a two dimensional innovation diffusion model for of...

Department of Operational Research, University of Delhi 93

Chapter 6

Two-Dimensional Model for Successive Generations of

Technology

Understanding and analyzing the rate of adoption of a new technology in the

marketplace is of principal significance for any organization to assess the suitability

of technology investment that can enhance the profitability and its market share in

long run. Successful new product introductions contribute substantially to long-term

financial success and are an effective strategy to increase primary demand. One major

way to increase the market presence is by improving the customer perspective about

the organization through periodic introduction of new products. Technological

breakthroughs are happening rapidly and these innovations often take form as new

products. Companies make a lot of investment in the research and development or in

technology acquisition and are therefore concerned about their success (Cooper and

Kleinschmidt 1986, Zirger and Maidique 1990, Cohen, Eliashberg and Ho 1996,

Winter and Sundqvist 2010). The concept of performance of a new technology

generation over its lifecycle has been explained by using the well known S-curve or

Sigmoid diffusion curve(Looy, Zimmermann and Ranga 2000);(Rodger, Pendharkar

and Bhatt 1996).

The diffusion of innovation refers to the tendency of new products, practices, or ideas

to spread among people. Diffusion models, like any other mathematical models are

simplification of reality. However they constitute a wide range of useful tools, in both

academic and business context. As describe in chaper1 (section 1.16.1) Bass model is

one of the pioneer mathematical model for diffusion of the product in the market.

This Chapter is based on the following research paper entitled:

1. Kapur, P. K., Aggarwal, A. G., Garmabaki, A. H. S., and Tandon, A. (2012), " Modelling

Innovation Diffusion by Incorporating Time & Price for Successive Generations of

Technologies," International Journal of Technology Transfer and Commercialization,

Communicated.

Chapter5 Two-Dimensional Model for Successive Generations of Technology

Department of Operational Research, University of Delhi Page 194

Norton and Bass (1987) proposed a model considering different generations of a

technology. Examples are the series of generations of mobile telephones and personal

computers. In the Norton–Bass model, each generation of the technology attracts

incremental population segments of potential adopters; in addition, later generations

may attract potential adopters of earlier generations. This modeling approach

effectively succeeded the models on technological substitution, where one technology

replaced its predecessor. Speece and MacLachlan (1995) demonstrated applicability

of the Norton–Bass model and forecasted the adoption of successive generations of

gallon milk containers. Mahajan and Muller (1996) extended the Norton–Bass model

to allow adopters of early generations to skip generations, for example, an adopter of

the first generation could replace it with third generation technology and second by

fourth, etc. They validated their model using generations of IBM mainframe

computers. Islam and Meade (1997) demonstrated that the assumption of constant

coefficients of innovation and imitation ( p and q ) over successive generations could

be relaxed. In a study of multi-national mobile telephone adoption, they demonstrated

that the coefficient of imitation ( q ) tended to increase from generation to generation.

Sohn and Ahn (2003) used the Norton–Bass(1987) model to demonstrate a cost–

benefit analysis of introducing a new generation of information technology. The

model proposed by Jiang (2010) is an extended multi generational diffusion model

that uniquely differentiates between the existence of switchers and substituters.

Norton-Bass model acknowledges the existence of substitution and switchers but does

not differentiate between the two mathematically.

6.1. Modeling Innovation Diffusion by incorporating Time & Price for

Successive Generations of Technologies

In this section, we propose a model that embeds a broader theoretical framework to

account for the interactions between different dimensions of adoption i.e. time and

price factors. Incorporating the dynamics of continuation time of the product in the

market and the price has allowed us to model the diffusion process in two-

dimensional framework. Technological adoptions and the role of other dimensions are

explicitly taken into consideration using the Cobb-Douglas production function. For

PhD Thesis Some Contributions to Multi-Release Problems in Software Reliability and Successive Generations of Technologies


many technology products customer interactions through word-of-mouth can reduce

the product-uncertainty and can play an effective role in the acceptance of the new

product. The other marketing-mix variable viz price also play important role in

diffusion of a technology product. A potential individual would become a purchaser

after satisfying himself of the utility of the product vis-à-vis the unit price. The model

proposed in this chapter is for successive generations of product. It decomposes the

total sales into first time purchasers, switchers and substitutes. Substitution and

switching are the two main factors of multi generational modeling. These assumptions

are intuitively consistent with the view that attitudes change towards technology as it

evolves from one generation to the next generation. The proposed model has been

compared with an established model. The result are encouraging and the findings are

consistent with the idea that attitudes of purchasers towards new generations change

from the previous one and it is imperative to identify a trend.

6.1.1. Modeling of the Two-Dimensional adoption process

In the chapter, we propose a mathematical structure which incorporates time and price

factors. For representation of the effect of these factors on the adoption process

simultaneously, we use the Cobb-Douglas production function (Cobb and Douglas

1928), as described in chapter 1 section 1.10.

The mathematical form of the production function is given as:

1t p (6.1)

Where is total production (the monetary value of all goods produced in a year) and

t, p are labor input and capital input respectively. Also is elasticity of labor. This

value is constant and determined by the available technology. In this paper t, p

represent time and price, respectively and is the degree of the impact of time to the

adoption process. By this technique, we are able to consider the effect of several

factors simultaneously and the impact of each factor on adoption process depends on

the value of elasticity parameter .

http://en.wikipedia.org/wiki/Labour_%28economics%29

http://en.wikipedia.org/wiki/Capital_%28economics%29



Figure 6.1: Cobb–Douglas Function Behavior

6.1.2. Bass model in Two-Dimensional framework

For modeling the innovation diffusion, usually an S-shaped function is used as the

basis of the diffusion equations. These diffusion equations generally have two kinds

of parameters: One measures the upper asymptote of the curve, i.e. the eventual level

or penetration of innovation’s diffusion, and the other measures the rate of adoption in

the social system. The other parameters due to marketing efforts are most of the cases

ignored. The two-dimensional innovation diffusion model proposed in this chapter

represents the adoption decision of an adopter that explicitly depends on the value of

the product introduced in the market. Value of the product can be determined by the

time for which the product is in the market and purchase price.

To formulate the functional relationship between the product continuation time in the

market and the price of the product is given by Cobb-Douglas production function.

Bass model for diffusion of innovation assumes that the potential adopter of a new

product can either be an innovator or an imitator depending upon how he/she makes

the purchase decision. The Bass model has been applied to a variety of new product

types introduced during past almost four decades. In spite of apparent limitations of

the model including that of non-inclusion of explanatory variables like price etc., most

of these applications have been successful (Bass, Krishnan and Jain 1994); (Mahajan

and Muller 1996); (Wilson and Norton 1989).



Recently Kapur, Singh, Chanda and Basirzadeh (2010) modified Bass(1969) model

into two dimensional frameworks.

Customers will buy a product according to their varied needs and expenditure

capacity. Thus, price plays a major role in the overall success of the product. A

potential individual would become a purchaser after satisfying himself of the utility of

the product vis-à-vis the unit price. Thus a target customer’s motivation towards a

new product will depend on its price. In their two-dimensional innovation diffusion

model it was assumed that value of the product as the major driver of diffusion and

can be classified into the following two main factors:

The factors which are related to the goodwill of the product, such as the

continuation time of the product in the market.

The factors which are responsive to the consumer buying decision, such as price.

The proposed innovation diffusion model depends on the above two factors. To

incorporate the effect of above two factors in the consumer decision making process,

measure of diffusion (i.e. time, for one-dimension) was extended in two dimensional

frames by means of value of the product in the model. Mathematical equation in Two-

Dimensional framework for the model of adoption due to innovation and imitation

can be expressed as:

1

1

, ,

1.

1

i i

i i

p q t p

t p t p

p q t pi

i

eN N F N

qe

p

(6.2)

Where

,t pN are the cumulative number of adopters in time t and price p.

N is the initial market size.

ip and iq are the innovation and imitation coefficients respectively.

p is the price

represent the effect of product valuation to the innovation diffusion model.



When 1 , adoption take place due to goodwill of the product and price has no

effect on the overall adoption.

When 0 , adoption take place due to the pricing effect of the product and

goodwill has no effect on the overall adoption.

The two dimensional adoption processes are dependent on unit price and time can be

viewed from the figure below:

Figure 2: Two Dimensional Adoption Process.

6.1.3. Proposed Model development

The proposed model considers three market conditions. It discusses the nature of

adoption when there is only a single product in the market and innovation gets

diffused in the population with time along with the effect of the price. In the second

situation, after a finite period of time a new product is launched in the market. The

new product competes with the existing product in the market and the two parallel

innovations exist in the market. Earlier, many models have been developed in the

literature to discuss this situation but in our model we have used a two dimensional

framework where both introduction time of the new innovation and also the price of

the product is taken into account to capture the adoption process. Finally, we study the

market situation when three generation of the product competes together

simultaneously in the two dimensional framework.



6.1.4. Assumptions of proposed model

The model is based on the following assumptions:

,( )j

i t pN is total number of adopters of thi generation when j number of

generations have been launched in the market at time t with price p.

Each generation of the product creates its own market potentialiN . Thus iN is the

eventual number of adopters of the thi generation if the next generation is not

introduced.

Let i be the introduction time of the

thi generation.

Let j be a fraction of those who substitute previous generations with the j

th

generation.

Once adopters adopt a new technology, she/he doesn't revert to earlier generation

technology.

When a new generation of a technology is introduced in the market, there would

be three types of purchasers. Those who are switching from the earlier generation,

(ii) those who adopt the new technology instead of the earlier one (substitutions),

(iii) potential customers who would only adopt the new technology.

Each adopter (New Purchasers) can purchase exactly one product unit and she/he

makes no further purchases of the product generations that they have adopted.

Also, each adopter after having made the first purchase may make a repeat

purchase of exactly one unit in the successive generation or they may skip the next

generation and wait for more advanced one.

A consumer's choice in each time period can be independent of her/his choice in

previous periods and depends on the utility of the new product.

Like the first time purchasers, switching purchasers (i.e. buyers who have also

purchased from previous generations) can also be influenced by the word-of-

mouth effect and as well as the marketing-effort made by the firm. All the two

kinds of the purchasers may be modeled according to the Bass model.

The model for different market situations can be built as follows:



6.1.4.1. The Formulation of Proposed Model

Case 1:

When there is only first generation (base product) in the market, the cumulative

number of adoption can be given by the model proposed by (Kapur, Singh, Chanda

and Basirzadeh 2010):

0

1

1 , 1 1 ,( ) ( )t p t pN N F (6.3)

Where 0 0t t and

(1 )1 1 0

0 (1 )1 1 0

( ).

1 ,( ).1

1

1( )

1 ( ).

p q t p

t pp q t p

eF

qe

p

(6.4)

In this case, note that there is no effect of other generation on diffusion of product.

Case 2: When two generations are in the market

Bass (1969) in his model framework assumed a constant potential market for

diffusion through the social network but in successive generation of technology

scenario is different. After introducing second generation in the market, there would

be a sizable portion of this population who are yet to purchase. So, when they have

the option to choose, they will go with the product having maximum utility. In this

situation, satisfied adopters can influence others, with positive word-of-mouth. Due to

the influence of promotional effort and word-of-mouth some potential buyers of

earlier generation technology who otherwise would adopt the older generation may

decide to purchase the new generation instead and some of adopters of first generation

upgrade their product and eventually become the potential repeat purchasers of the

later generation. Let fraction of the adopters of first generation who switch to the

second generation technology be 21,2 ,( )t pSw and

21,2 ,( )t pSub

be the fraction of

adopters of first generation who substituted their idea with new generation. Hence, the

number of adopters of first generation after second generation decreases and the



population of second generation increases. Thus in two generation market scenario,

cumulative number of adopters for two generations may be given as:

1 2

2

1 , 1 1 , 1,2 ,( ) ( ) ( )t p t p t pN N F Sub (6.5)

2 2 2

2

2 , 2 2 , 1,2 , 1,2 ,( ) ( ) ( ) ( )t p t p t p t pN N F Sub Sw

Where j jt t and ,( ) , 1,2, 1..2ji t pF i j is the cumulative adoption function

with model parameters ip and

iq , respectively.

(1 )

(1 )

( ).

,( ).

1( )

1 ( ).

i i j

ji i j

p q t p

i t pp q t pi

i

eF

qe

p

(6.6)

In the Eq. (6.5), 21,2 ,( )t pSub and

21,2 ,( )t pSw are the fractions of potential and actual

adopters of an earlier generation (first in this case) who buy the second generation.

The mathematical form can be given as follows:

2

21 1

2

1,2 ,

2 1 1 , 1 3,

0( )

. . ( ) ( )t p

t p P

tSub

N F F t

(6.7)

2

221

2

1,2 ,

1 1 2 , 3,

0( )

. ( ). ( )t p

t pP

tSw

N F F t

(6.8)

Where2

1 2 1 .

Case 3: When three generations of product are in the market place.

When a firm introduces its third generation product without withdrawing any of the

earlier generations, potential adopter of both the first and second generation come

under the influence of innovation and imitation effects of the latest one. As a result,

31,3 ,( )t pSub , 32,3 ,( )t pSub , are groups of potential adopters who substitute first and



second generation with the third one based on the substitution effect caused by

introduction of third generation in the market. This group otherwise would have

adopted the first and second generation technology. Similarly some of customers of

first generation substitute to second generation technology as 21,2 ,( )t pSub (as

discussed in earlier section). Finally, let 21,2 ,( )t pSw and

32,3 ,( )t pSw be the group of

people who adopted first and second generation and make decision for switching to

second and third generation, respectively. Thus in three generation market scenario,

cumulative number of adoption can be given as:

1 2 3

3

1 , 1 1 , 1,2 , 1,3 ,( ) ( ) ( ) ( )t p t p t p t pN N F Sub Sub

2 3 2 2

3

2 , 2 2 , 2,3 , 1,2 , 1,2 ,( ) ( ) ( ) ( ) ( )t p t p t p t p t pN N F Sub Sub Sw (6.9)

3 3 3 3

3

3 , 3 3 , 1,3 , 2,3 , 2,3 ,( ) ( ) ( ) ( ) ( )t p t p t p t p t pN N F Sub Sub Sw

Where j jt t and ,( ) , 1..3, j 1..3ji t pF i is the cumulative adoption function

with model parameters ip and

iq , respectively.

(1 )

(1 )

( ).

,( ).

1( )

1 ( ).

i i j

ji i j

p q t p

i t pp q t pi

i

eF

qe

p

In the above equations,21,2 ,( )t pSub ,

31,3 ,( )t pSub and 32,3 ,( )t pSub are fractions of

the potential and actual adopters of an earlier generation (first and second in this case)

who buy the second, third generation. The mathematical form may be given as

follows:

3

31 1

3

1,3 ,

3 1 1 , 1 3 4,

0( )

. . ( ) ( )t p

t p P

tSub

N F F t

(6.10)

3

32 2

3

2,3 ,

3 2 2 , 2 3 4,

0( )

. . ( ) ( )t p

t p P

tSub

N F F t

(6.11)



Where .j

i j i

Adoptions of generation i may also come from adopters of earlier generations who

switch to it (i.e., adopters of (i - 1) th

generation). The mathematical form may be

given by:

3

332

3

2,3 ,

2 2 3 , 3 4,

0( )

. ( ). ( )t p

t pP

tSw

N F F t

(6.12)

21,2 ,( )t pSub and 21,2 ,( )t pSw have been defined in previous case.

6.1.5. DATA

The proposed model has been validated on data collected from Dynamic Random

Access Memory (DRAM) computer chips, This company selects data of world-wide

dynamic random access memory (DRAM) shipments of six generations taken for a

period 1974 to 1992(Victor and Ausubel 2001). Estimation done on first three

generation as data set1( DS-1) and generations third and fourth as data set2(DS-2).

DS1 is related to 4K, 16K, 64K and Ds2 related to 64K and 256K chip generation of

DRAM, respectively.

6.1.5.1. Empirical analysis

The parameters of proposed model in the above sets of equation were estimated using

simultaneously nonlinear least squares (NLLS). Srinivasan and Mason (1986) by

software package SAS (SAS 2004). Estimated value of diffusion parameters of DS1

and DS2 are given in Table 6.1.

As expected, Table 6.1 shows all the estimated coefficients are highly significant and

asymptotic standard errors are very low and also α is highly significant for two

datasets.



Table 6.1: Parameter estimates of the two-dimensional proposed model.

Technology Generat

ions Parameters

jN jp jq

j

DRAM Company In

Three Generation(DS-1)

4k 376.36*

(20.41)

0.000026

(8.102E-6)

0.028817

(0.00369) -----

0.476363

(0.0197)

16k 1230.98*

(23.70)

6.526E-7

(5.122E-7)

0.04869

(0.00536)

0.1593

(0.0457)

0.476363

(0.0197)

64k 2639.96*

(23.89)

1.099E-8

(0)

0.064175

(0.00911)

0.0001

(0)

0.476363

(0.0197)

DRAM Company

In

Two Generation(DS-2)

64k 2658.776

(18.329)

1.01E-6

(0)

0.1340

(0.0221) -----

0.6273

(0.0224)

256K 4627.03

(24.03)

0.000023

(9.625E-6)

0.2051

(0.0178)

0.00001

(0)

0.6273

(0.0224)

Asymptotic Standard Errors are in parenthesis; (*)in millions (j = 1,2,3 )

In addition, Table 6.2 shows the values of SSE (Sum of Squares Error), MSE and

Adj-R2

of both the models that were applied on two sets viz; DS-1 and DS-2. To

establish the predictive validity of our model, we compared our model with the one

dimensional model. From Table 6.2 and Figures. 6.2, 6.3 and 6.4, it is clear that at any

point of time the proposed model will give more accurate and better fit to the data in

comparison to same situation in one-dimension model. It is also evident from Table

6.2 that the mean square error (MSE), Sum of Squares Error (SSE) and Adj-R2 of

two-dimensional model of all generations related to DS-1 and DS-2 (except second

generation in DS-1) is much better than one dimensional model.

The three dimensional curves of non cumulative and cumulative adopters on axis

time, price and adopters for expected number of cumulative and non-cumulative

adopters for 4 K, 16 K and 64 K in DRAM datasets are plotted in Figures 6.5, 6.6,

6.7, 6.8, 6.9 and 6.10, respectively.

The proposed model gives a clear view about the decomposition of total sales into

new adoptions and it also gives clear idea about the substitution and switching



behavior of purchasers in two dimensional framework. With this model we tried to

give a proper mathematical interpretation of the following:

1. Cumulative number of adoptions,

2. Substitution and switching behavior of adopters when technology comes in

multiple generations,

3. Influence of introduction of latest technology to the sales figure of earlier

generations,

4. Combined effect of time and price on the cumulative number of adopters for

successive generations of technologies.

Table 6.2: Comparison criterion of one-dimension and proposed model

Technology Generations Model

DRAM Company

In

Three Generation

One-Dimension 2-D Proposed model

SSE MSE Adj-R2 SSE MSE Adj-R

2

4k 94.633 5.4076 0.9995 47.0072 2.7383 0.9998

16k 5072.0 317.0 0.999 8423.2 537.6 0.9983

64k 63720.2 4720.0 0.9968 35327.6 2493.7 0.9983

DRAM Company

In

Two Generation

64k 72578 4147.3 0.997 49013.4 2800.8 0.9979

256k 56383.9 3888.5 0.9991 34707.4 2393.6 0.9994



Figure 6.2: simultaneous estimation of generation 4K, 16K and 64K

Figure 6.3: Simultaneous estimation of generations 64K and 256K.

0.00

500.00

1000.00

1500.00

2000.00

2500.00

3000.00

Cu

mla

tive

Nu

mb

er o

f A

do

up

ter

(in

mill

ion

)

Time

Cum Num of adopters for 4K

Exp Num of adopter for 4K


Exp Num of adopter for 16K


Exp Num of adopter for 64 K

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Cu

mu

lati

ve N

um

ber

of

Ad

ou

pte

r (i

n m

illio

n)

Time

Cum Num of adopters for 64 K


Cum Num of adopters for 256 K




Figure 6.4: Simultaneous estimation of generation 64K and 256K

Figure 6.5: Estimated adopters of for non-cumulative dataset of 4K DRAM Chipset

0

200

400

600

800

1000

1200

Act

ual

Nu

mb

er o

f A

do

up

ter

(in

mill

ion

)

Time

Actual Num of adopters for 64 K

Exp Actual Num of adopter for 64 K

Actual Num of adopters for 256 K

Exp Actual Num of adopter for 256 K

1.00

4.678.33

12.00

Time 389

2293

4197

6101

price0.45

24.74

49.02

73.30

sale



Figure 6.6: Output of the proposed model for cumulative data of 4K, DRAM.

Figure 6.7: Estimated adopters of for non-cumulative dataset of 16K DRAM .

6.1.6. Managerial Implication:

Cautious product planning is very essential, especially when product is launched in

the market in successive generations. Despite potential advantages of new

1.00

4.67

8.33

12.00

Time6101

8981

11861

14741

Price5

109

212

316

Sale

1.00

4.67

8.33

12.00

Time0

1119

2238

3357

Price0

93

187

280

Sale



technologies, their long-term viability is often unclear. Hence, before adopting an

innovation, decision makers need to answer following questions:

1. What is the target market of the product and its size?

2. How the innovation spreads through its target population?

3. How much is the effect of substitution and switchers on the sales pattern?

4. What and how much is the effect of marketing-mix variables such as price,

product etc on the diffusion of innovations within generations?

Also from managerial perspective, it is crucial to understand what product features to

offer, at what price, so as to help firms maximize their profits and market share, while

ensuring that the firm's customers are satisfied. Due to intense competition in the

market, it is important for a firm to understand and analyze the rate of adoption of

new technology that enhances profitability.

When a new technology is introduced in the market, it not only attracts first time

purchasers but also previous buyers who would upgrade their products. This factor

adds more complexity to the modeling. Innovator, Imitator, Substitution and Switcher

are the four major factors which influence the adoption process. Innovation diffusion

model can be of great help to the managers in understanding such decision making

process. The two dimensional model discussed in this paper helps the marketing

manager to figure out the number of innovators, imitators, substitution and switchers

separately, under the effect of time and price. In earlier research papers only one

aspect of adoption was considered i.e. time, but with the help of Cobb-Douglas

production function, the managers can confidently predict the adoption behavior of

the product that comes in generations, incorporating both time and price (at which

consumers are ready to adopt the product).



Figure 6.8: Output of the proposed model for cumulative dataset of 16K DRAM.

Figure 6.9: Estimated adopters of for non-cumulative dataset of 64 K DRAM.

1.00

4.67

8.33

12.00

Time0

2156

4311

6467

Price0

414

829

1243

Sale

1

7

13

19

Time0

924

1848

2771

Price0

222

444

667

Sale



Figure 6.10: Output of the proposed model for cumulative dataset of 64 K DRAM.

17

1319

Time 0

2065

4130

6195

Price0

879

1758

2638

Sale

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