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Innovation diffusion: Some newtechnological substitution modelsVinod Kumar a & Uma Kumar aa School of Business , Carleton University , Ottawa, Canada ,K1S 5B6Published online: 26 Aug 2010.

To cite this article: Vinod Kumar & Uma Kumar (1992) Innovation diffusion: Some newtechnological substitution models, The Journal of Mathematical Sociology, 17:2-3, 175-194,DOI: 10.1080/0022250X.1992.9990105

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Journal of Mathematical Sociology, 1992, Vol. 17(2-3), pp. 175-194Reprints available directly from the publisherPhotocopying permitted by license only© 1992 Gordon and Breach Science Publishers S.A.Printed in the United States of America

INNOVATION DIFFUSION: SOME NEWTECHNOLOGICAL SUBSTITUTION MODELS

VINOD KUMAR and UMA KUMAR

School of BusinessCarleton University

Ottawa, Canada K1S 5B6

March 20, 1991

Three innovation diffusion models of technology (KKKI-KKKIII) are proposed. The first two of thesemodels are motivated by Smith's model and von Bertalanffy's model already in use in population dy-namics. The first model also takes into account the internal and external influences of imitators andinnovators. The third model is an improvement over models by Floyd and Fisher-Pry. It can also belooked upon as a model which takes into account the effect of promotional subsidies on the innovationdiffusion. Empirical analyses using the data of three technological innovations show that the proposedmodels give comparable or better results.

1. INTRODUCTION

Technological substitution models (TSM's) attempt to model the time-dependent as-pects of the innovation diffusion process, that is, the process by which an innovationis spread through certain channels over time (and possibly over space) among themembers of a social system (Rogers 1983). The question of the spread of an inno-vation is of interest to all inquisitives in society. Particularly, in the era of technolog-ical change, an understanding of the way in which technologies that aim to improvethe socio-economic conditions can spread constitutes important knowledge of man.For example, the knowledge about the dynamics of the process of acceptance offertilizers, pesticides and new crop varieties in agriculture, acceptance of immuniza-tion and new family planning approaches in a health care system, growth of newcults in a sociological system, growth of advanced process technologies in manu-facturing, acceptance of innovative teaching aids in education system, etc. wouldbe extremely useful in rational development planning of social systems. Holt (1977)observes, "From a societal point of view the diffusion process is perhaps more im-portant than the innovation itself since the social and economic impacts are fullyrealised by all who adopt the innovation. In addition, this also becomes an impor-tant stimulus to new innovations."

The rich and multidisciplinary literature of the study of diffusion of innovationreflects in general two broad but distinct approaches adopted by researchers. The

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176 V. KUMAR AND U. KUMAR

first approach focusses on spatial aspects of the diffusion process and an examina-tion of the socio-economic factors that influence it. Geographers, sociologists, anddevelopment planners seem to be those most interested in this approach. In thesecond approach the major aspect of the diffusion process has been the study ofthe time pattern of the spread of innovation at a macro-level. This approach hasbeen adopted primarily by technology planners, market researchers, and industrialresearchers. It is the second approach where 5-shaped models, empirical as well ascausal, have been extensively used; the pioneer work of Ralph Lenz in this field goesback to the fifties. The first successful checks were obtained by Mansfield (1961)by fitting 5-shaped models with historical technological substitutions of railroads,coal, steel, and breweries. Blackman (1971, 1972) studied the innovation dynamicsin aircraft jet engine market and in electric utility and automotive sectors. Otherapplications include the study of industrial technology by Nevers (1972), medicalinnovations by Easingwood et al. (1981), energy-efficient innovations by Teotia andRaju (1986), telecommunication innovations by Bewley and Fiebig (1988), and agri-cultural innovations by McGowan (1986) and Dixon (1980). Merino (1990) noticedthat the performance of tire cord textile followed a 5-curve pattern when plottedagainst the effort spent.

The existing 5-shaped technology diffusion models can be broadly categorizedinto two fuzzy classes: i) models that consider diffusion in terms of adopters andnon-adopters of technology and ii) models that consider diffusion of technology asa process of substitution—an existing technology being replaced by a new technol-ogy. Both classes of models, however, enable estimation of future trends on thebasis of past data and give mathematically similar models. Some of the models arethose given by Mansfield (1961), Bass (1969), Floyd (1968), Sharif and Kabir (1976),Fisher and Pry (1971), Gompertz (1825), Jeuland (1981), Lilien, Rao and Kalish(1981), Norton and Bass (1987), Easingwood, Mahajan and Müller (1981, 1983),and Skiadas (1985, 1986). A review of most of these models has been given in Ma-hajan, Müller and Bass (1991). Stochastic models have been reviewed by Eliashbergand Chatterjee (1985).

All these models are given by

%-W) (D

where f(t) denotes the proportion of potential adopters which has adopted the in-novation till time t. From the model users point of view a technological substitutionmodel showing an 5-shaped pattern should have the following desirable character-istics:

(i) it should contain one or more parameters which can be adjusted to fit thedata, i.e., the model should be flexible;

(ii) it should exhibit a threshold point called the point of inflection which, fordifferent values of a parameter, can arise for any adoption level;

(iii) it should allow for both symmetric and nonsymmetric behaviour patterns fordifferent values of the parameter(s);

(iv) it should have a closed-form solution; and

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TECHNOLOGICAL SUBSTITUTION MODELS 177

(v) it should have as few parameters as possible, yet still be consistent with thepreceeding objectives.

From the mathematical point of view,

(i) <p(f) should be > 0, i.e., the technological innovation adoption level / shouldnever decrease;

(ii) <p(f) should be zero, when / = 1, since there can be no further adoption af-ter each of the potential adopters has adopted the technological innovation;

(iii) as / increases from 0 to 1, (f>'(f) should be positive in the beginning, thenshould become zero for some value of / between 0 and 1, and finally shouldbe negative thereafter, i.e., the / — t curve should change from convexity toconcavity for some value of / between 0 and 1.

Although it appears that there has been a proliferation of innovation diffusionmodels, many of these models work in some cases and do not perform well in oth-ers due to their prespecified restrictions on the shape of the diffusion patterns. Forexample, some models lack flexibility and do not perform well for the productswhere the growth shows a different pattern beyond their points of inflection; somemodels even have their points of inflection at a specific level of substitution leavinglittle maneuverability. In other cases, the model consists of a large number of pa-rameters making it unsolvable or computationally complex and inefficient. The needfor new mathematical models is well pronounced by Mahajan and Wind (1986), whoin a re-examination of these models concluded with the statement:

... we need to develop parsimonious flexible closed-form diffusion models that can accom-modate both symmetric and nonsymmetric diffusion patterns with a point of inflection thatcan occur at any stage of the diffusion process.

This statement is perhaps the main motivating force behind the present work. Here,in fact, we propose three new technological substitution models of innovation diffu-sion that exhibit almost all the desirable characteristics discussed above.

It is observed that in developing the existing models researchers, knowingly orunknowingly, have used a variety of approaches (Kumar and Kumar, 1991). How-ever, most researchers tend to claim adopting the conventional approach of mod-elling which consists of understanding the nature of the process, characterizing thesystem, defining its variables, formulating their relationships in the form of mathe-matical equations, and eventually validating that model by fitting a set of data to it.They tend to avoid discussing any other approach, even if they have considered one.Adopting a conventional modelling approach, however, has a certain disadvantage;it tends to exclude a number of existing or new and useful models from consider-ation. The intent of this argument is in no sense to undermine the importance ofconventional modelling approach but to signify that there are other ways of devel-oping models which can prove to be quiet useful in fitting the data and representingthe behaviour of the system. From a model user's point of view, what is perhapsneeded is the availability of a better and wider selection of models with certainstructures which could guide the user in choosing the appropriate model for his sit-uation (Kumar and Kumar, 1992). Here, we adopt a mixed approach in developingthe new models.

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178 V. KUMAR AND U. KUMAR

In this paper, three S-shaped diffusion models of technological substitution havingmost of the above desirable characteristics are proposed; for the purpose of identifi-cation, they are called KKKI to KKKIII models. The motivation and the theoreticaladvances associated with each of them are discussed along with their presentationrespectively. The next three sections, Sections 2-4, present these three models. Thebehaviour and some important mathematical properties of these models are alsodiscussed in these sections. The KKKI model is motivated by Smith's (1963) originalmodel in population dynamics which has been further elaborated by Kapur (1985).KKKII is motivated by another generalized population dynamics model given byKapur (1985, 1988) which has its original motivation in the work of von Bertalanffy(1957). KKKIII is motivated by the limitations of two important models, the modelsgiven by Fisher-Pry and by Floyd. It can also be looked upon as a model whichtakes into account the effect of promotional subsidies on the growth of innovationdiffusion. Section 5 provides the results of empirical testing of the three proposedand several other existing models using three sets of innovation data and discussestheir relative performance. Finally, the paper concludes with brief remarks.

2. THE KKKI MODEL

There are many parallels between innovation diffusion models and the models ofpopulation dynamics, ecology, and epidemic theory. As an example, the Fisher-Prymodel

% (2)

is essentially the same as the logistic model in population dynamics.

(3)

In (2), (1 — / ) denotes the fraction of potential adopters which have yet to adoptthe innovation, while in (3), ((a/fc) - x) represents the size of the population whichcan still be supported by the environment. In one case, the upper limit to / existsbecause the number of potential adopters is assumed to be fixed; in the other case,the upper limit to x exists because the environmental resources are limited and cansupport only a limited size of the population.

Another example of parallelism is a simple deterministic epidemic model withoutremoval

§ l->0 (4)Here again, y(t) represents the number of infected persons at time t, and N + l — ydenotes the number of susceptibles who have yet to be infected; there is an upperlimit N +1 to the number of persons who can be infected.

Due to similarity in the phenomena, it might be worthwhile to look again at exist-ing models of population dynamics, epidemic theory, ecology, etc., both determinis-tic and stochastic, for their possible adoption in the field of technological innovationdiffusion and forecasting. In fact, a few of the existing models have been adoptedfrom these fields. However, during the adoption process, one should see that not

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TECHNOLOGICAL SUBSTITUTION MODELS 179

only does the model have the necessary characteristics presented in Section 1 but italso provides physical insight into innovation diffusion processes.

The KKKI modeldf_ _ (p + qf)(l-f) ,~dt ~ 1 + 6/ w

has been motivated by the Smith (1963) model in population dynamics,

dM r K-M-dT = rM [k + (r/c)M

but has been adjusted to meet the desired needs by using / to represent the poten-tial adopters which have adopted by time t, instead of population size, and intro-ducing the internal and external coefficients q and p, respectively.

Smith studied the Verhulst-Pearl logistic:

^ -N/K) (7)

which mainly is the product of three factors: (1) a rate constant (the intrinsic rate ofincrease); (2) a measure of population density; and (3) a measure of the portion ofavailable limiting factors not yet utilized by the population. He argued that the logis-tic model contains no provisions for time lags in the interactions among the threefactors. In order to compare it with any real system, either lags must be introducedinto the model, or time-free data must be extracted from the system. On experi-mentation, Smith modified the logistic to his model (6). The major difference in (6)from the logistic is that the relation of the specific growth rate, (l/M)(dM/dt), tomass, M, is not a straight line but a concave curve. The degree of concavity dependsupon the size of the ratio, r/c.

One important consideration for determining the range of parameter, b, is that0 < /* < 1. For /* to be greater than 0, b should be smaller than ((I/a) -1 ) , wherea = p/q and at ft = I/a -1, /* will be 0. At b = 0, /* = (\ - \a). In fact, the rangeof b can be extended to have negative values of b > - 1 . For these values, /* willvary from (^ — |a) to 1. Thus, for — 1 < b < (I/a — 1), the point of inflection variesbetween 1 and 0. This is, of course, a highly desirable property of the model.

The KKKI model can be thus compared with the Bass model, in that the (1 —/ ) of the Bass model has been replaced by (l-f)/(l + bf), so that if Z>>0, itrepresents an inhibiting effect on the rate of diffusion, and if b < 0, it represents apositive effect on this rate.

KKKI can also be compared with the GRMII model of Skiadas (1985)

i = ¡F7(Tr^7<W), o«r<i (8)

When n = 1, this givesd± = c / ( I - / ) f9.

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180 V. KUMAR AND U. KUMAR

This is a special case of the KKKI model, where the external influences are ne-glected and b = (1 - a)/a > 0. The Skiadas model contains three parameters, c, nand a, and the KKKI model also contains three parameters, p, q, and b.

KKKI has a closed-form solution which, as detailed in Appendix I, is given by

q b

where K = C - (A/q)\nb + B lnfe;

q — pb b + 1A = T5——r, ß = -r-, r, C = constant of integration.

For the point of influence of this model,

" / \ * ~ L-J J\ r 1J • 1 1J ) "\f ' 1J J\* JJ I _ n

or

3. THE KKKII MODELKKKII is primarily motivated from models in population dynamics. When the pro-portion of population which has adopted a technological innovation is / , 1 —/represents those, in proportion, who have yet to adopt it. This (1 — / ) fraction ofpopulation is a mix of potential adopters who, in fact, are at different stages ofthe adoption process. Not all are totally ignorant of the new innovation. Lekvalland Wahlbin (1973) give five stages of the adoption process: ignorance, awareness,interest, trial, and adoption. Potential adopters in this subset have varying degreesof technological backgrounds and this advanced nature of the technological back-ground facilitates the adoption of a new technological innovation. This, as viewedby Bundgaard-Nielson (1976), could imply that late adopters will adopt the innova-tion faster than the earlier ones, as they are in a better position to assess the newtechnology than the earlier ones. The state of the technological background of the'yet to adopt' population has a definite impact on the rate of adoption. However, itis not convenient to determine the exactness of this influence. As such, the 1 — / ofthe Fisher-Pry model is replaced by 1 —fa, where the parameter a captures suchan impact.

Another viewpoint could be borrowed from von Bertalanffy (1957). Technologicalgrowth, or the speed at which an innovation is adopted, is affected by two oppositeforces, one trying to increase the adoption rate and the other trying to slow theprocess. Some factors which help in the diffusion rate, for example, are (i) thatchanges in design and new applications may be possible; (ii) that there exists a ballrolling effect or a bandwagon effect; (iii) that the initial performance is likely to besuperior to the existing techniques. On the other hand, some factors which becomebarriers, hence slowing the adoption rate, could include (i) competition from othertechniques or products; (ii) poor market economic conditions; (iii) the fact that

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TECHNOLOGICAL SUBSTITUTION MODELS 181

most of the potential adopters have already adopted the innovation; (iv) shortage ofcapital, and bottlenecks in production, which may limit the supply; (v) the existingtechniques may be quite new for replacement; and (vi) the anticipation, that in thefuture, because of learning, the costs of production and thus the price may go down,and quality may improve.

The growth can be considered a result of a counteraction of both these forces.There will be growth as long as the positive forces prevail over the negative forces.The von Bertalanffy general weight growth equation for a type 2 metabolic growthtype is given by the expression:

von Bertalanffy found that the change of body weight W is given by the differencebetween the processes of building up and breaking down; 77 and k are constantsof anabolism and catabolism respectively and 77 indicates that catabolism is propor-tional to some power of body weight W. The KKKII model can also be viewed ashaving been developed from this concept.

Tt äalso has a closed-form solution as detailed in Appendix II, and is given by

fa faJ - JO cQt

In (12), if a = 1, the logistic law results. The behaviour of this model is exhibited bythe following theorems.

THEOREM KKKII model can be transformed to the Fisher-Pry model by substituting

Proof Substituting fa = z in (12), we get

which is the logistic, Blackman, Mansfield, or Fisher-Pry (1971) model. D

THEOREM For the KKKII model, (i) the point of inflection occurs at f* = (1 +a)-1/» and this increases monotonicalty from Oto 1 as a increases from —1 to 00; (ii)as a increases both f(t) and df/dt decrease; (iii) the curve corresponding to a = Oisthe Gompertz curve (1825), and the curve corresponding to a = 1 is the logistic curve.

Proof Since

there is a point of inflection when /* = (1 + a)~lla. It is proved in Appendix IIIthat /* is a monotonie increasing function. It increases from 0 to 1 as a increasesfrom — 1 to 00.

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182 V. KUMAR AND U. KUMAR

Again,

à (df\_ a(-/°ln/)da\dt ) qj a-

It is proved in Appendix III that this is always negative, so that df /dt and / aredecreasing functions of a.

Further,

which corresponds to the Gompertz curve, and

lim = o/(l — / ) (14)a—i a j-> \ •> / 1

which correspond to the logistic curve. D

4. THE KKKIII MODEL

The two 5-shaped models of technological substitution that provide the backgroundand the motivation of the KKKIII model are the Fisher-Pry model (1971) and theFloyd model (1968) given below respectively:

-jr-=qf(\-f) (2)

%-qm-ff (15)

Fisher-Pry model is a special case of Blackman's model of technological substitu-tion as it considers the upper limit of the market share, that an innovation cancapture, in the long run to be 100 percent. Martino (1983) discusses the problem ofoverestimation and underestimation of these models. Except for the situation wheresubstitution has reached near completion at the time of the forecast, Fisher-Prymodel gives an overestimation of the forecast while Floyd's model gives an under-estimation of the forecast. What is needed is a smooth S-shaped curve which lieswithin the region bounded by the Fisher-Pry and Floyd's curves which eliminatesthe optimism of the former and the pessimism of the latter. Kapur et al. (1991) sug-gested an approach of developing new models by combining them linearly in certainfashion. This approach is based on the following theorem:

THEOREM Let

df df

(where <pi(f), <h(f) are functions satisfying conditions given in Section 2 representtwo IDM's; then

a>0, b>0 (17)

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TECHNOLOGICAL SUBSTITUTION MODELS 183

represents an IDM with its point of inflection f* tying between the point of inflection/j* and /2* for the two given models, provided that

has onty one real root between 0 and 1.

ILLUSTRATION Obtained from the linear combination of the models by Floyd andFisher-Pry,

. "ij v* ./ ,/ • "ij v j j ~ "J \*- j jy.* H^I " - e - I (.1°)

where fi = fc/(a + ft) and c = (a + b)q.

This IDM is the KKKIII model.Similar to the model given by Sharif-Kabir which also lies between the region

bounded by the Fisher-Pry and Floyd's curves, the point of inflection of The KKKIIIlies between | and j . The parameter /¿ can be called the coefficient of pessimismor the delay coefficient. Similar to a parameter of the Sharif-Kabir model, fi repre-sents data scatterness, data extent, market share, and the effective life span.

An Alternate Derivation of the KKKIII Model

Let the government set aside a total sum c for subsidizing adoption. The model is

-j- = f (amount of remaining subsidy)

The impact of c on the growth rate will now be considered.Let us take a special case when the subsidy per adoption is constant, so that when

a fraction / of the potential adopters has adopted the innovation, an amount cfout of the total subsidy will have been disbursed and the remaining amount will bec — cf, so that the model becomes the Fisher-Pry model, with a new interpretation.

Further, let the subsidy decline at a constant rate relative to the proportion ofinnovation adopters, so that if s is the subsidy given,

ds— = -A:, s = so - kf.

Since the total subsidy to be given is c, we have

y1 kc= (so-kf)df = so--r

Jo ¿Thus,

k kso = c + - and ^ = SfVmï> = s0-^. (19)

The total subsidy spent until a fraction /

rf 1

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184 V. KUMAR AND U. KUMAR

and the remaining subsidy is c-sof + \kf2 = c — (c + k/2)f + \kf2. Thus, themodel becomes

From (19), it is obvious that c > k/2 or k/2c < 1. If k/2c = 1, then the KKKIIImodel becomes the Floyd model.

If k/2c = \i < 1, it leads to a generalized model

^ = c / ( l - / ) ( l - / x / ) , 0</x<l . (20)

This is the KKKIII model.Consideration is given to the point of inflection of this KKKIII model given by

(20).Here

Sf/dt1 = 0(J) = 3/x/2-Sf/dt1 = 0(J) = 3/x/2 - 2/(/i + 1) + 1 = 0.

Thus, the point of inflection arises when

f

Here

so that the quadratic equation of 0(/) = 0 has two roots, one between 0 and 1, andthe other between 1 and oo. For current purposes only the root between 0 and 1 isconsidered. Thus the point of inflection arises when

If ¡i = 0, the KKKIII model reduces to the logistic model, and has a point ofinflection at /* = | .

If /i = 1, the KKKIII model reduces to the Floyd model, and has a point of in-flection at /* = | .

If 0 < fi < 1, the point of inflection lies between 5 and | . Thus all KKKIII curveswill lie between the logistic and the Floyd curves.

5. THREE (MODEL) ILLUSTRATIONS

To determine the applicability of the three substitution models of innovation diffu-sion, the diffusion of the following three technological innovations (TIs) is consid-

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TECHNOLOGICAL SUBSTITUTION MODELS 185

ered:

TI-1: Color televisionTI-2: Nylon tire cordTI-3: Ultrasound equipment

Of these, TI-1 is a consumer durable innovations; TI-2, an industrial innovation; andTI-3, a medical innovation. Time series data for the diffusion of each innovation arefitted on the three proposed models (KKKI-III) of this study as well as on some ofthe better known existing models.

Following the criteria proposed in Kumar and Kumar (1992) (i.e., the modelsshould be relatively independent to each other and contain not more than fourparameters) for choosing the models for comparison, we came up with the followingfour existing models: the Sharif-Kabir (1976), the NUI (Easingwood et al, 1983),the Floyd (1968), and the GRMI (Skiadas, 1986). To this list the following twomodifications have been made: First, we have added the Fisher-Pry (or the logistic)model (1971) since it is perhaps the most popular model, and second, the GRMImodel was replaced by the Generalized model (also given by Skiadas, 1986), sinceSkiadas illustrated in his paper that the generalized model gives better performancethan the GRMI. The final list of the five existing models selected for the purposeof data fitting are given below:

(i) the Fisher-Pry modeldf A (22)

(ii) the Generalized model

—j- = q- (1 — / ) , (23)

(iii) the NUI model

dt (P + 9 / ) ( l - / ) (24)

(iv) the Sharif-Kabir model

Tt = l-){\-a) (25)

(v) The Floyd model

^- = qf{l-ff (26)

Here, p, q, g, ô, and a are the model parameters.A nonlinear regression analysis algorithm was used to define the parameters of

various models for the time series data of each of the three technological innova-tions. Predictions for a few years ahead in each case were also made. For assessingthe fit of the model, an emphasis has been placed on the explanatory ability of themodels used, and for this purpose the root mean squared error (RMSE), mean ab-solute deviation (MAD), and the coefficient of determination R2 were estimated.

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186 V. KUMAR AND U. KUMAR

The RMSE and MAD between actual and predicted values were also computed.The better model was considered to be the one which showed low RMSE andMAD.

What follows now in this section are the summary results of testing of the timeseries data of three TIs on five of the existing and the three proposed models.

TI-1: The Case of Color Television

Since the evolution of the color TV over thirty five years ago, its market followeda typical S-shaped pattern exhibited by many new consumer durables. Its low initialsales in the U.S. for approximately ten years was followed by a sharp upward trendin the mid-sixties. By 1973, almost 70 percent of households owned at least one colorTV. Since 1973, a generally declining first-time buyer sales has been observed. Bayuset al. (1989), the source of first time buyer data for our analysis, pointed out thata single approach or a model was inadequate to accurately forecast sales over theentire period since the introduction of color TV. They advocated for disaggregatemodelling approaches where sales to each of the major buyer type (e.g., first-timebuyers, replacement buyers, additional-set buyers, institutional buyers) are estimatedseparately. Our concern is only with the first-time buyers' data.

Empirical testing results of time series data (1956-1978) of first-time color TVbuyers and predictions for four years ahead (1979-1982) as shown in Table 1 andTable 4 respectively indicate that the KKKI, the KKKII, and the NUI models havecomparable results and are more appropriate than the others. Their respectiveRMSE (388.57, 462.06, 437.23) and MAD (283.94, 393.75, 363.31) are quite lowand close to each other. Coefficients of determination are high (.9539, .9349, .9416).These three models also give good predictions.

TI-2: The Case of Nylon Tire Cord

Historically, the products used as tire cord textiles are cotton, rayon, nylon, polyes-ter, fiberglass, and Kevlar. Reliable data on the diffusion of these products in theU.S. is available from the Textile Economic Bureau. Kovac (1969) in a comprehen-sive study "Tire Reinforcing System" estimated the life cycle of a tire cord to beabout 35 years. In addition, Kovac estimated that a new tire fabric is introducedevery 10 to 15 years. The Nylon-tire cord was introduced in the late forties butpeaked about 1967 and is declining. It substituted rayon and was substituted pri-marily by polyester. In fact, the four technologies, viz. polyester, fiberglass, steeland Kevlar which were introduced after Nylon are still increasing at the time of thisstudy. Merino (1990) is the primary source of data on the nylon tire cord used inour analysis.

The results of empirical testing of time series data (1947-1972) and predictionsfor four years ahead (1973-1976) as shown in Tables 2 and 4 respectively indicatethat the KKKI, KKKII, and the NUI models have comparable results and are moreappropriate than the others. Their respective RMSE (10.58, 9.78, 10.21) and MAD(8.75, 7.11, 6.99) are quiet low and close to each other. Coefficients of determinationare high (.9539, .9349, .9416). These three models also give good predictions.

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TECHNOLOGICAL SUBSTITUTION MODELS

TABLE 1Parameter Estimates, Fit Statistics: The Case of Diffusion of Color TV* (1956-1978)

187

Model

Fisher-Pry

Generalized

NUI

Sharif-Kabir

Floyd

KKKI

KKKII

KKKIII

Parameter Value

q = .3641Ñ= 57392

q = .3384g = 1874N = 58168

p = .0181q = .2082S = .4172N = 64716

q = .5100<7 = .9800N = 70473

q = .5118F = 70665

p = -.0029q = 1.2844b = 9.9557N = 72751

q = .1919a = -.1631N = 64291

q = .5117p. = .98TV = 69970

MAD

616.43

645.41

363.31

467.01

464.91

283.94

393.75

465.00

RMSE

875.27

785.43

437.23

643.84

641.35

388.57

46106

641.45

.7662

.8117

.9416

.8735

.8744

.9539

.9349

.8744

*First-time buyers in thousands of units

Parameter, W is the upper limit of the potential adopters

TI-3: The Case of Ultrasound

Ultrasound is one of the important radiology innovations. Data for ultrasound arenot from census and were originally obtained from a survey of 209 hospitals through-out the U.S. (Schmittlein and Mahajan, 1982). Surveyed hospitals were asked toidentify themselves as adopters or nonadopters of ultrasound technology. If adoptersthey were asked to provide the date of adoption. The survey indicated that by 1978,of the 209 hospitals, 168 hospitals (80.4%) had adopted this innovation.

Empirical testing results of time-series data (1965-1975) and predictions for twoyears ahead (1977-1978) as shown in Table 3 and 4 respectively show that the threeproposed models KKKI, KKKII, and KKKIII give comparable results relative to theother analyzed models.

In summary, the three sample innovations taken in this study represented a di-versity of innovations and data types (data for consumer durable color TV are totalsales to the U.S. first-time buyer households, the data for industrial innovation of

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188 V. KUMAR AND U. KUMAR

TABLE 2Parameter Estimates, Fit Statistics: The Case of Diffusion of Nylon Tire Cord* (1947-1972)

Model Parameter 'Value MAD RMSE R2

Fisher-Pry

Generalized

NUI

Sharif-Kabir

Floyd

KKKI

KKKII

KKKIII

*In million of pounds

q = .2563W =4981

q = .2377g =92.112

N = 5179

p = -.0013q = .1156ó = .6N = 7511.5

q = .2738<r = .5N = 6358.2

^ = .2895ÏV = 7222.6

p = .0009q = .3736b = 5.3891N = 9712.3

q = .1042a = -.0305N = 7597.2

q = .2893/i = .9O00W = 6873.1

17.82 2Z07 .9630

14.86 17.19 .9775

6.99 10.21 .9920

15.25 19.43 .9713

13.87 17.53 .9766

8.75 10.58 .9915

7.11 9.78 .9927

17.56 13.89 .9766

nylon tire cord are the amount of the nylon used by the U.S. tire manufacturers, andthe data for ultrasound, a medical innovation, were based on a survey of hospitals).The empirical testing of various diffusion models on these innovations seems toindicate that the proposed three models give better or comparable results than mostof the quiet popular models.

6. CONCLUDING REMARKS

In this paper, three substitution models of technological innovation diffusion havebeen suggested which satisfy almost all the desirable conditions set forth. The KKKImodel, is quite flexible, since the point of inflection for this model can occur any-where between 0 and 1. All three models admit a closed-form solution and canaccommodate both symmetric and nonsymmetric behavior patterns. These modelsrequire two or three parameters. In this sense, these proposed models are parsimo-nious models which have most of the desired characteristics of a diffusion model.Each model has been formulated with a definite objective and motivation. The

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TECHNOLOGICAL SUBSTITUTION MODELS 189

TABLE 3Parameter Estimates, Fit Statistics: The Case of Diffusion of Ultrasound (1965-1975)

Model Parameter 'S&lue MAD RMSE R2

Fisher-Pry

Generalized

NUI

Sharif-Kabir

Floyd

KKKI

KKKII

KKKIII

q = .478477 = 208

q = .5037g = -1.853777 = 200

p = -.0247q = 1.15386 = 1.8713AT = 152

q = .4784(7 = .0200N=212

q = .493977 = 338

p = -.0028q =.44866_= - .55

q = .8654a_=2.2

p = -.0052q = .5175

2.6039 3.4078 .8585

2.6757 3.3798 .8608

2.2953 2.8215 .9030

2.6046 3.4085 .8584

2.7110 3.543 .8585

2.4476 3.1377 .8800

2.4062

2.7647

3.0605

3.4821

.8858

.8523

77 = 195

TABLE 4Forecast Performance of the Models

Model

Fisher-PryGeneralizedNUISharif-KabirFloydKKKIKKKIIKKKIII

Color TV(1979-1982)

MAD

180Z661651.80687.31

1025.251015.82741.19883.77

1016.44

RMSE

1806.241652.86704.82

1026.051016.63966.95885.96

1017.22

Technological Innovation

Nylon Tire Cord(1973-1976)

MAD

90.9573.1814.4162.1047.0219.2714.2736.35

RMSE

94.3076.9918.6264.9250.1921.6717.5540.31

Ultrasound(1977-1978)

MAD

&177.043318.21

12.251.811.315.62

RMSE

8.237.074348.26

12.462.22IM5.66

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190 V. KUMAR AND U. KUMAR

significant properties of each of the three models are discussed. The KKKII andKKKIII models in the current form do not take into account the external influ-ence. However, this influence can be easily considered by a simple modification ofthe models. These three suggested models will undoubtedly give a wider choice ofmodels to the users.

ACKNOWLEDGMENT

The authors are grateful for the helpful comments of Patrick Doreian and the refer-ees. Authors also thank Dr. J. N. Kapur for helpful, positive comments and sugges-tions on the above models.

APPENDIX I

The closed-form solution of the KKKI model:df _ (p + qf)(l-f)

Tdt (1

Let 1 + bf = y or / = (y - T)/b and

Tt » ¿I ' ^Substituting (A.2) in (A.I), we get

hyTt =(Pb~4+qy)(.b-y +*)or

yTT) = yThe left hand side of (A.3) can be written as

L.H.S. = ^(/Jô-ç + çy) ( 6 - y + l)"

Using the cover-up rule

(A.4)

Substituting the above values of A and B in (A.3),

(q-pb)dy + (fc+l)dy = di ( A ^

Integrating (A.5), we get

— ln(p + qf)-Bln(l-f) = -+K (A.6)

where .y — / " • " ^ i** i* i D i« i*

A. = c — — lno + i>]no9

and where A and B are given by the equation (A.4).

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TECHNOLOGICAL SUBSTITUTION MODELS 191

APPENDIX II

The closed-form solution of the KKKII model:

dt aIntegrating (B.I)

Let fa = z, then / = z1/", and df = {l/a)zifa-1dz. Substituting in (B.2)

/ZO

= In In1 - z 1 - z0

a 1 _ fa— 2 / o

APPENDIX III

Some Properties of the KKKII Model

H e r e ,

and the point of inflection

r = (l + a)-V<*=h(a) (say). (C.2)

Now

\nh(a)) =ln(l + a ) - a Jk(a)

a2 a(l + a) a2(l + a) a2(l -

For a -» 0

1 1,. ( l + a ) l n ( l + a ) a ,. l n ( l + a ) ,.h m ^ hr^ xr = lim —±z = hm — = - .a-.o a2(l + a) a-o 2a a-o2(l + a) 2

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192 V. KUMAR AND U. KUMAR

Since k'(a) = ln(l + a) = 0 according as a = 0; a > —1, k(a) is an increasing func-tion of a for a > 0, it is a decreasing function of a for a < 0. Also k(0) = 0 from(C.3). It can be easily shown by plotting a graph of k(a) versus a that k(a) > 0.

. ' . ^(lnfc(a)) > 0 for a = 0

.". ln/i(a) is an increasing function of a

Therefore /* is a monotonie increasing function of a.Further for a —• — 1, /* -> 0; for a = 0, /* = 1/e; and for a —> oo, /* -> 1, i.e., as

a increases from —1 to oo, the point of inflection occurs between 0 and 1 for higherand higher values of / .

Also, from (C.1)

Now, g(x) < 0 when x > 0

. . ^ - J u negate

or — is a decreasing function of a

or f(t) is also a decreasing function of a.

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