Review of Innovation Diffusion Models

Abstract

This paper tries to review innovation diffusion models which are used in market research and diffusion of innovation.

Full text

Working paper No. 1/2015-01
Review of Innovation Diffusion Models
i
Naresh Kumar
CSIR-National Institute of Science Technology & Development Studies
K. S. Krishnan Marg, Pusa Gate, New Delhi-110012, India
Email: nareshqumar@yahoo.com
Section 1
1.1 Introduction
Innovation is a driving force for economic growth and its speed is affected by
accessibility of the new product. The word ‘diffusion’ has become a well-known
connotation in the context of diffusion of innovation. The diffusion of innovation provides
a conceptual paradigm for understanding the process of diffusion and social change. The
study of innovation diffusion is important as innovations without diffusion do not survive
and have no impact on the society or economy. Diffusion is imperative in the process of
innovation growth and technological change. A wide spread diffusion of an innovation
determines the rate of technological change, and is a measure of its effect on productivity.
Its efficacy has prompted the scholars about the features and phases that characterize
diffusion process.
The idea of diffusion was first introduced by Tarde (1903) in his book “The Laws
of Imitation”, which is now known as law of adoption. Schumpeter (1939) further extended
the idea and classified the phases of technological change into three categories namely:
(i) invention (ii) innovation and (iii) diffusion. Present study pertains to the process of
diffusion, where new products come into widespread use. Here an attempt has been
made to understand the pace and framework of innovation diffusion with the help of
innovation diffusion models. Innovation diffusion models describe the time dependent
aspect of innovation growth process that explains how an innovation spreads in a social
system through certain channels of communication over time and space.
Innovation diffusion models have been extensively used in many contexts
including social, political, economical, and technological and particularly for innovation
diffusion. Scholars have developed various theories and models to describe the diffusion

phenomenon and hence applied in different disciplines. Among them sociologists and
economists applied diffusion models to study the spread of new agricultural techniques
(Katz et al, 1963, Malecki et al, 1975, and Ryan and Gross, 1943). Social anthropologists
are interested in studying the spread of ideas and techniques within different societies
and cultures (Edmonson, 1961, Harland and Zohary, 1966, Sauer, 1952 and Seidenberg,
1960). Further, medical researchers used these models for diffusion of new drugs and
family planning techniques in a society (Anderson, 1974, Belcher, 1958, and Roberto,
1976). All these studies prove the applicability/versatility of diffusion models. In addition,
marketing and industrial researchers have tried to investigate the diffusion process of new
technologies and diffusion of new consumer durables (Bass, 1969, 1978, Blackman,
1972, Lakhani, 1975 & Bundgaard-Nelson, 1976,). Thus, over the years the researchers,
scholars, and technologists have shown deep interests in this area and as a result several
diffusion models are available in the literature to describe the growth phenomenon of an
innovation.
Innovation diffusion is a dynamic phenomenon. It is important to understand its
nature not only for policy makers but entrepreneurs as well. As innovation diffusion is
related to the costs, benefits, purchasing power and adoption level, the theory of
innovation diffusion plays an important role in determining future penetration of innovation
or product by understanding its characteristics. The nature of diffusion may be linear or
non-linear and depends upon its accessibility and behavior of the potential adopters. In
general the process of innovation diffusion follows sigmoid curves (Mansfield, 1961). The
sigmoid or S-shaped curve describes the diffusion process and growth pattern of all the
pervasive technologies and diffusion models set down alone as a self-sufficient topic.
There is an immense and rich amount of literature dealing with this theme.
1.1.1 Theoretical Beginning of Innovation Diffusion
Contemporary concepts of innovation diffusion originated through sociological
studies particularly from rural sociological literature on the adoption of new farming
technologies (Ryan and Gross, 1943). Since then these theories have been applied and
developed in many other fields resulting in a technique that is methodologically and
empirically mature (Rogers, 1995). In addition to research on the adoption of an

innovation that developed the interest in diffusion displayed by marketers (Parker, 1994),
diffusion of innovations also suggests processes of pre-launch data collection as well as
identification and control of key variables. This specifies that diffusion process is more
significant and stimulating since the social and economic aspects are realized by all the
adopters of innovation.
The multidisciplinary literature of the study of diffusion of innovation reflects two
approaches, which are generally adopted by the researchers and scientists. The first
approach is based on the micro level socio economic factors. This spatial and traditional
approach of innovation diffusion starts with the assumption that the diffusion process is
similar to the spread of species or/and a disease among a given homogeneous population
and is affected by the social and economic structure. Sociologists, geographers, social
anthropologists and development planners seem to apply this approach. The second
approach studies the pattern of the spread of an innovation at macro level. The
technology planners, market and industrial researchers that investigate the time pattern
of a spread of innovation at a macro level adopt this approach. These approaches can be
better understood through diffusion modelling.
1.1.2 Mathematical Models
Mathematical modelling has been an important feature of the scientific enquiry and
interpretation. Mathematical models are abstract representations of real world objects,
systems, or processes. They are used most often for illustrating theoretical concepts but
are increasingly being used in more applied situations such as predicting future
outcomes, or simulation experimentation etc. These can be a simple mathematical
equation that defines a relationship between a variable and system and are applied in
different areas. Mathematical models, also, try to capture qualitative as well as
quantitative aspects of different systems and are applied to study the dynamics of the
behaviour of social, political, economic and technical systems.
1.3.1 Innovation Diffusion Models
The growth and diffusion are synonym of each other. Usually, the term growth is
used for increase in number in a particular population while the term diffusion is applied

to processes that involve some mechanism of transfer of information such as the spread
of disease, sales of a new product or the adoption of new technology. For this diffusion
has become a key concept in technology diffusion research and marketing studies.
Hence, both concepts are essentially concerned with the process to represent continuous
and gradual increase of a certain quantity over a period of time e.g. the number of cells
or people who acquire a new product (Berny, 1994). Carrillo and Gonzalez (2002) pointed
out that growth curves are applied to describe how a variable increases over a particular
time interval, until it approaches its saturation level. Due to several external constraints
indefinite growth is not possible consequently, a curve representing growth or diffusion
process characterises S-shape, generally known as sigmoidal curve. The methodology
used in growth curve modelling is based on tendency analysis. Fitting one or other of the
known sigmoidal models to the available historical series (Carrillo & Gonzalez, 2002). The
most common techniques used without doubt are logistic and Gompertz models (Berny,
1994), whereas in the fields of technology diffusion and marketing research Bass diffusion
model, which is a combination of both logistic and modified exponential model (Jain &
Rao, 1990 and Bhargava & Jain, 1991) is widely used.
Innovation diffusion models have been extensively applied to explain the
dissemination of new ideas, practices and services in a social system. Chaos theory, non-
linear techniques and other stochastic methods are the major constituents in the
development of innovation diffusion models, which try to capture the dissemination
phenomenon through interactive mechanism. S-curve is another widely applied and
empirically studied model. It is a combination of both exponential and reverse exponential
growth models. Its applicability in technological and economical perspectives was first
tested with historical technological substitution of railroads, coal, steel and breweries
(Mansfield, 1961). Blackman (1972) applied the innovation dynamics in aircraft jet engine
market and automobiles industries. Nevers (1972) studied the industrial technologies by
fitting this model. Other well known studies include analysis of medical innovations
(Easingwood et al, 1981), energy-efficiency (Teotia and Raju, 1986), telecommunication
innovations (Bewley and Fiebig, 1988), electronics consumer products (Bewley and
Griffiths, 2002), agricultural innovations (McGowan, 1986), diffusion of consumer
durables (Jain, Rai, Sharma and Bhargava, 1991), growth of information and scientific

knowledge (Karmeshu and Pathria, 1980) and social and technical system (Karmeshu,
1998). Diffusion models were also used for forecasting the demand of consumer durables
(Kumar and Rai, 1998) and growth of S&T manpower in India (Rai, Kumar and Madan,
2001).
The diffusion of an innovation may also be considered as an evolutionary process,
where an old technology is replaced by a new technology i.e. technological substitution
takes place. To analyse such cases, besides diffusion model, technology substitution
models are also used. Some of the well-known models in these categories were proposed
by Verhulst (1838), Coleman, Katz and Menzel (1966), Floyd (1968), Fisher and Pry
(1971), Sharif and Kabir (1976), Jeuland (1981), Lilien, Rao and Kalish (1981),
Easingwood, Mahajan and Muller (1981, 1983), Skiadas (1985, 1986), Norton and Bass
(1987), and Rai (1999). Most of the available models are basically extensions and
generalizations of the fundamental models proposed by Verhulst (1838), Gompertz
(1825) and Bass (1969).
Innovation diffusion models are used for forecasting future market demand and
penetration of new product in the market. Diffusion models provide a simplified
mathematical representation of diffusion processes that attempts to incorporate several
influence factors. Diffusion of innovation is affected by a number of ingredients. Rogers
(1983) has identified following seven factors, which affect the diffusion process. These
factors are; a) innovation, b) communication channel, c) social system, d) time, e) change
agents, f) space and g) adopters. Diffusion models generally incorporate one or more of
these factors explicitly and are based on certain basic assumptions which include:
 the potential number of adopters may not be in each case the whole population
under view.
 the way in which innovation is spread may be heterogeneous and not be always
in uniform pattern.
 repeat purchase is not allowed.
 the probability to optimize the innovation is dependent of market perspectives.

Most of the models discussed above exhibit the non-linearity and follow the S-shaped
growth pattern, popularly known as ‘logistic model’. Due to its mathematical simplicity and
relatively wide applicability, the logistic curve itself has many derivatives. It can be
generalized by adding more parameters to fit the empirical data. This model has been
extensively applied in the innovation diffusion and marketing research.
1.3.2 The Logistic Model
The logistic law of growth assumes that a system grows exponentially until an upper
limit inherent in the system is approached, at which point the growth rate slows and
eventually saturates, characterizing S-shaped curve (Stone 1980), which may be
symmetrical or non-symmetrical as shown in Figure 1.1 and Figure 1.2. It is a basic model
in innovation diffusion literature and has broad range of applicability. S-curve has been
found to be very versatile and hence was applied by scientists and researchers to analyze
innumerable cases of various kinds. In general this model can describe development-
taking place in any area including biology, socio-economic and technology. However,
most of its applications found in the literature centered around in the following fields:
 Growth of human population
 Development of organisms
 Spread of new technology, technique and many more.

Figure 1.1: Symmetrical S-curve
Notwithstanding, since its inception the logistic model has undergone for various
criticisms by economists, statisticians and biologists for validity and generalization of its
application. Smith (1952) first tried to justify logistic curve as a model and law of growth.
(Marchetti and Nakicenovic, 1979, Nakicenovic, 1987, Grubler, 1990a, 1990b) have done
a pioneering work to generalize and promote logistic model and they have attempted to
raise the logistic function to the status of a natural law of growth. As a result this curve
has been used as a forecasting tool in several fields and disciplines. In the following
section these issues are discussed in detail.

Figure 1.2: Non-symmetrical S-curve
1.4 Genesis of the Logistic Model
The logistic model is a function of time and used to model natural systems involving
growth with natural resources. This simple model along with differential equation with its
general S-shaped curve is familiar to mathematicians, natural and social scientists alike.
The credit of logistic model goes to the Belgium mathematician Pierre-François Vurhulst
(1838). He argued that in the early stage of growth a population would increase
exponentially until such time when the resources are not crucial. He assumed that rate of
growth is retarded by some function linearly proportionally to the size of the maximum
carrying capacity, and developed a differential equation for symmetrical sigmoidal curve
of growth, which he called logistic curve. Apart from natural phenomenon the non-linear
S-shaped logistic pattern is also observed for technologies and new product diffusion.
The Logistic curve has a long history of development and several parametric
developments have taken place on introducing new parameters and terms to the original
equation. The selection of an S-shaped model is a common step in attempts to model

and forecast the diffusion of innovations. From the innovation diffusion literature on model
selection, forecasting, and the uncertainties associated with forecasts, the following
assumptions have been drawn:
1. No single diffusion model is suitable for all processes.
2. Unconditional forecasts based on a database estimate of a fixed saturation level
form a difficult benchmark to beat.
3. Simpler diffusion models tend to forecast better than more complex ones.
4. Short-term forecasts are good indicators of the appropriateness of diffusion
models.
Practitioners and researchers have discussed the implications of these
assumptions. In 1798, the Reverend Thomas Robert Malthus (1798) did a major work in
his "An Essay on the Principle of Population”, and analyzed the problem of an
exponentially growing population with a limited food supply. He concluded that this
situation led to much human misery and it was termed as Malthusian growth. Later on
Gompertz (1825) (Gompertzian growth) and Verhulst (1838, 1845) (Logistic growth),
modified the Malthus's work. The Gompertz Growth Law (1825) in particular has been
used to very good effect to describe the growth rate of a solid, a vascular tumor at the
population level and has had some practical application in determining chemotherapy
regimes. However, it is known that different tumor with different cell kinetics, many
different sub-populations of cells, can produce the same end result, and this poses a
challenge to the mathematical modeler to help elucidate and explain these observations.
In spite of this all three types of growth law (Malthusian, Gompertzian, and Verhulstian)
have been analyzed and applied extensively to model many diverse biological systems
as well as to model technological growth and diffusion. Carlos (1913) applied for studying
growth of yeast and Pearl (1927, 1930) modeled US population by the curve. Morgan
(1976) made use of the Pearl model to explain herding behavior of African elephants and
Krebs (1985) employed the logistic curve to fit Peruvian anchovies. Further, Skiadas
(1985), Sharif-Kabir (1976), Grübler (1990) and Rai and Ghosal (1986) have extensively

applied the logistic model in different forms to define and describe the pattern of
technological diffusion as a means of population size.
1.4.1 Derivatives of the Logistic model
This section is devoted to analyze logistic model with its derivatives to provide an
overview of what is currently going on in the field-a bringing-up-to-date of the qualitative
and quantitative mathematical modelling being done. The population growth is like a cell
growth which models the interactions of the cells of the body's immune system with tumor
cells, using methods similar to those used in nonlinear differential equations. To develop
and model the logistic function based on the analogy of cell growth or epidemic spread,
integral or partial differential equations are used. This model is more physically based
and takes into account such effects as surface tension between cells, internal pressure
that puts forward the hypothesis that internal stresses on the cells within the tumor cause
rupturing of the cells and hence death. In the following paragraphs some of the well-
known derivatives of the logistic function are being discussed.
Taking N as the population size at time t, the rate of change in the population at
any time t, is proportional to the quantity N(t) and is represented as
)(
)( trN
dttdN 
(1.1)
Separating the variables to solve for N(t) with initial value N(t=0)=0, it gives
rt
eNtN 0
)( 
for t

0 (1.2)
where r is a growth constant. Equation (1.2) exhibits unbounded population growth at an
exponential rate assuming unlimited resources.
Basically logistic growth is the extension of exponential equation and limits the size
of population. Here, if K is the maximum sustainable population size or the carrying
capacity and forms upper bound on the growth size, then (K-N) is the remaining potential
size and
 
K
N
1
is the percentage of K available for population growth. Thus logistic
population growth can be denoted mathematically in the following way:
N
K
N
r
dt
dN 




 1
(1.3)

which on integration gives
 
00
0
)( NeNK
KN
tN rt 

(1.4)
with initial condition N0 = N(t=0),
The above equation may be written in the following form
kt
Ce
K
tN 

1
)(
(1.5)
where
0
0
N
NK
C

here K is referred as carrying capacity or saturation level. In the above equation N(t)
converges towards K at an exponential rate as
.t
It is parameterized by the initial
population size, the initial growth rate is such that the point of inflection of this model
occurs at
2
K
. The logistic model is also known as the Verhulst-Pearl model in the
literature. The general characteristics of the logistic growth are
 N(t) = K, the population will reach its carrying capacity as
.t
 The inflection point of the logistic growth occurs at
2
K
.
 The relative growth rate declines linearly with increasing population size.
Due to its generality and wide applicability several other derivatives of the logistic models
are available in literature. Some well-known modifications and extensions of the logistic
model are given below.
(i) Standard logistic form (1838)





 K
N
Nr
dt
dN 1
(1.6)
(ii) Logistic growth with time lag















 


K
N
rN
dt
dN t
1
(1.7)

(iii) Discrete logistic equation






K
N
NN t
tt 1
1

(1.8)
(iv) Quadratic discrete logistic equation
2
1ttt N
K
NN 









(1.9)
(v) Yule’s growth equation (1925)


t
e
L
y

1
(1.10)
(vi) Ricker logistic equation (1954)











K
N
rNN t
tt 1exp 01
(1.11)
(vii) Von Bertalanffy’ logistic model (1957)














 3
1
3
21K
N
rN
dt
dN
(1.12)
(viii) Pielou logistic growth model (1969)
  
NNK
Ntr
dt
dN 


(1.13)
where
)(t

is stochastic variable.
(ix) Turner et al model (1976)
















  K
N
rN
dt
dN 1
)1(1
(1.14)
(x) Jantsch’s logistic Model (1980)
bt
K
ya
101
(1.15)
From the above equations it may be noticed that in all the above cases upper limit
has been fixed at K. To relax this criteria Ghosal (1972) proposed a dynamic logistic
model where the market potential (K) varies with time. Accordingly, he formulated the rate
of change of demand with the help of differential equation

 
ttt
tNKN
dt
dN 

(1.16)
which on integration yields,
t
tK
t
t
teC
K
N



1
(1.17)
where
tt CandK,
are upgraded from time to time. This model was further extended by
Rai and Ghosal (1986) by introducing lag period (

) in it as follows:
 



 tt
tNK
dt
Ndlog
(1.18)
which in its integral form, under certain assumptions, is written as
 
 


aKt
t
t
tt
eC
aK
N


1
(1.19)
They also proposed another logistic model with reverse Lag as given below
)(
log



 tt
tNK
dtNd
(1.20)
The above differential equation on integration yield
 
 


aKt
t
t
tt
eC
aK
N


1
(1.21)
where constants have their usual meaning.
Meyer (1994) and Meyer & Ausubel (1999) have derived logistic function exhibiting
Bi-logistic growth to analyze growth of pulses. The Bi-logistic growth model is represented
in the following equation:


















)(
)81(
exp1)(
)81(
exp1
)(
2
2
2
1
1
1
mm tt
t
In
k
tt
t
In
k
tN
(1.22)
where k1 and k2 are two level of saturation for ∆t1 and ∆t2 respectively.
Recently, Norden-Raleigh (2002), proposed another kind of logistic model in the following
form, which is known after their name.













2
2
2
01p
t
t
evv
(1.23)
where ‘v’ is earned value and t is the time and these two parameters determine the
function. Apart from the various derivatives discussed here, there are some special cases
of logistic function, which deserve special mention and hence are discussed below
separately.
1.5. Gompertz Model
Benjamin Gompertz (1825), an English mathematician, proposed a model in 1825,
which is typically used for forecasting market penetration. This model is practically useful
for analyzing many empirical cases as it assumes an asymmetric growth pattern. In this
model maturity phase is longer as compared to other phases viz. introduction, take off
and growth. The standard equation of the Gompertz model is
kt
be
LeY


(1.24)
where Y is the forecast variable representing performance, L is the upper asymptote and
b, k are the coefficients.
The choice of a curve depends on the underlying dynamics of the process being
modeled. The slope of the logistic curve is a function of both the present level of the
technology and the difference between the present level and the upper limit. In contrast,
the slope of the Gompertz curve (1825) for large values of Y (

N) is a function only of the
difference between the present value and the upper limit as shown in Figure 1.2. The
Gompertz curve is, more typically applied to forecast the absolute technical performance
such as speed, storage density, temperature, and so on. The value of the upper limit, L,
will have a large impact on the coefficients estimated hence this value must be selected
with care. For example, underestimating L, when using the logistic curve, will result in a
curve that rises too steeply, thereby providing too optimistic a forecast of technological
advance. If the upper limit is overestimated, the curve will rise too slowly and result in an
overly conservative forecast. L must be determined exogenously from the data based on
our theoretical understanding of the physical limitations of the technology.

1.6 Substitution Models
Diffusion of an innovation can also be described as an evolutionary process, where
an old technology is replaced by a new one for solving similar problems or accomplishing
similar objectives. The theory of substitution is based on the competition between old and
new technology or simply it may be put as the process by which the adoption of a new
innovation or product spreads and grows to replace an existing innovation or product. The
interest in technology substitution modelling is not very old as it started in the second half
oh the twentieth century. In biological parlance it is defined in the form of a prey-predator
equation as mathematically formulated by Lotka-Volterra (1925, 1926). From
technological diffusion viewpoint it begans with Mansfield (1961) and Fisher-Pry (1971).
However, over the years a number of substitution models have been proposed to describe
and represent the substitution process over time.
Modelling efforts in this field have generally resulted in deterministic interpretation of
the time dependent aspect of technological substitution. Mansfield (1961) did a pioneering
work in the development of substitution models. He developed a model to describe the
substitution process in industrial sectors. Fisher-Pry (1971), Blackman (1972) and
Coleman et al (1966) have subsequently revised and applied the technology substitution
models. Floyd (1968), Sharif-Kabir (1976) and Jeuland (1981) have further extended
these models to represent a general model for improving forecasting of technology
substitution. In a trend setting approach recently Rai (1999) proposed a class of
substitution models by incorporating time dependent influence factor to get more flexible
forms. Generally substitution models are based on the following three assumptions
(Fisher-Pry 1971), which were further elaborated by Marchetti and Nakicenovic (1979):
 Frequent technological advances are measured in terms of competitive
substitution and grow at logistic rates that tend to saturate the market at any
given time.
 The substitution process leads to competition that connects the period of
growth to its subsequent period of decline and,
 Rate of fractional substitution of new for old is directly proportional to the
remaining fraction of the old to be substituted and declining technologies die

away gradually at logistic rates uninfluenced by competition by new
technologies.
The first assumption implies that a new product at the time of introduction is not well
established than the old product with which it is competing, whereas it has greater impact
on improvement and for reduction in cost. The second assumption articulates that the
substitution has gained a few percent of the available market and has shown economic
viability with increasing volume. The last one allows us to determine saturation behavior
by competition from emerging technologies Linstone and Sahal (1976).
1.6.1 Mathematical Formulations
The simplest formulation for substitution models is based on again logistic differential
equation. If f(t) denotes the fraction affected individuals in the total population, we obtain
the following logistic differential equation
)1( fbf
dt
df 
, (1.25)
which on integration yields
)}(exp{1 1
)( bta
tf 

(1.26)
The above equation may be rewritten in the following form:
bta
tftf
In 








)(1 )(
(1.27)
where a and b are constants. The above model (equation 1.27) is known as the Fisher-
Pry (1971) model for explaining the substitution process. Many scholars have proposed
several modifications of the above model by relaxing its assumptions to make it more
flexible. For example, Blackman (1972) assumed the upper limit of the market to be F
share and proposed the following model:
tba
tfF tf
In 








)(
)(
(1.28)
For F = 1, the Blackman model turns to the Fisher-Pry model. Similarly, earlier Floyd
(1968) also proposed another modified form of the Fisher-Pry (1971) model by adding to

the Fisher-Pry (1971) solution the second, analogous term but with the limit F and without
the natural logarithm. His model may be written in the following form:
 
tba
tfF F
tfF tf
In 










)(
)(
(1.29)
Further, it was extended and modified by Sharif and Kabir (1976) to get more
generalized form of the model for technological substitution. It was based on the
assumptions that the linear combination of the Blackman and Floyd’s models can give
correct estimations. This model is more flexible and can be given in the following
mathematical form:
 
tba
tfF F
tftf
In 












)(1 )(
(1.30)
Defining growth rate






dt
df
as a function of market share and time and introducing
a time dependent influence factor Rai (1999) proposed following four technological
substitution models. These modified models are given below in their integral forms:
(i) Parabolic
   
ffbta
dt
df  1
(1.31)
or
2
)(1 )( ctbta
tftf
In 

(1.32)
(ii) Log-log
 
ff
t
b
dt
df 






1
(1.33)
or
)(
)(1 )( tInba
tftf
In 

(1.34)
(iii) Power
 
ffat
dt
df r 1
(1.35)

or
1
)(1 )( 


r
bta
tftf
In
(1.36)
(iv) Exponential
   
ffbt
ae
dt
df 
1
(1.37)
or
)exp()(
)(1 )( btbac
tftf
In 

(1.38)
Rai argues that earlier models [equations (1.27) to (1.30)] follow a linear trend, whereas
in real life applications we observe different patterns such as exponential, parabolic, etc.
His models are non-linear in nature, and hence can describe the non-linear trends in a
better way.
All the substitution models discussed above deal with two competing technologies
only and do not describe the multi-substitution process. Peterka and Fleck (1978) and
Marchetti and Nakićenović (1979) tried to propose multi-substitution models. Marchetti-
Nakićenović (1979) assumed that each technology passes three distinct phases, as
measured by its market share fi, leading to logistic growth, non-logistic saturation and
finally logistic decline. This multi-substitution model with market share fi is written as:
tba
tf
tf
In ii
i
i








)(1
)(
(1.39)
If fj is the share of saturating innovation or technology, the residual share of the dying
technology will follow the following path
)(1)( tftf ji ij 


. (1.40)
Peterka and Fleck (1978) proposed another model of multivariate substitution
process. They considered and assumed that the capital

i needed to increase the
production of a commodity i by a unit (and is called specific investment), next taking into
account specific production cost ci of the commodity i and the market price for the

commodities. In terms of investment and capital Peterka and Fleck (1978) derived a
sophisticated differential equation describing the substitution of n commodities







 

n
jijji
icfcf
dt
df
1
1

(1.41)
where fi and fj have been used in their usual notation. Kwasnicka, Raman & Kwasnicki
(1983) also attempted to develop substitution models for asexual biological populations.
The limiting point of all the models approached along a sigmoidal kind of curve i.e.
logistic in nature. Basic substitution equation describes competition between the old and
new innovations and equilibrium of the substitution curve is determined by the diffusion
coefficients (Cameron and Metcalfe, 1987) that determine the growth rate. Moreover,
modelling of substitution process tries to study the changes in parameters by comparing
the different substitution process. It means that in broad-spectrum the rate of change
models for better forecast both for the shorter horizon and longer horizon as well (Young,
1993). The models integrate different patterns of technological substitution frequently
made in the forecasting techniques that allow the S-curve to be symmetrical and
nonsymmetrical in nature.
1.7 Discussions and Remarks
The relevance of these functions is significant in innovation diffusion research that
determines future trajectory of an innovation. Most of the functions have been developed
to represent the spread of an innovation among prospective adopters in a social system
in terms of a simple mathematical function of time. These functions such as Gompertz,
logistic and other distribution functions are deterministic in nature. However, modern work
is being done to explain law of diffusion by developing more flexible models. The diffusion
theory is based on the principle of spread of epidemic approach where spread of
information mechanism is a motivating catalyst between non-user and user. These
functions are incompatible for different empirical data sets thereby necessitating the
unification of existing substitution and diffusion models by incorporating such factors that
may relax underlying assumptions. It may help in enhancing the compatibility and permit
flexibility over time of the existing functions in explaining the diffusion process.

Section 2
1.2 Innovation Diffusion: Theoretical Approach
2.1 Introduction
Management of innovation diffusion in the form of new or improved technology is
very crucial for entrepreneurs and policy makers. Theories of innovation diffusion have
generally aimed to explain the future direction of new product diffusion. Consequently,
the area of diffusion of technology has exaggerated and theories developed have
established multidisciplinary applications. Since, process of innovation diffusion is
concerned with the investigation of diffusion trends of new product development, which
could arise from the relationship of various factors such as public concerns, structure of
the potential adopters and nature of new product etc. These factors serve as a catalyst
for growth and spread of innovations in a social system. Therefore, the objectives of
studying technology diffusion are to integrate and investigate the wide array of factors
that affect the diffusion phenomenon, by developing suitable methodology and
mathematical models. Technology diffusion models can provide theoretical as well as
practical tools to assess the pace and speed of technology diffusion.
2.2 What is Innovation Diffusion?
Innovation diffusion is a widely used term to state ‘flow of information, ideas,
knowledge in any given system’. Everett Rogers (1983) has defined diffusion as the
process by which an innovation is communicated through certain channels over time
among the members of a social system. Channels of communication, in which
participants create and share information with one another in order to reach a mutual
understanding. Rogers' definition consists of following four elements that are present in
the diffusion process:
(1) social system - a set of interrelated units that are engaged in joint problem
solving to accomplish a common goal.

(2) communication channels - the means by which messages get from one
individual to another. These are of two types (i) vertical channel for example
mass media and (ii) horizontal channel such as word of mouth.
(3) time – it contains three time factors. These factors are: (i) rate of adoption (ii)
relative time with which an innovation is adopted by an individual or group (iii)
innovation-decision process.
(4) innovation - an idea, practice, or object that is perceived as new by an
individual or other unit of adoption.
2.2.1 History of Innovation Diffusion
The anthropologists normally used the term ‘diffusion’ to study cross-cultural
diffusion. However, in the technological parlance the concept of innovation diffusion is not
very old; it originated in the beginning of the twentieth century with the writings of Tarde
(1903), who propounded the law of imitations. He is considered as the founding father of
diffusion research in the contemporary diffusion research, who plotted the original S-
shaped diffusion curve by presenting theory of imitation. Tarde’s original work on S-
shaped curve is still relevant because most of the innovations follow an S-shaped
adoption pattern (Rogers, 1983). There may be some variance in the slope of the S-curve
because some new innovations diffuse rapidly creating a steep S-curve; other innovations
have a slower rate of adoption, creating a more gradual slope of the S-curve (non-
symmetrical). Hence a study of the rate of diffusion is important area of research to the
researchers of different disciplines including marketing research.
Bryce Ryan and Neal Gross (1940), sociologists by occupation, further studied the
diffusion of innovation process. They studied diffusion of hybrid seed among Iowa farmers
renewing interest in the diffusion of innovation. This recognised diffusion research as a
renewed wave of research, in which they showed that the rate of adoption of the
agricultural innovations followed an S-shaped curve when plotted on a cumulative basis
over time. This rate of adoption curve was similar to that of Tarde’s S-shaped diffusion
curve. In their study of diffusion of hybrid corn Ryan and Gross (1940) classified the

different segments of Iowa farmers in relation to the amount of time it took them to adopt
the innovation.
2.3 Applications of Diffusion Research
Innovation diffusion is concerned worldwide and is the product of accumulation of
knowledge from different fields and disciplines. Innovation diffusion research has wide
spectrum of applications in different areas where it studies effects of innovations and
spread and adoption of a new technology in a social system. The chronological behaviour
of aggregates is driven by individuals’ efforts to innovate and/or make use of others'
innovations. Bearing in mind the reality that innovation diffusion studies are carried out to
study technology diffusion in general, the application of diffusion research has wide gamut
including the diffusion of agricultural technology and ideas, teaching and learning
innovations, medical and health ideas, news events, and consumer durable products etc.
Due to its interdisciplinary nature of applications, innovation diffusion research has major
potential of application in the following areas:
(i) Areas of Innovation Diffusion Research
a. Anthropology
b. Early sociology
c. Sociology
d. Education
e. Public health and medical sociology
f. Communication
g. Marketing research
h. Geography
i. Economics
j. Other traditions
Above-mentioned applications, however, are not exhaustive. A brief descriptive
summary of selected diffusion studies conducted in the past is provided in Table-2.1.

Table 2.1:A brief summary of applications of diffusion models
S.
No.
Study by
Diffusion
model used
Applications
1.
Tarde (1903)
Logistic
Law of imitations
2.
Ryan and Gross(1940)
Logistic
Hybrid seeds
3.
Griliches (1957)
Logistic
Hybrid corns
4.
Bass (1969)
Bass
Consumer durables
5.
Fisher & Pry (1971)
Fisher & Pry
Technology substitution
6.
Blackman Jr.
Blackman
Technology diffusion
7.
Rogers (1985)
Rogers
Innovation diffusion
8.
Olshavasky (1980)
Mansfield
Consumer durables
9.
Kobrin (1985)
Bass
Oil production
10.
Shrivastava et al. (1985)
Bass
Financial investment
11.
Mahajan et al. (1988)
Bass
Adoption process of
technologies
12.
Modis & Debecker
(1988)
Mansfield
Growth patterns
13.
Mansfield
Mansfield
Infrastructure
14.
Rao & Yamada (1988)
Lillien, Rao &
Kalish
Diffusion of drugs
15.
Takada & Jain (1988)
Bass
Consumer products
16.
Grubler (1991)
Technologies diffusion
17.
Meyer (1994)
Bi-logistic
Population dynamics
18.
Gatignon et al. (1989)
Bass
Consumer durable
19.
Ghosal & Rai (1986)
Dynamic logistic
Time lag
20.
Karmeshu (1988, 1998)
Stochastic
Diffusion of information
and consumer durable
21.
Jain, Rai & Bhargav
(1991)
Bass, Fisher-pry
Consumer durable,
technology substitution
22.
Rai (1999)
Rai
Technology substitution

23.
Rai & Kumar (1998,
2002)
Bass, Fisher-Pry
and Rai
Innovation diffusion and
substitution
24
Marchetti (1980, 1989)
Fisher-Pry
Technology substitution
2.4. The Nature and Characteristics of Adoption Process
The basic principle of adoption process is based on rejection and adoption (may
be repetitive) of an innovation at the time of its launch. In general, diffusion models do not
allow repeat purchases by the population during diffusion process. Rogers divided the
total population in two groups namely adopters and non-adopters. He further classified
adopters into five categories viz. (i) innovators, (ii) early adopters, (iii) early majority, (iv)
late majority and (v) laggards as given in Figure 2.1.
Source: http://www.technology-marketing.com/pages/Ideas/adcurve.html

Figure 2.1: Main adoption categories based on innovativeness
Dowling and Walsh (1990) studied the behavioural characteristics of different
adopters categories. These adoption categories have been derived from Bass diffusion
model (1969). Following Mahajan et al (1990) the mathematical expression for all the
adopter’s categories can be derived from Bass model (1969) as follows
)1,0(],][)([  qpNMN
M
q
p
dt
dN
(2.1)
where N is the cumulative number of adopters at time t, p, q are the coefficients of
external and internal influence respectively and M is the total number of adopters. Integral
solution of equation (2.1), may be written as
,
1
1
)(
)( M
e
p
q
e
fMN
tqp
tqp



























(2.2)
with initial conditions
.0)0( M
N
fandtN 
Using equation (2.2), Rai (1994 )
obtained following expressions for the point of inflection T*, the extent of penetration f(T*)
at T* and the rate of diffusion
dt
Tdf )( *
at T*.















 q
p
In
qp
T)( 1
*
(2.3)
and
q
p
Tf ,
22
1
)( *
(2.4)
2
*)(
4
1
)( qp
qdt
Tdf 









(2.5)
Further the time span of diffusion has been divided by him in four parts namely 0 to T1,
T1 to T*, T* to T2 and T2 to

as illustrated in figure 2.2 and written in Table 2.2.
Mathematical expressions for estimating T1 and T2 are given below:

 
 















































q
p
In
qp
T
and
qp
In
qp
T
32
11
2
)32(
)( 1
1
(2.6)
Figure 2.2: Distribution of adopters
Using the values of T1, T2 and T* the percentage of different categories of
adopters in any social system can be calculated with the help of mathematical
expressions listed in Table-2.3 (Rai, 1994). Significant conclusions regarding the
behaviour of adopters in different economies may be drawn with the help of above adopter
categorisation.

Table 2.2: Time span for different adopter categories
Adopter category
Time interval
Mathematical expression for time
interval
Innovators
0 to ∞
-
Early adopters
0 to T1
















qp
In
qp )32(
1
Early majority
T1 to T*
 
32
1








In
qp
Late majority
T* to T2
 
32
1








In
qp
Laggards
T2 to ∞
-
Table 2.3: Size of various adopter categories
Adopter category
% Adopters
Innovators
















p
q
In
q
p1
Early adopters






























































33
1
1
12
1
1
2
1p
q
In
q
p
qp
qp
Early majority









































233
1
12
1In
q
p
q
p
Late majority
 
 

































33
322
1
12
1In
q
p
q
p
Laggards
 
















































)32(
33
1
12
1
1
2
1In
q
p
q
p
q
p
Attempts have also been made to develop diffusion theory from temporal diffusion
of innovations to analyse the time pattern of innovation diffusion. Similarly, to understand
the diffusion process from the economic perspective there are two popular approaches.

First approach is based on demand aspects of new technology called adoption
perspective approach (Hagerstrand, 1967, Robson, 1973). Here it is assumed implicitly
that all adopters have an equal opportunity to adopt the new product or innovation. The
second approach is based on supply aspects of an innovation comprising two
components - market and infrastructure. It shows that initially innovation is diffused
through diffusion agencies and then strategy is executed by each agency to include
adoption among the population. So, demand and supply aspects are two major factors
that influence innovation diffusion.
The diffusion pattern of an innovation may be linked to communicative, economic
and market perspectives as shown in Table-2.4. The Bass theory of innovation diffusion
spells out that rate of diffusion is a linear function of previous adopters while epidemic
theories of growth and diffusion advocate that the diffusion is non-linear in nature. The
key development in diffusion theory and process over the period has been in the area of
understanding the nature of innovations and adopter categories (Figure 2.2). Generally
innovation diffusion theories attempt to integrate market mix variables to develop flexible
diffusion models and enhance their applicability and scope.
Table 2.4: A summary of perspectives on technology diffusion
Perspective
Discipline
Basic
assumption
Unit of
analysis
Factor of
significance
Process
Communicative
perspective
Rural
sociology,
communicatio
n studies,
geography,
marketing
Majority of
individuals
are risk
averse
The
adopter
Adopters’
uncertainty
and perceived
risks/benefits
of adoption
Uncertainty
reduction;
particularly
through
referencing
Economic
history
perspective
Economic
history, public
policy
Individuals
are rational
economic
agents
The new
and old
technology
Declining
costs and
improved
performance
Technological
problem-
solving

of the new
versus old
technology
Development
perspective
Economics,
development
studies,
agricultural
economics
Unequal
distribution
resources in
society
The
adopter
Adopters’
relative
purchasing
power
Access to
resources,
particularly
money and
credit
Market
infrastructure
perspective
Geography
Opportunity
to adopt is
unequal
The
diffusion
agency
Availability of
the new
technology
Diffusion
agency
establishment
and actions
Source: Adapted from Miller, D. and Garnsey, E, Entrepreneurs and Technology
Diffusion: How Diffusion Research can benefit from a Greater Understanding of
Entrepreneurship, Technology in Society Vol. 22, 2000, pp. 449.
2.5 Parameter Estimation
A number of techniques are reported in diffusion literature to estimate parameters.
Particularly following four techniques have been recommended (Mahajan and Wind,
1986, and Rai, 1994) and are widely employed to estimate the model parameters. These
techniques are briefly discussed below.
(i) Ordinary Least Square (OLS) method,
(ii) Non-linear Least Square (NLS) method,
(iii) Maximum Likelihood Estimate (MSE), and
(iv) Algebraic Estimate (AE).
Out of these methods, Non-linear Least Square (NLS) method has been found to be
more accurate as compared to others (Bass 1969, Mahajan and Peterson 1985, Rai
1994). However, this method has the problem of local and global minima. Hence to
overcome this weakness, a suitable starting value in this procedure is required. For this
purpose Ordinary Least Square (OLS) method is used first, as suggested by Bass (1969),

and Jain et al (1991) and then these parameter estimates are used as starting values in
Non-linear Least Square (NLS) method and final estimates are obtained (Mahajan and
Peterson, 1990).
For innovation diffusion studies Bass model has significant implications for
forecasting, decision about new product viability and product launch performance
tracking. Some of the key behavioural and mathematical assumptions in the Bass
diffusion model, as described by Bass (1969), Nevers (1972), Dodds (1973) & Rai (1994)
are as follows:
 Diffusion process is binary (consumer either adopts, or waits to adopt)
 Constant maximum potential number of buyers are (M)
 Eventually, M will buy the product
 No repeat purchase, or replacement purchase is allowed
 The impact of the word-of-mouth is independent of adoption time
 The marketing strategies supporting the innovation are not explicitly included
2.6 Conclusions
In this chapter a variety of phenomenon involved in innovation diffusion have been
presented. Several issues pertaining to different aspects of innovation diffusion have
been discussed and several other related determinants of diffusion process have also
been analysed. The findings that at continuous time, the instantaneous rate of adoption
for individuals depends both on their intrinsic rate of adoption and their infections rate of
adoption are important to understand adoption of a new product. The first inference shows
the linearity on relevant attributes of the individual and the second is assumed to depend
linearity on factors affecting the individual’s likelihood of contacts with previous adopters.
From the above discussions it may be noticed that various theories have been put
forwarded to understand and describe innovation diffusion process but undoubtedly there
is still lack of a viable diffusion theory. The important aspect of diffusion theory is the bio-
economic interactions between costs and prices on one hand and biological rates and
carrying capacity on the other hand. Moreover, some questions relating to the factors that
affect the diffusion process are yet to be analysed. These factors include repeat

purchasing, nature of innovation, adopter characteristics and socio-economic issues.
Under these circumstances an alternative approach is proposed here to integrate
diffusion theory, which may further enhance the foundation of diffusion research in a
broader perspective.
Section 3
1.3 Models of Technology Diffusion
3.1 Introduction
In innovation diffusion theory, growth and diffusion are used for different reasons.
Generally the term growth is used for augmentation in number in a particular population
while the term diffusion is applied to processes that involve some mechanism of transfer
of information such as the spread of disease, sales of a new product or the adoption of
new technology. For example, the height of any individual or any plant / tree is a function
of time and similarly the rate of transfer or diffusion of an innovation / new product is
proportional to the potential adopters and remaining adopters. For that reason, the
mathematical methodology of innovation diffusion is employed to analyse and describe
the statistical data with the objective how the consequential information can be interpreted
and predicted. Mathematical methodology is evolved in term of mathematical diffusion
models. Innovation diffusion models are thus abstract forms of mathematical formulation,
which, describe the pattern of new product diffusion. The models are formulated and
developed to measure the level of spread of innovation among the population of a social
system.
Several studies in different areas and contexts have been carried out to give an
integrated theory of diffusion and to develop diffusion models. Innovation diffusion
modeling is a time dependent process, based on binary notion; researchers, however,
have raised many questions about the binary nature of the adoption process. As a result
diffusion models have a few limitations in capturing the dynamics of innovation diffusion
(Grübler, 1996). Further, diffusion models may have several variables that can be time
dependent and / or density dependent. Diffusion models accordingly help to predict the
potential of the new products by giving the theoretical explanation of trajectory of new

technology. These models have their pedigree in biology from the study of epidemic
spread.
Diffusion of innovation generally tends to follow sigmoid pattern as stated before.
Recently Sawhney and Eliashberg (1996) proposed a parsimonious model, which shows
various patterns. Similarly Moe and Fader (1998) proposed a joint segmental model of
consumer products. Furukawa, Kato and Yamada (2002) proposed a model describing
adoption of new products that can take different pattern from S-shaped to J-shaped in a
continuous manner. In addition technology diffusion modelling ranges on a scale from
technological determinism to technological dynamics. Independence and continuity are
the key issues in the theoretical debate between determinists and dynamic modelling.
Determinist view of diffusion of innovation is seen as an autonomous force where
parameters are static. Determinists also view the expansion of technology as
discontinuous. They assume that technological growth is not a gradual, evolutionary
process, but it is a series of revolutionary forward. While dynamic innovation diffusion
theory perceive technological growth as the ultimate culmination of a long history of slow,
gradual adoption of an innovation.
Researchers have proposed and suggested a number of innovation diffusion
models to study new product diffusion that explains the nature of diffusion process.
Despite the number of factors that may affect new product acceptance, there have been
a large number of mathematical models over the last five decades, which have been
designed to characterise certain regularities in the diffusion patterns, product penetration
curves etc. Diffusion models have been applied successfully for industrial and market
penetration forecasting.
3.2 A framework of diffusion modelling
There are many factors to consider when we consider for describing innovation
diffusion. For example, one could consider the following factors, which may be
incorporated in innovation diffusion theory.
1. What type of forecasts is needed (for example, market share, sales, marketing
costs)?

2. What data are relevant and available?
3. What type of forecast horizon is appropriate (for example, short-term, medium-
term, long-term and current status?
4. When is the forecast needed?
5. Who will use the forecast and in what manner?
6. What situation exists (for example, stage of the product life cycle, and state of the
economy etc.)?
This is not the comprehensive listing and certainly there are other relevant factors, but
these serve to illustrate the complexity involved. All these factors are very significant in
innovation diffusion modelling. Innovation diffusion models are based on adoption
analogy of an innovation.
3.3 The Basic Diffusion Models
The concept of innovation diffusion refers to the pattern of new product adoption
by a population over time and it is postulated that at each point of time, the instantaneous
rate of adoption of a product depends upon their intrinsic rate of adoption and their
infectious rate of adoption. The first is assumed to depend linearly on relevant attributes
of individual and the second to depend linearly on factors affecting the individual’s
contacts with previous adopters. These propositions have been extensively discussed by
Mahajan and Peterson (1985) and can be expressed in the form of following equation:
)]()[(
)( tNNtg
dttdN 
(3.1)
with the boundary conditions
N(t = t0) = N0
where
N(t) = cumulative number of adopters at time t


t
t
tndttntN
0
)(,)()({
being the noncumulative number of adopters at time}
N
total number of potential adopters at time t,

dttdN )(
rate of diffusion at time t,
g(t) = coefficient of diffusion, and

N0 = cumulative number of adopters at time t0.
The above model shows that the rate of diffusion of an innovation at any time t is a
function of the difference between the total number of potential adopters existing at that
time and the number of previous adopters at that time
)]([ tNN 
. The nature of the
relationship between the rate of diffusion and the number of potential adopters existing at
t,
)]([ tNN 
, is controlled by g(t), the coefficient of diffusion. The value of g(t) which is
interpreted as the probability of an adoption at time t, depends upon the nature of
innovation and communication channels etc. In such a condition
).(tg
)]([ tNN 
stands
for the expected number of adopters at time t, n(t). Here n(t), can be viewed as the new
adopters at time t which have been transferred from potential adopters. So, the g(t) can
also be phrased as conversion coefficient.
As a conversion coefficient, following two approaches have been used to define g(t):
(i) g(t) as a function of time
)()( tftg 
and
(ii) g(t) as a function of the number of previous adopters.
)(tg
f {N(t)}
Generally g(t) is defined as function of N(t) in the following form;
...)()()( 2 tcNtbNatg
(3.2)
To interpret the parameters and for analytical view, g(t) can be taken as:
)()(
),()(
,)(
tbNatg
ortbNtg
atg



where a and b are model parameters.
3.3.1 External Influence Model
Taking g(t) = a, the basic diffusion model can be written as,
)]([
)( tNNa
dttdN 
(3.3)

the constant ‘a’ serves as a change agent, which are influenced by vertical
communication channels. Integrating equation (3.3), the cumulative number of adopters
at time t, N(t) can be obtained as:
]1[)( at
eNtN 

(3.4)
or
at
N
tN
In 












)(
1{
1
(3.5)
with initial conditions
.0)0( tN
The above model is known as external influence diffusion model. Plotting time
against cumulative adopters the general shape of the curve follows the path as illustrated
in Figure 3.1. It shows that cumulative number increases but at a decreasing rate i. e. the
model represents the decaying exponential diffusion curve. The model is based on the
assumption that the diffusion rate at time ‘t’ is the function of only the remaining potential
number of adopters in the social system at time ‘t’ and is not the function of prior adopters.
Figure 3.1: External-influence diffusion curve

It can be argued at this juncture that the external-influence diffusion model is
appropriate when potential adopters are isolated. Using this model Coleman et al (1966)
studied the diffusion of medical innovations and Hamblin et al (1973) applied this model
to analyse labour strikes and political assassinations in developing nations. In both the
cases there was a lack of interaction among the members of the social system and the
innovations were separated with some exceptions.
3.3.2 Internal Influence Diffusion Model
The internal influence diffusion model is based on the assumption that there is a
continuos interaction among adopters and non-adopters of an innovation. The conjecture
is well incorporated in the following differential equation signifying the internal influence
diffusion model.
)]()[(
)( tNNtbN
dttdN 
(3.6)
In internal influence diffusion model the diffusion rate is a function of both prior
adopters and potential adopters in a system. Interaction among the adopters also
represents the imitating behaviour of the adopters. On integrating above equation (3.6),
we get the cumulative number of adopters as follow:
)]([
0
00
)(
1
)(
ttNb
e
N
NN
N
tN




(3.7)
or
)(
)(
)( 0
0
0ttNb
NN
N
In
tNN
tN
In 

















(3.8)
with initial conditions
 
.
00 NttN 
In the above equation ‘b’ is defined as a coefficient of internal influence or a
coefficient of imitation and implies that there is a continuos interpersonal interaction
between prior adopters and potential adopters represented by
)}().(({ tNNtN 
. Here it

may be noticed that equation (3.8), corresponds to a logistic curve as shown in Figure
3.2.
Figure 3.2: Internal-influence diffusion curve
The first application of this model can be cited from the work of Griliches (1957).
He studied the diffusion of hybrid corn among farmers in USA. Other examples can be
mentioned from the studies of Mnasfield (1961), who conducted study on diffusion of
infrastructures. Gray (1973), has applied the model for diffusion study of public policy
innovations in USA. Of late, Rai (1996) and Jain & Rai (1988) have conducted several
innovation diffusion studies applying this model in Indian economy.
3.3.3 Mixed influence Diffusion Model
Mixed influence diffusion model, which is defined by taking
)()( tbNatg 
, takes
into account both external influence as well as internal influence. It is represented by
following differential equation:
)](}[({
)( tNNtbNa
dttdN 
(3.9)
The integral form of the model can be expressed in the following form:

)])(([
0
0
)])(([
0
0
0
0
)(
)(
1
)(
)(
)(
ttNba
ttNba
e
bNa
NNb
e
bNa
NNa
N
tN









(3.10)
where
., 00 ttatNN 
.
The earliest application of mixed-influence diffusion model is found in the studies
of Bass (1969), who applied this model to forecast the sales of consumer durables.
Mahajan and Muller (1979) and Mahajan and Peterson (1985) have reviewed it
extensively. Recently this model has been analysed at length and was successfully
applied for sales forecasting of different consumables in different economies by Jain et al
(1991) and Rai and Kumar (2002).
More than 4000 research papers (Goldenberg et al 2000) have been published till
now describing and explaining the innovation diffusion process. Besides applications,
many researchers proposed new diffusion models by relaxing assumptions or
incorporating additional parameters to extend the domain of the existing models. In this
series some of the models deserve special mention and are accordingly being briefly
described below.
(1) Mansfield Model
Mansfield (1961) had proposed a diffusion model to describe the diffusion patterns
by relating the diffusion rate to the profitability of adopting an innovation. He assumed
that the innovation is a function of (i) the proportion of firms that have already introduced
the innovation, (ii) the profitability of the innovation relative to possible investment and (iii)
the size of the investment. This relationship of the variables can be put mathematically
as:
   
.
,.....,, 








ijijij
ij
iij Sn
tm
ft


(3.11)
where mij, nij, πij, and Sij represent number of firms, total number of firms, possible
investment and investment size respectively. The above equation on approximation yields

 
1
(
1)( 

 tl
ijij ijij
entm

(3.12)
The above equation represents the logistic function in Mansfield notation where φij is the
growth rate.
(2) Fourt and Woodlock Model
Fourt and Woodlock (1960) proposed one of the most primitive new product growth
models for market penetration. They assumed that the conditional probability of adopting
a new product over time is constant. The proposed model by Fourt and Woodlock (1960)
may be written in the following form:
)](1[)(
)(1 )(
tFatf
or
a
tF
tf



(3.13)
where a is the growth rate, f(t) is the density function of potential adopters and F(t) is its
cumulative distribution function.
(3) Bass Diffusion Model
The available literature reveals the fact that Bass (1969) diffusion model is
extensively used for forecasting purposes. The model is generally written in the following
mathematical form:
)10,1,0(
),()(









qpandqp
NMN
M
q
p
dt
dN
(3.14)
The Bass model contains two different terms, the first term











N
M
q
p
,
denotes the diffusion effects and the second term
 
NM 
denotes the saturation
effects. On integrating equation (3.14), we get:




















))(exp(1
))(exp(1
)( tqp
p
qtqp
MtN
(3.15)
where p, q are the coefficient of external and internal influence and M is the total market
potential and N stands for the cumulative number of adopters at time t.
Many attempts have been made (Mahajan and Paterson 1978, Majhajan et al.
1979, Jeuland 1981, Easingwood et al. 1983) to modify the Bass model. These attempts
include more complex marketing issues like supply restrictions (Jain et al. 1991),
substitutes and successive product generations (Paterson and Mahajan 1978; Eliasberg
and Jeuland 1986; Norton and Bass 1987 & Bayus 1993), pricing and income effects
(Robinson and Lakhani 1975, Rao and Bass 1985 and Kamakura and Balasubramanian
1988), brand level diffusion (Parker and Gatignon 1994), and price expectations
(Narasimhan 1989). Quite a few studies have been conducted by incorporating other
marketing decision variables (Bass and Krishnan 1992). All these studies were to test
research hypothesis providing valuable information on diffusion process. To make the
model more flexible, some researchers have proposed and suggested that parameters
should reflect stochastic processes (Lilien et al 1981; Eliasberg and Chatterjee 1986).
However these approaches require more parameters that make the model sophisticated
and cumbersome.
(4) Other Well-known Models
Several studies have been conducted to explain and support the findings about
the diffusion process. Consequently, the literature dealing with diffusion, in the area of
new product diffusion, marketing research, technological and economic development,
has grown-up enormously. Linstone & Sahal (1976), Hurter & Rubinstein (1978), Mahajan
& Paterson (1985), Karmeshu (1988, 1998) and Rai & Ghosal (1986) have extensively
reviewed and studied these models. Several other Studies (Blackman 1974, Simon, 1975,
Romeo, 1977, and Jain et al, 1991) had been made to illustrate diffusion process
assuming different hypothesis. Further, Metcalfe (1970), Nasbeth & Ray (1974) and

Davies (1980), also conducted to realize the influencing role of profitability, size of
investment etc. Here some of the well known mathematical models are listed below for
the benefit of the readers.
(i) Floyd (1962)
 
2
)(
)( tNM
MtN
b
dt
dN 







(3.16)
(ii) Nelder (1962)
 
h
tNM
MtN
b
dt
dN )(
)( 







(3.17)
(iii) Young and Ord (1985)












)(
)( tN
InM
MtN
b
dt
dN
(3.18)
(iv) Teotia and Raju (1986)
 
)(
)( tNM
MtN
t
b
dt
dN 













(3.19)
3.4 Mathematical Properties of Diffusion Models
Properties and behaviour of some of the well-known diffusion models are listed in
Table 3.1.
Table 3.1: Properties of diffusion models
Model
Differential form
Integral form
P. I.
Sym.
Application
Gompertz
(1825)






F
qFIn 1
 
qtce
e
0.37
NS
Agricultural/consumer
products
Mansfield
(1961)
 
FqF 1
 
qtc
e
11
0.5
S
Industrial innovations

Bass (1969)
  
FqFp 1
 
 
tqp
tqp
e
p
q
e




1
1
0.0-0.5
NS
Consumer durable,
industrial innovations
Floyd (1962)
 
2
1FqF 
0.33
NS
Industrial innovations
Fisher and
Pry (1971)
qf(1-f)
 
qtc
e
11
0.5
S
Consumer durables
Sharif and
Kabir (1976)
 
 



11 12
FFqF
0.33-
0.5
NS/S
Industrial innovations
Jeuland
(1981)
  


 1
1FqFp
0.0-0.5
NS/S
Consumer durables
NSRL (1981)
 
FqF 1

0.0-1.0
NS/S
Medical innovations
NUI (1983)
 
 
FqFp 1

0.0-1.0
NS/S
Retail services
consumer durables
Bertalanffy
(1957)
 





1
1
1FF
q
 
 



1
1
1qtc
e
0.0-1.0
NS/S
Teotia and
Raju (1986)
 
FF
t
q1
0.0-0.5
NS
Energy innovations
Bewley and
Fiebig (1988)
 
 


k
ktq1
1
 
 

,
11
qtc
e

0.0-1.0
NS/S
IT related innovations
Rai (1998)
(i)
   
ffbta 1
(ii)
 
ff
t
b





1
(iii)
 
ffatr1
(iv)
   
ffbt
ae 1
(i)
 
2
1
1
ctbta
e

(ii)
  
Intba
e
11
(iii)
 
1
1
1


r
bta
e
(iv)
 
 
 
bt
e
b
a
c
e
1
1
NS
NS
NS
NS
Consumer durables
3.5 Comments and conclusions
From the preceding deliberations it may be noticed that not only diffusion modelling
has travelled a long way since its beginning but research concerning diffusion of

innovation process has increased significantly during the past several decades due to its'
interdisciplinary approach and usefulness. Of late the area of diffusion research has also
emerged in the field of marketing research and forecasting. During the course of the
twentieth century the diffusion of innovation theory has proven to be versatile and
universal with multidisciplinary approaches, which is more relevant to the contemporary
diffusion studies. Presently several diffusion models exist in the literature and their
properties have been extensively investigated. However most of the available diffusion
models are deterministic in nature, which do not take into account different stages of
diffusion. This gap needs to be bridged by incorporating appropriate change agents and
relaxing some of the assumptions for example fixed upper ceiling and dynamic parameter
values. Moreover, existing models consider diffusion as a binary process and do not allow
repeat purchasing. Under such conditions it has been observed that in many cases
diffusion models fail to work, hence these issues need to be addressed to make diffusion
models more reliable and robust. Besides many issues such as linearity, parameter
estimations, etc. are yet to be tackled. Therefore an attempt is made here to introduce an
alternative approach to diffusion modelling.
Section 4
1.4 Technology Substitution Models
4.1 Introduction
Innovation diffusion is very important in accounting for business cycles (Helpman
and Trajtenberg, 1996) and market penetration. One aspect of innovation diffusion is
based on the proposition of aggregate adoption, while other aspect is based on the
competitive advantage of budding technologies generally known as substitution of
technologies. In some of the literature, substitution of technologies is generated by
changes in technology progress. This is because economic agents divert more resources
to learning and adoption of new technology to implement new technology after an
innovation occurs which in turn characterises competitive behaviour of the emerging
technologies. This implies that an innovation in the near future will diffuse in terms of
technology substitution, which leads to recessions in the present and subsequent
modification and expansion in future. Different types of diffusion processes can generate

dynamics of aggregate variables, which include business cycle asymmetries sprouting
technological substitution. It means that diffusion of an innovation can also be envisaged
in terms of technological competition i.e. substitution of a new technique for the old one.
The succession of technology depends upon difference in initial fraction between
adoptive agents. The difference may be in stipulations of either superior or inferior
products. Many examples may be cited of technological substitutions such as
automobiles vs. horses, TV vs. radio, colour TV vs. black & white TV, VCRs vs. movie
theatres, ballpoint pens vs. fountain pens, mobiles vs. fixed telephones, digital watches
vs. analog watches, synthetic clothes vs. cotton clothes, detergents vs. soap, vacuum
tubes vs. solid state components, distance learning vs. universities, cyber brokers vs.
brokers, online advertising vs. offline advertising, and online grocers vs. supermarkets
etc. The above list is not exhaustive, however, evidently shows the importance of
technological substitution and application of substitution models in technology diffusion
and forecasting.
Substitution of an innovation explains the main feature of the diffusion process in
two ways. (i) An innovation rarely remains unchanged during the course of its adoption.
The adoption rate of an innovation depends upon the improvement of its functional
aspects for new applications. These improvements and changes can be quantitative or
qualitative and are interconnected and (ii) Adoption of a new innovation is often
associated to the nature and importance of some analogous and comparable older
innovation in use. It connotes, if comparable innovations exist they generally tend to affect
the other and hence as a result substitution takes place as shown in Figure 4.1.

Figure 4.1: Process of technology substitution
4.2 Nature of competition
The idea of competition and substitution is initially taken from the interaction of two
species in biological context such as prey-predator process, discussed and analysed by
Lotka (1925), Volterra (1926). Further several scholars including Armstrong and
McGehee (1980), Ayala, Gilpin and Ehrenfeld (1973), Connell (1961), Bremermannn and
Thiene (1989) and Bulmer (1976) studied thoroughly the process of competition and
predator-prey process. The competition can occur between two technologies or multiple
technologies resulting in the form of substitution. A substantial literature concerned with
technological substitution exists within the domain of several products and technologies.
The process of simple one to one substitution is shown in Figure 4.1.
Substitution process, as a result, examines the dynamic sales behaviour of
different successive generations of products using varying forms of a multigenerational
diffusion model. The multigenerational diffusion models encompass the elements of
diffusion and substitution modelling, assuming that a new product will diffuse through a
population of potential consumers over time and that market share competition will be
introduced with successive generations of the new products launched. Substitution

effects differentially affect market share between various products though diffusion of a
product may also act as complements rather than as substitutes to one another in some
cases.
4.3 Multiple technology substitution
The saturation phase of the existing technology represents a barrier to further
adoption of technology. Consequently, new phase starts in the process of technology
diffusion. The evolution of parallel technologies may result in multiple substitutions
process. This process of substitution in terms of aggregate adoption and in terms of
multiple substitutions is illustrated in Figures 4.2 and 4.3 respectively. Competition among
the technologies is attracted into the market when products with similar characteristics
become more feasible and profitable.
Figure 4.2: Substitution in terms of capability enhancement

Figure 4.3: multiple substitutions
4.4 Modelling Technological Substitution
Technology substitution models consider diffusion of technology as a process of
substitution where an existing technology or innovation is replaced by a new technology.
Further, substitution models attempt to model time-dependent aspects of the innovation,
where introduction of a new product influences the sale of the previous generation. The
new technology often provides a diverse set of profit and a different set of end-use
applications. In such circumstances forecasting sales of a new product is very complex
as products have highly positive or negative influence due to substitution and competition
effect (Jun et al, 2002).
The idea of competition in technology diffusion is originally taken from the
interaction of two species in biological sciences. Lotka (1925) and Volterra (1926)
developed a mathematical model illustrating the predator-prey relationship. The model is
well known as Lotka-Volterra model. Lotka-Volterra model given in equation (4.1) is based
on the competitive environment of predator-prey relationship. Here it may be noticed that
increase in the size of either population tends to influence the growth rate of the other
population.










mPbHP
dt
dP
aHPrH
dt
dH
(4.1)
It has two variables (P, H) and parameters ‘r’, ‘a’, ‘b’ and ‘m’ that are defined below:
H = density of prey
P = density of predators
r = intrinsic rate of prey population increase
a = predation rate coefficient
b = reproduction rate of predators per 1 prey eaten
m = predator mortality rate
Two types of technological substitution models are found in the literature. First,
substitution models focusing on the spatial aspect of the substitution process and second,
substitution models focusing on the temporal aspect of the substitution process. These
models are complementary to each other. Biologists describing the morphological
changes in organisms (Reeve & Huxley, 1945) proposed originally the spatial category of
models. These models have been hardly applied in the diffusion studies. The temporal
type of model was first proposed by Fisher-Pry (1971) with several applications. Some of
the well-known substitution models available in the literature are briefly discussed below:
(i) Fisher-Pry Model
Fisher-Pry Model may be written in the following mathematical form:
 
fk
dt
df
f 12
1
(4.2)
where f is the fractional substitution and 2k is the annual growth rate. On integrating
equation (4.2), the following expression is obtained.
ckt
f
f
In 
2
1
(4.3)

where c is the constant of integration. Take over time, which may be defined as the time
taken by the new technology to penetrate from 10% to 90% of the market share, can be
estimated as:
k
ttt 2.2
1.09.0 
.
It shows that 'take over time' is inversely proportional to the annual rate of growth
(k) and represents internal influence. This model has been widely studied and applied to
explain the substitution process and is based on the following assumptions:
1. Many technological advances can be considered as competitive substitutions of
one method of satisfying a need for another.
2. If a substitution has progressed as far as a few percent, it will proceed to
competition.
3. The fractional rate of fractional substitution of new for old is proportional to the
remaining amount of the old technology left to be substituted.
(ii) Coleman Model (1957):
Coleman proposed a technology substitution model that incorporates external
influence. This model describes that social process is intervened between the initial trials
of a new innovation by a few and its final adoption by virtually the whole community.
Accordingly Coleman et al proposed the following mathematical expression for
substitution model.
 
fa
dt
df  1
(4.4)
or
cat
f
In 
11
(4.5)
(iii) Blackman model (1972):
By introducing the upper limit F, he proposed a substitution model, which
incorporates measures of uncertainty. The model can be derived in the following way:
Let f(t) = market share of new technology at time t and F = upper limit, then the
incremental change in the market share, ∆f, between time t and t+1 may be expressed
as

   
 
tfF tftf
tf 

 1
)(
, (4.6)
where
 
 
IRFtftf ,,/)()(


. (4.7)
Here
F
tf )(
is the figure of merit and
)(tf
is a function of profit R and investment I. Using
Taylor expansion and considering up to second order we obtain the following expression.
)()1( tftf
dt
df 
=
 
 





 F
tf
tfF )(
)(

(4.8)
where

is the sum of all terms not containing
F
tf )(
and

is the sum of all terms
involving
F
tf )(
in the Taylor’s expansion. On simplification of equation (4.8) following
solution is obtained.
  
 
 
 
tC
tCF
tf






exp1
exp
)(
(4.9)
where C is a constant of integration. By applying the boundary conditions, the general
solution of the above equation may be obtained in the following form,
,bta
fF f
In 

(4.10)
where a and b are constants.
(iv) Floyd Model (1968)
Floyd (1968) proposed a substitution model, which exhibits a slower rate of
technology development. The model is expressed as
bta
fF F
fF f
In 



, (4.11)
where symbols have their usual meaning.
Here it may be mentioned that most of the substitution models have been derived
by adjusting the conversion factor. For example defining substitution rate as a function of
the share of new and old technologies, we can write it as

   
fGfA
dt
df  1
. (4.12)
Here A(f) is the conversion factor, which has been defined in different ways to derive other
well-known substitution models (Rai, 1999). The mechanism is demonstrated below:
A(f) Model
a Coleman
bf Fisher-Pry
 
 
efbf  11
Sharif-Kabir
 
 
e
feb 1
Von-Bertalanffy
bf Mansfield, Floyd
d
af
Easingwood-Mahajan-Muller
bfa
Jeuland
 

bfa
NUI
af
Nedler
g
gf
b

1
Generalized
 
eff
af
1
GRM-I
effe af

GRM-II








bf
r
ae
af
fb
bfa
Rai
 
 












bff
a
rfbfa
1
2
Jain, et al

Mathematical properties of the models, which include point of inflection and symmetry,
may be analysed in detail. Point of inflection occurs when substitution rate is maximum
while symmetry specifies the shape of the contour about y-axis.
4.5 Conclusions
Technology substitution models are methodologies that provide an insight into
competitive advantages in the adoption of a new product. Although there exists several
substitution models, which are widely applied to forecast market penetration of different
technologies, but all are linear in nature. In the past many efforts have been made to
extend or modify these models to improve their applicability. However, it is felt (Rai, 1999)
that there is a need to integrate time dependant element also to improve their capability
for analysing the complex dynamics of the substitution process. Further, the propensity
of adoption of any innovation is linked with heterogeneity of potential adopters and
involves behaviour of the potential adopters. In addition the takeover time of complete
substitution varies from technology to technology. Thus we find that many issues related
to technology substitution still remain unaddressed and call for relaxing the existing
assumptions in general and incorporating time dependent variables in particular.
Therefore, a different approach is necessary to explain the problem of technological
substitution, which may extend the flexibility of the traditional substitution models.
Induced by the limitations of existing models, an attempt is made here to elaborate
assumptions and principles of substitution models and the forces driving the substitution
processes. Some of the issues, as mentioned earlier, have been worked out and new
models for technology diffusion and substitution are proposed in the next chapters. The
proposed models have been tested and validated for their applicability by analysing a
number of cases pertaining to technology diffusion and substitution. On comparing the
results of these newly proposed models with those of well-established models in the fields
it may be concluded that these models are comparatively more flexible and can be applied
to analyse empirical data with a better fit over the past data.

ii
References are given in the text.