases canconstitute a significant proportion of sales even in the early stages of product diffusion, thus
limiting the application of the Bass Model. When significant repeat purchases exist, thefitting and forecasting performance of the Bass Model may degrade, particularly when thesales trend begins to assume an asymmetrical shape after the peak point of diffusion. Even
if the model fits and predicts well in some situations despite the existence of considerable
repeat purchases, the parameter estimates will be different from those obtained using only
initial purchases data.

In the prior literature, researchers have proposed model extensions to account for repeat
product purchases made to replace existing product units or for adopting multiple productunits. Olson and Choi (1985) propose a model assuming that sales comprise only adoptions
and replacements, and that the replacement hazard function follows the Rayleighdistribution (Papoulis and Pillai 2002). The Olson-Choi Model is developed for cases in which
the number of products in use is available.

Kamakura and Balasubramanian (1987) propose a similar model that generates long-term
forecasts by incorporating the adoption and replacement components of sales. Benefiting

12

from a more flexible hazard function to capture replacement purchases, their model isapplicable with or without data for replacement sales. Their model uses information fromsimilar products when data for replacement purchases is not available. Steffens andBalasubramanian (1998) further advance the modeling of replacement sales by allowing the
distribution of the service life of replaced products to change over time.

Focusing on the PC processor industry, Gordon (2009) introduces a dynamic structuralmodel that explicitly considers product replacement decisions under uncertain future
product quality and price. Gordon’s model concentrates specifically on productreplacements due to obsolescence as a result of product upgrade releases. The model
requires highly specific data to function as it uses a composite dataset including sales,
ownership, price, and product quality. Due to data limitations, Gordon’s model does notconsider multi-unit ownership purchases.

The aforementioned models have their focus on adoptions and replacement purchases.
There are other models from the prior literature that have considered other components ofsales, in particular multi-unit ownership purchases, in addition to adoptions andreplacements. Dodson and Muller (1978) propose a model that captures the type ofasymmetric sales trend illustrated by the solid curve in Figure 2 without decomposing sales
into multiple components. The market they consider is composed of three groups, i.e.,
those who are not aware of the product, those who are aware of the product but have notmade a purchase, and those who have purchased the product. Although the portrayedinteraction between adopters and non-adopters is insightful, their model cannot be
operationalized if the data for the different market divisions is unattainable.

Figure 2. Sales vs. Adoptions
Sales (including initial and repeat purchases)

Adoptions (initial purchases only)13Based on econometric and simulation models, Bayus, Hong, and Labe (1989) incorporate
first time sales, replacement sales, additional-unit sales, and institutional sales into theiranalysis of color television set sales. As stated by Steffens (2003), Bayus et al.’s model isdeveloped to perform well over short terms.

Steffens (2003) presents a model for sales resulting from multi-unit ownership purchases
based on the Bass Model. His model differs from Bayus et al.’s in that it imposes asaturation level on multi-unit ownerships, which makes the model applicable for longer timeframes.

Researchers have also developed models for repeat purchases of products for a specificindustry. For instance, Lilien, Rao, and Kalish (1981) propose a highly specialized model toproject sales of prescription drugs as a function of a focal pharmaceutical company’s own
detailing effect, competitors’ detailing effect, and word of mouth. In a subsequent study,
Rao and Yamada (1988) provide further empirical support for the model by Lilien, Rao, andKalish, then propose an alternative method for developing priors for the new drug’s
parameters, and demonstrate how the parameters can be updated after sales data
becomes available. Because Lilien, Rao, and Kalish’s and Rao and Yamada’s models are
specifically designed for prescription drugs, they cannot be used to predict sales of generalproduct categories.

A detailed review of the literature suggests that there is a need for a comprehensive modelthat can be fit to aggregate sales data that records both initial purchases and repeatpurchases including both replacements and multi-unit ownerships.

Jiang, Aslan and Jain (2019) proposed a model termed as the Generalized Diffusion Modelwith Repeat Purchases (GDMR). The development of this model is based on a branch ofmathematics named fractional calculus. Specifically, the model generalizes the fundamental
differential equation governing the Bass Model, and employs a non-integer integral
operator with flexible order, thereby rendering the Bass Model a special case of theextended model. The GDMR adopts an approach that is different from those in the prior14

literature in that it captures the sales growth rate using a non-integer order integral
equation, rather than an integer-order differential equation as used in the prior literature.

Multi-Generation Diffusion of Innovations Models
Technological advances fuel the development of new products and services. Examples areabundant. Decades ago, black-and-white TV was replaced by color TV, which is now ceding
market share to HDTV. Even within the HDTV category, newer models are continuing toemerge, with the most recent variety 3D-capable, due to even more technologies. In thecellular phone market, the earliest generation was only equipped with basic calling features,the following generation enhanced to include cameras, media players, etc., while thenewest generation, called smart phones, allow users to surf the Web, and run moresophisticated applications. The same phenomenon also exists in the software market, where
vendors keep releasing new versions to meet users’ ever-increasing appetite forfunctionalities and take advantage of improvements in hardware technologies. The
Microsoft Windows and Office lines of products are two well-known examples, with newversion typically introduced every few years.

The diffusion of successive product generations has been well studied in the prior literature.
Most of the existing multigeneration diffusion models are inspired by the seminal Bassmodel (Bass 1969). Among them, the model proposed by Norton and Bass (1987) (NB model
for short) is often credited as the pioneering work in describing multigenerational diffusion.
The NB model assumes that each generation has its own market potential and market
penetration process, and adopters of earlier generations can shift to newer generations.
After Norton and Bass, several other notable multigeneration diffusion models have been
proposed. Speece and MacLachlan (1995) extend the NB model to incorporate the influenceof pricing and test it with multigenerational data for fluid milk packaging technologies.
Mahajan and Miller (1996) develop a model that captures the number of systems-in-use foreach generation and use it to study the optimal market entry timing for successive
generations. Jun and Park (1999) combine the diffusion effects and choice effects andpropose two integrated models: the Type 1 model distinguishes first-purchased demand
and upgrade demand while the Type 2 model does not. Kim et al. (2000) propose a dynamic
market growth model that captures not only the diffusion of multiple generations within the15

same product category, but also the complementarity and competition present by related
product categories. Danaher et al. (2001) develop a two-generation model that includes
both first-time sales and periodic renewals. By selecting appropriate adoption time
distributions, their model can also incorporate the impact of market mix variables. Morerecently, Jiang (2010) proposes a simple two-generation model to analyze the optimal freeoffer policy for successive software versions.

Despite the progress made in the last two decades, a review of the literature reveals that
the NB model remains the most tested and extended multigeneration diffusion model todate. We believe that the desirable mathematical properties (e.g., offering closed-form
expressions, parsimonious, and continuous-time based) of the model plays a key role
behind its popularity. However, the NB model is not applicable to all business scenarios. Thisis primarily because when counting the number of adopters who substitute an old
generation with a new generation, the NB model does not differentiate those who havealready adopted an earlier generation and those who are first-time adopters of any
generation. In their study, Norton and Bass do acknowledge the existence of two different
types of substitutions but admit that their model does not differentiate them (Norton andBass 1987, p. 1074). Without such differentiation, the NB model cannot be used to estimate
the number of cross-generation repeat purchases, nor can it help forecast future demand or
project revenue for certain business scenarios (e.g., when revenue is generated throughboth product sale and after-sale service). Jiang and Jain (2010) propose a Generalized
Norton-Bass (GNB) model that overcomes this limitation while retaining the desirable
mathematical properties of the NB model. As we will demonstrate later, the proposedmodel offers greater flexibility in parameter estimation, forecasting, and revenue projection
for a wider range of scenarios.
A couple of other areas that need more attention are a) multinational diffusion models b)
stochastic diffusion models and c) successive new product generations models. We
summarize the research work in these areas in Tables (2), (3) and (4), respectively.
Researchers interested in these areas can build on the work highlighted in the respective
tables.

16

Table 2: Multinational Diffusion Models
Research Article

Focus

Gatignon, Eliashberg and Robertson (1989)

Modeling Multinational Diffusion Patterns

Takada and Jain (1991)

Cross-National Analysis of Diffusion ofConsumer Variables in Pacific Rim Countries

DeKimpe, Parker and Sarvary (1998)

Estimation of International Diffusion
ModelsEliashberg and Helson (1996)

Modeling Lead-lag Effects in Global
Markets

Jain and Maesincee (1997)

Lead-lag Time Effects on Global ProductDiffusion

DeKimpe, Parker and Sarvary (2000)

Multimarket and Global Diffusion

Talukdar, Sudhir and Ainslie (2002)

Investigating New Product Diffusion Across
Products and Countries

Kuman and Krishnan (2002)

Multinational Diffusion Models

Table 3: Stochastic Diffusion Models
Research Article

Focus

Eliashberg and Chatterjee (1986)

Stochastic Issues in Innovation Diffusion
ModelsBoker (1987)

A Stochastic First Purchase Diffusion Model

Niu (2002)

A Stochastic Formulation of the Bass Modelof New-Product Diffusion

Niu (2006)

A Piecewise-Diffusion of New-Product
Demands

Raman and Jain (2010)

A Stochastic New Product Growth Model

17

Table 4: Results of Successive New Product Generations Research
Research Studies

Research Focus

Product/Service Analyzed(Examples)

Wilson and Norton (1989)

Moorthy and Png (1992)

Mahajan and Muller (1996)

Morgan, Morgan and
Moore (2001)

Prasad, Bronnenberg and
Mahajan (2004)
Krankel, Duenyas andKapuscinski (2006)

Mehra, Seidmann and
Mojumder (2014)
Jiang, Qu and Jain (2019)

Optimal entry timing for a
Publishing firm’s decision online extension of a new
introducing hard cover andproduct
paperback versions
Sequential vs. Simultaneousa. IBM PS/1, PS/2 Game
introduction of new productComputerextensions
b. Minolta single lensreflex camerasc. Volvo 6-cyllinder 760
and 4-cyllinder 740
modelsOptimal timing ofd. Four successive
successive generations ofgenerations of IBM
technological innovations
mainframecomputers
Quality and Time to Market Auto, medical devices,
Trade-offs Made forcellular phones andMultiple Productintegrated circuit chips
Generation Launches
Timing strategies for new
Timing of video release after
product versions througha theatrical release of amultiple channels
movie
Successive Product
IBM’s mainframe computer
Introductions Timings under
Demand Diffusion andTechnology Improvements
Optimal upgrade intervals
Product lifecycle
for successive software
management of packaged
versions
software
Optimal market entry
Technology products (e.g.
timing for successive
computers, Microsoft Officegenerations of technological 365, software licenses)
innovations18

3: Estimation Methods
However logically correct a theory is, the world would not accept it wholeheartedly unless itgets empirical support. This is very much true for a marketing model like the Bass model
because of its practical application in business. Empirical support means that a proposed
theoretical model is able to explain the underlying real phenomenon happening in themarketplace. In order to do that, we first gather data of the phenomenon from the realworld, and statistically see if the proposed model “fits” with the data. We do the empiricaltest with data sets from multiple categories. In this chapter, we will analyze the various
statistical procedures proposed in the literature to empirically test the Bass model and a few
other extended models.

3.1 OLS (Ordinary Least Squares)
Bass (1969) proposed the fundamental diffusion model in a differential and solved theresulting differential equation to get a closed form expression for the Sales in time domain.
However, for empirical testing he chose to use equation [1] and used Ordinary Least
Squares (OLS) method after converting equation [1] to a linear model as follows.

𝑓(𝑡)= 𝑝 + 𝑞𝐹(𝑡)1 − 𝐹(𝑡)𝑆 (𝑡) = 𝑚 𝑓 (𝑡) = 𝑚 [1 − 𝐹 (𝑡)][𝑝 + 𝑞𝐹 (𝑡)]𝑆 (𝑡) = 𝑚𝑝 + (𝑞 − 𝑝)𝐶𝑆(𝑡) − (𝑞/𝑚) 𝐶𝑆 2 (𝑡)Note that in equation [9], CS(t) is cumulative sales at t and is m F(t), while sales is S(t) which
is m f(t). It is easy to see that expression [9] is a linear equation where the dependent
variable is S(t) and the independent variable is CS(t) and that the intercept is “m p” and thetwo slope coefficients are “q-p” and “-q/m”.

𝑺(𝒕) = 𝜶 + 𝜷𝑪𝑺(𝒕) + 𝜸𝑪𝑺𝟐 (𝒕)Where 𝜶 = 𝒎𝒑, 𝜷 = 𝒒 − 𝒑 𝒂𝒏𝒅 𝜸 = −𝒒/𝒎
Solving 𝜶, 𝜷 𝒂𝒏𝒅 𝜸 𝒊𝒏 𝒕𝒆𝒓𝒎𝒔 𝒐𝒇 𝒑, 𝒒 𝒂𝒏𝒅 𝒎𝒎=−𝜷 ± √𝜷𝟐 − 𝟒𝜶𝜸
𝟐𝜸19𝒒 = −𝜸𝒎𝜶𝒂𝒏𝒅 𝒑 =𝒎Bass fitted this linear model on 11 new products’ sales data and found excellent fit. By using
sales data of 11 new products to offer empirical support for his model, Bass (1969) showed
the versatility of the proposed model to track sales growth of different new products andthe robustness of the model. Note that OLS was not, as researchers later showed, a good
estimation technique for the inherently non-linear Bass model and yet the fit was excellent,
and the parameter estimates had plausible values with expected signs. This further
demonstrated the flexibility and robustness of the Bass model.

3.2 NLS Estimation
In their 1986 paper, Srinivasan and Mason (SM) offered an elegant non-linear regression toestimate the Bass model. The non-linear least squares (NLS) method used the closed form
solution of the Bass model (equations [2] and [4]) i.e. the sales in time domain as follows.
The equations are reproduced here for convenience.
𝑆(𝑡 − 1, 𝑡) = 𝐶𝑆(𝑡) − 𝐶𝑆(𝑡 − 1)
where,𝐶𝑆 (𝑡) = 𝑚

1 − exp[−(𝑝 + 𝑞 )𝑡]𝑞
1 + 𝑝 exp[−(𝑝 + 𝑞 )𝑡]

The NLS method showed remarkable improvement in the fit and the parameter estimates(less standard error). Most importantly, it enabled much better forecasting ability with theBass model, which we will explain now.

Jain and Rao proposed the following non-linear formulation of the Bass model
𝑆 (𝑡 − 1, 𝑡) = (𝑚 − 𝐶𝑆 (𝑡 − 1))[𝐶𝑆(𝑡) − 𝐶𝑆 (𝑡 − 1)][1 − 𝐶𝑆(𝑡 − 1)]

Where CS(t) is given in equation (4). NLS method was used to estimate the three model
parameters, p, q and m.

20

A model’s empirical performance depends on how cleverly it uses the information
contained in the data to estimate its parameters. In that context, Jain and Rao (1990) model
outperforms many other models. It also uses NLS but in a much more clever way and hence
yields better estimates.

3.3 Bias in Estimates
Van den Bulte and Lilien (1997) embarked on a rather unique research. They tested if theBass model parameters estimated from data suffered from any systematic bias induced by
the number of data points used in the data set. They used NLS-SM estimation method andused 11 real data sets and many simulated data sets to show that as the number of data
points used for estimation increased the m (market potential parameter) estimate wassystematically obtaining a higher value, and the q (WOM parameter) was obtaining a lower
value. These findings are naturally important from forecasting perspective. However,Krishnan and Feng (2010) showed that inappropriate pooling of various data in the Van den
Bulte and Lilien (1997) study might have led to their conclusion on the systematic bias.
More research needs to be done in this area to figure out if the problem is for real, and, if
yes, if the cause was estimation procedure or something else.

3.4 MLE (Schmittlein and Mahajan, 1982):
Noting that each adoption is an outcome of a binomial choice –adopt or not- at a given time
for a given individual, they formulated a likelihood function from the Bass model and used
MLE to estimate the model parameters from a given data set. Although many researchers
prefer to use NLS over MLE mostly for simplicity, the discussion of which estimation
procedure is better is still going on. For example, Hardie, Fader and Wisniewski (1998) foundmaximum likelihood to be noticeably better than NLS when applied to period-by-period
sales. However, over shorter series, they found NLS applied to cumulative sales comparablewith MLE.

3.5 Augmented Kalman Filter with Continuous State variable and Discrete observations:
AKF(C-D): AKF(C-D) was borrowed from engineering and introduced by Xie, Song, SIrbu andWang (1997) to offer another estimation procedure for the Bass model. Interestingly, thismethod uses the differential equation form of the Bass Model and not its closed-form
21

solution, although it is not clear if the superior performance would remain the same if one
use AKF(C-D) on the latter form. AKF(C-D) acts on a closed-loop feedback mechanism, where
each of the three parameters get updated by the difference between the predicted onestep ahead forecast and the actual sales in the following period. At the start, eachparameter is given a prior chosen judiciously inferred from the diffusion of previous
analogous products or from meta-analytic studies like Sultan, Farley and Lehmann (1990).
These are used on the Bass Model to forecast the first year’s sales, and the difference
between this forecasted first year sales and the actual first year sales, the parameters get
updated through an updating function. And the process gets repeated year by year until all
the observations are used up. The authors show the superior performance of the AKF(C-D)
by comparing it with OLS, NLS-on-SM and MLE. However, the authors use the accuracy ofthe one-step ahead forecast as the metric for this comparison, which is a little unfair
because other estimation procedures use the “maximizing the overall fit of the model withthe data” as their objective while AKF(C-D) uses exactly the one-step ahead forecast
accuracy as its objective to update the parameters. Further, the AKF(C-D) procedureassumes observation error and parameter estimates’ errors, and it is not clear how far theseassumptions affect the performance. Also, if a manager wants to do multi-period ahead
forecasting, it might be difficult to employ this procedure because it’s basic premise is
updating the next period performance. A comprehensive list of estimation procedures ispresented in Table 2.
Table 5
Diffusion Models: Parameters Estimation Methods
Estimation Methods
Ordinary Least Squares (OLS)

Maximum Likelihood Estimation (MLE)

Research Articles
Bass (1969)
Bass (1980)
Horsky and Simon (1983)
Norton and Bass (1987)
Simon and Sebastian (1987)
Kamakura and Balasubramaniam (1988)
Horsky (1990)
Jones and Ritz (1991)
Bass, Krishnan and Jain (1994)
Schmittlein and Mahajan (1982)
Hardie, Fader and Wisniewiski (1998)22

Non-linear Least Squares (NLS)

Hierarchical Bayes (HB)
Augmented Kalman Filter (AKF)

Srinivasan and Mason (1986)
Jain and Rao (1990)
Van den Bulte and Lilien (1997)
Bass, Jain and Krishnan (2000)
Lenk and Rao (1990)
Xie, Song, Sirbu and Wang (1997)

4: Normative Implications of Diffusion Models
Theoretical models like diffusion models will not be appreciated by practitioners unless they
could be put to use to address some of the practical managerial issues. Diffusion models canbe ideally used to address two such managerially relevant problems: forecasting andoptimal marketing mix (pricing and advertising) policy decisions.

4.1 ForecastingFrankly speaking, forecasting is as much an art as it is science. It is especially true with a new
product where the consumer himself carries a lot of uncertainty with the product. Further,
the sales growth of a new product very much depends on how the product manager
employs the strategy in the marketplace, and a forecaster will not have access to thosestrategies. Having said that, managers would find it immensely useful if we could provide
them with a fairly reliable tool to forecast the sales of a new product, stating therein theassumptions made. To that extent, let us now see how we can use the Bass model for theforecasting purposes.

If a new product finds some traction in the marketplace, the product manager might like toknow how the sales would grow over time so she could plan the production and distribution
activities appropriately. Diffusion models perform their best in this forecasting exercise.
Taking a simple approach, if we could somehow get the three main parameters of the Bassmodel, namely, {p, q, m}, for the given new product, we can easily draw out the annual salesof the product using expressions [4] and [6]. Where do we get these three parameters?

Initially from some analogous products introduced earlier in the same marketplace to thesame target segment, and later when some actual sales number of first few periods arecollected, using those sales numbers one can getter better estimates for the three23

parameters. One may wonder why the Bass model is the preferred model for forecasting?
The answer to this question is rather straightforward. Recognizing that the Bass model issimple, theory-based, robust and is driven by only one independent variable, time, we can
comfortably say that the Bass model is the best model to use for forecasting purposes. We
may have other models like GBM that have successfully incorporated the influence of priceand advertising, but if we want to use them for forecasting purposes, we have to necessarily
predict how those additional variables, price and advertising, also would change over time.
On the other hand, Bass model does not require them. Its parameters already reflect theirimpact implicitly. Secondly, the Bass model is a very robust model. Krishnan and Bassmeasured this robustness and compared that with robustness of other models.

When the product shows clear signs of success after the first 3 or 4 periods, the manager
would be interested to know when the sales growth would hit the peak. i.e. Peak-Sales Time(T*), as presented in equation (5). Knowing the T* for the given product can help a manager
to (a) plan the production capacity and schedule, (b) know when to invest in the next new
product and (c) decide on the right pricing and promotion strategies to face the competition
that is likely to arise near the peak sales time. In other words, forecasting the T* using earlyyears data might be the most important forecasting exercise for a product manager toundertake. Krishnan, Feng and Jain (2020) show how one could go about doing this forecast.

4.2 Optimal Pricing
Of all the marketing mix variables, perhaps the max attention is paid to ‘price’ because it is
one element that directly and immediately affects the sales and, further, the impact can bemeasured. Starting with Robinson and Lakhani (1975), many optimal pricing paths havebeen advanced in the literature. These include the articles presented in Table 6.
Table 6
Diffusion Models: Normative Implications
Marketing Mix Variable
Price(Reference Price, Experience
Curve)

Research Articles
Robinson and Lakhani (1975)
Bass (1980)
Bass and Bultez (1982)
Dolan and Jeuland (1981)

Normative Findings
Optimal Pricing Policy

24Advertising(Cumulative Advertising)Price and Advertising

Jeuland and Dolan (1982)
Kalish (1983)
Kalish (1985)
Horsky (1990)
Dodson and Muller (1978)
Horsky and Simon (1983)
Simon and Sebastian (1987)
Krishnan and Jain (2006)
Kalish (1985)
Krishnan, Jain and Bass
(1999)

Optimal Advertising
PolicyOptimal Pricing andAdvertising Policies

As mentioned earlier, many of the earlier models were developed by including price variable
into the Bass model in a plausible and logical way but without providing empirical support
for the same. For example, the optimal policy derived from these models point tosomething interesting. They recommend that the price should increase as long as the sales
keeps increasing and should start decreasing after the sales goes beyond the peak anddecrease. In other words, they derive an optimal policy that mimics the sales pattern. It ofcourse makes sense because the sales anyway keep increasing until the peak and hence itwould seem prudent to increase the price that time. However, simple economic theory
suggests that price increase or decrease should be a function of how sensitive the demand
is to price changes. In terms of diffusion, it would mean that price change should depend on
how effective the price is in affecting the rate of increase in the sales. Recognizing this,Krishnan, Bass and Jain (1997) used the GBM (1994) and derived the dynamic optimal
pricing for a new product. Their results suggest that optimal price would increase initially (asothers had suggested) but would start decreasing after some time, not necessarily after
peak as others had suggested. The transition time could be even zero implying that the
optimal price would decline monotonically right from the introduction time. When exactly
the price would transit from increasing to declining trend is a function of sensitivity ofdemand to price and the future-discount rate.

4.3. Optimal Advertising
Compared to the optimal pricing literature, number of researchers worked on deriving
optimal advertising is rather limited. There are only three distinct articles in principle: Teng25

and Thomson (1983), Dockner and Jorgensen (1988) and Krishnan and Jain (2006) 6. Of thethree, the first two papers used the following extended Bass model for deriving theiroptimal advertising policies.

𝑆(𝑡) = [𝑝1 + 𝑞1 𝐹(𝑡) + {𝑝2 + 𝑞2𝐹 (𝑡)}𝑓(𝑎)][𝑀 − 𝐶𝑆(𝑡)](12)In equation [12], f(a) is a function of advertising which acts on one part of the diffusion force
represented by innovation and imitation effect, and the rest of the parameters are basicallythe Bass model. If f(a) is zero, we get the Bass model. Teng and Thomson (1983) used alinear function for f(a) while Dockner and Jorgensen (1988) let it free and be a concave
function. Depending upon the relative values of {p2, q2} with respect to {p1, q1}, theauthors claim that optimal advertising policy is determined.

Teng and Thomson (1983) results suggest that if advertising is more impactful on imitators
optimal advertising should be minimum in the early and final periods and maximum in the
middle periods i.e. Min-Max-Min optimal policy. Under the same condition of advertisingmore impactful on imitators, Dockner and Jorgensen (1988) results suggest a monotonicallyincreasing advertising policy. If the advertising is more impactful on innovators, Dockner andJorgensen (1988) suggest a monotonically declining advertising policy. A main shortcomingof these results is that the model [equation 10] used was not empirically tested and hence,as mentioned earlier, managers would find it difficult to follow them.

Krishnan and Jain (2006) used the empirically proved GBM to derive their optimal
advertising policy. Their results suggest a variety of paths for optimal advertising that
depend on how effective is advertising on the diffusion, future discount rate and the ratio ofadvertising to sales ratio (A/S ratio). They show how different combinations of these three
forces result in either increase-decrease or decrease-increase policies, where monotonicallyincreasing is a special case of the increase-decrease policy and the monotonically declining
policy is a special case of the decrease-increase policy.

6

Horsky and Simon (1983) is a special case of the Dockner and Jorgensen (1988) and hence is not consideredseparately. Similarly, Thomson and Teng (1984) has similar model and results as those of Teng and Thomson(1983) and hence is not considered separately. We will discuss these two papers later, however.

26

4.4. Joint Optimal Pricing and Advertising
It is very challenging to derive jointly optimal price and optimal advertising policies. In
Section 4.2 where we discussed optimal pricing policies, all of the results assume thatadvertising does not interfere with the price-to-sales impact directly, and probably
advertising influence the price sensitivity. In Section 4.3 where we discussed optimal
advertising policies, the researchers had assumed constant price or price declining at aconstant rate. Haodong and Krishnan (2012) derived joint optimal pricing and advertising
policies.

5. Summary and Directions for Future Research
The purpose of this article is to highlight the developments in the area of diffusion of
innovations and its contributions to the new product marketing literature. Specifically, we
focus on:•model formulations that capture the global diffusion dynamics•

robust estimation procedures for empirical analysis•

normative models that yield optimal marketing mix policies

Moving forward, we would like to develop models that take into considerationa) The role social media plays in the marketplace: new product diffusion models needto incorporate word of mouse in addition to word of mouth that researchers havedealt with extensively in previous work.
b) The market entry timing decisions for introducing successive generations: an areaof strategic importance for firms because among other reasons, the introduction of a
new generation has the potential to cannibalize the sales of current offerings,
thereby affecting the firm’s total revenue model. (e.g., Apple’s iPhone, Samsung’s
Galaxy)c) Market entry timing strategy for new products starting with quality variation: newversion of better quality than previous version (e.g., high-definition TV vs. standardTV) or new version of poorer quality than previous one (e.g., hard cover textbooks
first and paperback edition later)27d) Product lifecycles are getting shorter: Given the large investments in developing a
new product, there might be financial risks associated with its premature or delayed
introduction.
e) New technological developments: Impact of artificial intelligence (AI), machinelearning (ML), and internet of things (IoT) on new product diffusions.

Successful launch and management of successive generations of technological products isimportant for firms to sustain its revenue stream and profitability as well as maintain a loyal
customer base. Although, product lifecycles may be getting shorter due to rapid
technological advancements, but customer lifecycles are becoming longer due to enhanced
longevity resulting from health science research and advanced medical innovations.
Enriching total customer experience requires firms to focus on R&D for new products andhave a proper plan for entry timing for successive product generations. Successful launch of
successive product generations also effects product branding as well as corporate brandingin achieving a favorable, innovative image in the marketplace.

28