ISSN:2249-877X Vol. 6 Issue 5 May 2016 Impact Factor: SJIF 2013=4.748
P ublis he d b y: S out h A s ia n A c ade m ic R es e arc h J our nals
SAJMMR:
South Asian Journal of
Marketing & Management
Research
( A D o u b le B l i n d R e fe r e e d & R e v ie we d I nt e r na t io na l J o ur na l)
A REVIEW ON THE INNOVATION DIFFUSION MODELS FROM 19602010 WITH THEIR APPLICATION ON DRAM INDUSTRY
Geet Kalra*; Nirali Kansara**
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*
Birla Institute of Technology
and Science, Pilani.
**
Birla Institute of Technology
and Science, Pilani.
______________________________________________________________________________
ABSTRACT
The cases of Television sets generations(LCD, LED), computer generations and many more, the
penetration of such products in their target market follows a nonlinear trend with time. It can be
seen as function of set of numerous variables affecting price, time, placement and publicity.The
modelling of diffusion of innovations has remained the hot topic for market researchers since
1960’s when Fourt and Woodlock(1960) gave their very first, primitive market research tool on
the volume of consumer purchases per year. Bass model has been considered as the most basic
model which was later modified to include several other factors in the basic model. In this paper
different cases of models have also been considered which includes diffusion of single innovation
in a market, multiple market and diffusion across generations. Different type of S-shaped
functions have also been mentioned keeping into consideration the S-shaped nature of the
cumulative adoption function of an innovation. Four different S-shaped functions which includes
Gompertz model, Log-logistic function and Generalized bass model have been used to find the
best fit curve in the case of DRAM industry. Generalized Bass model has been found to be the
best among the other models with high R-squared values across all the generations of the DRAM
products.
KEYWORDS: Innovation, Diffusion, Bass model, DRAM, Gompertz, Log-logistic, Pricing
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INTRODUCTION
At the end of 2014, almost 5 billion people are using mobile phones. Being invented in
Scandinavian countries, this innovation is said to have its first purchasers located in these
countries only. In many of the developed countries, the mobile phones have penetrated to such
an extent that every unit of the population carries at least one mobile phone containing
sometimes more than one sim cards. The case of mobile phones is not an exception. We have
seen the cases of Television sets generations(LCD, LED), computer generations and many more.
The penetration of such products in their target market follows a nonlinear trend with time. It can
be seen as function of set of numerous variables affecting price, time, placement and publicity.
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The modelling of diffusion of innovations has been a topic of practical and academic interest
since 1960‟s when Fourt and Woodlock(1960) gave their very first, primitive market research
tool on the volume of consumer purchases per year based on the households which make trial
purchases of a product and households which make a repeat purchase within a first year.
The model given by the authors is expressed as
𝑉 = (𝐻𝐻 𝑋 𝑇𝑅 𝑋 𝑇𝑈) + (𝐻𝐻 𝑋 𝑇𝑅 𝑋 𝑀𝑅 𝑋 𝑅𝑅 𝑋 𝑅𝑈)
HH : Total number of households MR : Measured Repeat
RR : Repeat RepeatersTR : Trial rate
TU : Trial Units
RU : Repeating Units
This model marked the beginning of the more complex models of innovation, revisiting
and revamping many of them in the subsequent years. The knowledge of Innovation Diffusion is
important for a firm to decide various factors and determinants related to the product release. The
previous models were made keeping in mind the assumption of homogeneous target markets.
However, with the course time new models were developed that revisited each of these
assumptions along with the inclusion of parameters which directly or indirectly affect the
diffusion process.
Consumer Psychology
Any innovation can be divided into two categories.
1) Forced innovation – It is a case where the innovation is forced upon the consumers by a
third party. For example –Upgradation of windows version in a school laboratory from
Windows 8 to Windows 10.
2) Convenient adoption of an innovation – It is a case where a consumer deliberately adopt
an innovation without any force from anyone through any mode of communication.
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There has been many theories of psychology on consumer learning, one of the most used is that
of Horward(1963) who proposed the following consumer brand choice learning function.
𝑃𝑎 = 𝑀(1 − 𝑒 −𝑘𝑡 )
Pa = Probability of Purchasing brand T = number of reinforced trials
M = Maximum attainable loyalty
K = learning rate
The exponential function stated above has various implications related to the volume of sales of
a particular product. However, the learning rate varies for an individual to individual but in our
case it is safe to assume a constant learning rate for all the individuals within a population to aid
our further research.
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LITERATURE REVIEW
The introduction of models for the diffusion of innovations marked their beginning
back in 1960 when Fourt and Woodlock gave their market research tool to predict volume of
sales based on households. Then there were several many models that tried to explain the
phenomenon in the subsequent years. Some of them were Mansfield(1961), Floyd(1962) and
finally Bass(1969). Till 1969, there were primitive models which wither completely eliminated
with the introduction of new models or not have a significant existence these days.
Bass Model(1969)
“Bass model is one of the most quoted aggregated model used in marketing literature which has
a large acceptance in the field of innovation diffusion” (Mahajan, Muller and Winf, 2000). There
have been several attempts to revamp the basic model to a more appropriate model by the
inclusion of desired marketing variables in the past 30 years. The basic Bass model assumes two
kind of population within a target market. Innovators and Imitators. Bass is known to have
followed the concept given by Everett Rogers and gave mathematical formulation to it. The basic
model formulation can be given as
𝑑
𝑁 𝑡 =𝑝 𝑚−𝑁 𝑡
𝑑𝑡
+𝑞
𝑁(𝑡)
[𝑚 − 𝑁 𝑡 ]
𝑚
Where N(t) is the cumulative number of adopters till time „t‟, m is the initial market size, p and q
are the innovation and imitation coefficients respectively. Few literature studies also depict Bass
model in the form of market proportions which can be illustrated as
𝐹 𝑡 =
𝑁(𝑡)
𝑚
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Using this model number/proportion of the adopters can be calculated at any time „t‟. The value
of the coefficients of innovation and imitation can be calculated based on the historical data of
the same or the similar products. Many studies safely assume the coefficients to be constant with
successive product generations within a same product category(Norton and Bass model) which
gives similar shapes for the diffusion of products in the successive generations. However, this
assumption is contradicted based on various factors related to product placement, profits and
investment for a consumer(Meade 1985). Mansfield(1961) had also said that the technology
which involves higher profit and lower investment for a consumer diffuses much faster and this
usually happens in the products of later generations than earlier generations.
There have been many generalizations related to the value of p and q. Lawrence and
Lawton(1990) said that the value of p+q always lies within the range of 0.3-0.7. Studies of
Sultan(1990) have reported that the average value of p is approximately around 0.03 and that of
q is 0.38. The coefficients of innovation itself tells many a things about the proportion of the
population which would like to innovate with the new technology. Similarly bass model
classified the entire population into five adopter categories namely Innovators, early adopters,
early majority, late majority and laggards. The curve is said to follow normal distribution with
successive standard deviations dividing the entire region into adopter categories (Rogers 1983).
The diffusion of single innovation in a market
After Bass model, which was purely based on homogenous assumption of the market, many
studies did follow which assumed target markets to be heterogeneous in terms of various factors
including behaviour and income. Bonus(1973) suggested that the given the income distribution
within a population is bell shaped and price decreases monotonically, then this would result in S
shaped distribution curve which is in accordance to the Bass and earlier models. Many authors
also did try to relate the model with the Gini curve of income distribution to obtain an S shaped
diffusion curve with various assumptions.
There have been theories that suggest that for an innovation to be self-sustainable, it must be
adopted at least by the critical mass. The definition of critical mass varies for different kind of
products. Incubation time of a product is defined as the time interval between the completion of
the product development and beginning of the substantial sales. Kohli, Lehmann and Pae(1999)
hypothesized the incubation time with various factors related to sales and diffusion determinants.
They found a positive association between incubation time and time to peak sales but a negative
association between incubation time and coefficient of innovation.
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Similarly heterogeneity in terms of geographical locations, socio-economic factors also gathered
attention after the works of Goldenberg et al(2000). All these models were more or less the
extensions of the basic Bass model.
Authors have also tried to include various environmental variables like GDP per capita(Tanner
1974), advertising and pricing (Mahajan and Peterson, 1978) into the basic Bass model.
Robinson and Lakhani(1975)for the very first time included pricing to the Bass model. They
formulated it as
𝑑𝐹(𝑡)
= 𝑝 + 𝑞 − 𝑝 𝐹 𝑡 − 𝑞𝐹 𝑡
𝑑𝑡
2
. 𝑒 −𝑘𝑃𝑟 (𝑡)
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where Pr(t) is the price. K= coefficient and rest is the same as was used in the Generalized basic
Bass model.
In 1994, the latest Generalized Bass model with pricing was developed by three researchers
named Frank Bass, Trichy Krishnan and Dipak Jain.
𝑓(𝑡)
= 𝑝 + 𝑞𝐹 𝑡 𝑥(𝑡)
1 − 𝐹(𝑡)
x(t) is the function of percentage of price change or other variables that majorly included pricing
and advertisement.
Similarly Kalish et al(1983) modelled the effects of the factors such as advertising on the basic
Bass model. They revamped it as
ℎ 𝑡 = 𝛽0 + 𝛽1 𝐹 𝑡
+ 𝛽ln⁡
(𝐴 𝑡 )
here the β0 corresponds to the effects of publicity whereas β1 and β word of mouth and
advertising respectively.
Modelling diffusions across nationalboundaries
Generally for modelling diffusion in different countries, data pertaining to the leading country is
used to estimate the coefficients of the lagging country. For example – If mobile phones were
first purchased in South Korea and after 2 years of adoption within the country it was adopted in
Scandinavian countries, then the model is expected to replicate in the case of Scandinavian
countries with small changes with respect to the national differences. There have been many
studies conducted by authors like Takada and Jain(1991), Ganesh and Kumar(1996) and Kumar
and Krishnan(2002). They all had concluded that the success of the adoption of the new product
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depends upon the performance of the same product in the leading countries. If the product has
performed well in leading countries then the risk associated with innovation in the lagging
countries would automatically gets reduced.
Gatignon(1989) also modelled the effects on the innovation and imitation parameters for a
particular product in different countries which can be illustrated as.
𝑝𝑖 = 𝛽𝑝,𝑖,0 +
𝛽𝑝,𝑖.𝑘 𝑍𝑖,𝑘 + 𝑒𝑝,𝑖
𝑘
Where pi is the coefficient of innovation, Zi,k represents the cultural variable and ep,I is the
disturbance term.
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Modelling of diffusion across generations of technology
The Fisher-Pry substitution model(1971) is one of the earliest models to study the demand
relationship across different generations of a product. The Fisher and Pry made three basic
assumptions and expressed the relationship as
𝑑𝑓
=𝑏 1−𝑓 𝑓
𝑑𝑡
Where f corresponds to the proportion of the market already captured by the new market whereas
(1-f) depicts the proportion of the market still held by the older technology. b is the initial rate of
adoption associated with particular technology. The model was created keeping in mind the basic
assumption of natural competition, market interventions were assumed to be absent.
Norton Bass model(1987) is one of the most popular models in the marketing literature to model
demand in successive technological generations. It was an extension of Bass(1969) model and is
able to predict demand across successive generations quite satisfactorily(Johnson and Bhatia
1997). For a single generation with no successor, the sales model is given by simple Bass model
𝑆 𝑡 = 𝑚𝐹 𝑡
Assuming the sales of ith generation product in the time period „t‟ is given by S i(t), the
mathematical form can be given as
𝑆1 𝑡 = 𝐹1 𝑡 𝑚1 [1 − 𝐹2 𝑡 − 𝜏2 ]
Norton and Bass model suggests that the growth phase of the later generations are lesser as
compared to the early generations. But one of the major limitations of the Norton Bass model
was that it assumed the coefficients of innovation and imitation to be constant with the
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successive generations. However, the subsequent models like Islam and Meade(1997) relaxed
the constant coefficient assumption across generations. Mahajan and Muller(1996) introduced
the concept of skipped generation i.e. the adopters of first technology would skip the second
generation and would adopt the third generation directly.
Pricing across generations
Padmanabhan and Bass(1993) studied optimal pricing across generations and have found that
skimming price policy was consistent with the Bass parameters1. If the price of the product
during earlier generations is reduced, this leads to the greater take up of the earlier as well as
later generations. Similarly the bell shaped curve of the pricing during a particular generation
increases the sales for a particular generation and adds value to the successive generations.
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ALTERNATE S-SHAPED DIFFUSION MODELS
Xt is a function of cumulative number of adopters at time t. a denotes saturation level for a case.
The additional parameters which are adopted by b and c.
1. Cumulative lognormal
𝑡
𝑋𝑡 = 𝑎
0
ln 𝑦 − 𝜇2
exp⁡
(
𝑑𝑦
2𝜎 2
𝑦 2𝜋𝜎 2
1
2. Cumulative normal
𝑡
𝑋𝑡 = 𝑎
0
y − 𝜇2
exp⁡
(
𝑑𝑦
2𝜎 2
2𝜋𝜎 2
1
3. Gompertz
𝑋𝑡 = 𝑎𝑒𝑥𝑝(−𝑐 exp −𝑏𝑡 )
4. Log Reciprocal
1
𝑋𝑡 = 𝑎𝑒𝑥𝑝( )
𝑏𝑡
5. Logistic
1
Meade, N.. "Modelling and forecasting the diffusion of innovation - A 25-year review", International Journal of
Forecasting, 2006
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ISSN:2249-877X Vol. 6 Issue 5 May 2016 Impact Factor: SJIF 2013=4.748
𝑋𝑡 =
𝑎
1 + 𝑐𝑒𝑥𝑝(−𝑏𝑡)
6. Log-logistic
𝑋𝑡 =
𝑎
1 + 𝑐𝑒𝑥𝑝(−𝑏𝑙𝑛 𝑡 )
7. Flexible logistic models
𝑋𝑡 =
𝑎
1 + 𝑐𝑒𝑥𝑝(−𝐵 𝑡 )
8. Modified exponential
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𝑋𝑡 = 𝑎 − 𝑐𝑒𝑥𝑝 −𝑏𝑡
9. Weibull
𝑋𝑡 = 𝑎(1 − exp
𝑡𝑏
𝑐
)
APPLICATION OF MODELS ON THE SALES DATA OF DRAM INDUSTRY
DRAM (Dynamic Random Access Memory) is a type of random access memory that
stores every bit of data in a separate capacitor which can be charged or discharged in an
integrated circuit. It is also known as the main memory(RAM) in the case of personal computers.
The manufacturing of DRAM requires very advanced technologies and significant capital
expenditures. The industry has seen many up as well as down trends with Samsung being the
biggest player in the market since 1990. 2005-06 onwards has been a depressing decade for the
DRAM industry with its ever decreasing sales. For our analysis we have considered the time
period from 1974-1998. This is so because during this period there were not many complex
factors that governed the sales of a product in the global market.
In this paper, we will compare different models on the sales data of the Dynamic
Random Access Memory industry and would try to find out the best nonlinear curve fit. For
convenience we have considered four types of models in our analysis namely Gompertz curve,
log-logistic curve and Generalized bass model. The cumulative adoption data of dram
industry(4k,16k,64k,256k,1M and 4M) have been taken into consideration with time period
ranging from 1974-1998. The data was used for nonlinear curve fitting using the above
mentioned curves. R-square and other parameters were calculated for each generation and
analyzed with other generations to draw various conclusions.
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Gompertz curve estimation
Type
Parameter
Estimate
Std. Error
95% Confidence Interval
R-Squared
Lower Bound
b
16506383.77
0.008881763
3763202022
0.128222926
-8368430250
-0.276816719
8401443017
0.294580245
c
72626.71081
16157915.55
-37187593.36
37332846.78
b
0.00576051
0.12939018
-0.292613779
0.3041348
c
928036.9671
125142924
-269426813.5
271282887.5
64k
b
c
256k
b
0.007169013
42245.15765
0.005462912
0.080513467
7687285.844
0.112411408
-0.166769758
-17347603.58
-0.248829361
0.181107784
17432093.89
0.259755184
c
51794.03421
6761610.344
-14680489.33
14784077.4
b
0.005612986
0.078109935
-0.164573942
0.175799914
c
23040.00138
4377330.831
-9879170.292
9925250.295
b
0.005038422
0.107120665
-0.237285357
0.247362201
c
4k
16k
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Upper Bound
4M
0.04
0.03
0.045
0.042
0.039
0.029
Table 1 : Gompertz curve parameters estimation
Log-logistic curve estimation
Parameter Estimates
Parameter
Estimate
Std. Error
95% Confidence Interval
Lower Bound
4k
16k
64k
256k
1M
4M
Upper Bound
c
3898.950017
8398983.622
-18710202.78
18718000.68
b
1.192990593
307.8561404
-684.7532366
687.1392178
c
703.0566638
2021187.213
-4660163.014
4661569.127
b
0.834195642
378.7139128
-872.4816534
874.1500447
c
1060.410516
1765139.61
-3812291.876
3814412.697
b
0.939481851
219.2150622
-472.6458675
474.5248312
c
908.6811399
2064388.178
-4669061.823
4670879.185
b
0.863009883
333.0662778
-752.5852562
754.3112759
c
499.2232672
756576.4468
-1647939.246
1648937.692
b
0.8085999
222.2516366
-483.4361174
485.0533172
c
671.9402839
1569602.856
-3550016.403
3551360.283
b
0.784110553
343.9117063
-777.1982192
778.7664403
R-squared
0.002
0.002
0.002
0.002
0.002
0.001
Table 2 : Log-logistic curve parameters estimation
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High standard errors for the estimates of parameters in the case of Gompertz curve explains the
insignificance of these estimates. Also a highly insignificant R-square value suggests that this
model cannot satisfactorily explain the cumulative diffusion of products in the DRAM industry.
Bass model curve
Parameter Estimates
Parameter
Estimate
Std. Error
95% Confidence Interval
Lower Bound
4k
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16k
64k
256k
1M
4M
R-squared
Upper Bound
p
1.181144495
2.089334597
-3.474183096
5.836472086
q
304.8008215
17.67443269
265.4197313
344.1819116
p
q
p
0.029317321
0.710749039
-0.004750578
0.018055101
0.133061961
0.006936546
-0.012317816
0.403907607
-0.019736074
0.070952458
1.017590471
0.010234918
q
1.089508043
0.066965061
0.944838824
1.234177261
p
0.018684515
0.010250128
-0.004502886
0.041871916
q
0.780731581
0.078892767
0.602263742
0.959199419
p
q
p
q
0.012417995
0.602846881
0.021091299
0.798084519
0.004935949
0.036904156
0.016536925
0.131767111
0.001663486
0.522439632
-0.016317824
0.500006604
0.023172505
0.68325413
0.058500422
1.096162433
0.967
0.781
0.953
0.916
0.957
0.803
Table 3: Bass model curve parameters estimation
CONCLUSIONS AND FURTHER RESEARCH:
The studies in the past 50 years have resulted in hundreds of models that differ in objectives and
its parameters. Each and every model is unique in its kind and calls for further revamp as a
broader research topic. The models like Bass and Norton-Bass have gathered recognition and
forms the basis of many subsequent models. With the changing scenario, there have been
inclusion of many external and environmental variables to forecast the sales of the products. If
not the deep details of the forecast, but many models apply in different situations to estimate the
time of peak sales and this helps in optimal pricing of the products and the right time of release.
The marketing managers have been using such models for many years to make estimates for
different quantities under sales forecast.
The further research on the topic would include studying of the models in various fields based on
different approaches and objectives. The final objective is to find the correlation between value
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ISSN:2249-877X Vol. 6 Issue 5 May 2016 Impact Factor: SJIF 2013=4.748
creation and innovation diffusion. What are the impending effects of innovation diffusion on the
creation of value for a firm/product.
In the case of DRAM industry Generalized Bass model has been found to fit much better than
Gompertz and Log logistic curves with parameter estimations depicted in the table above. The Rsquared for Generalized Bass model has decreased in the successive generations which is mainly
because of the new factors in the market that affect the sales of the products directly or indirectly
which majorly includes pricing and advertisement.
Based on our analysis, we can conclude that Generalized Bass model can best predict the sales in
the case of DRAM industry. To consider the effect of other variables Generalized bass model can
be used as basis to modify the model accordingly.
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