Log-logistic model

Here, Equation (28) is the log-logistic form seen in Meade and Islam (2010b) and as initially proposed by Tanner (1978):

𝑌(𝑡)=

𝐿 1 + 𝑎𝑒(−𝑏𝑙𝑛(𝑡))

(28)

This function has the b in the simple logistic replaced by the natural log of b thus providing an accelerating impact of that parameter with time. Like the Gompertz, it was originally envisaged as a mortality model, and it brings an inflection point that is flexible between 0 and 0.5 of share.

3.3.7.

Extending the logistic to substitution situations

The type of technological decline of technology investigated in this thesis is that linked to the diffusion of new technology with superior utility. Two well-known models are linked in the literature to this substitution effect. The Fisher and Pry (1971) model described in this section and the Norton and Bass (1987) model described in Section Error! Reference s ource not found., which follows on from this section. Both models link diffusion and decline directly. Fisher and Pry (1971) working with the linear form of the simple logistic model, developed the first generally recognised model of substitution of one technology for another. Their model was an internally influenced model unlike the Bass model’s mixed- influence foundations, so was less linked to Rogers’ theories than the later Norton and Bass model built on the mixed-influence Bass model. They demonstrated useful accuracy and consistency of performance in technology underpinning product categories such as paint, cleaning products, and building materials. Fisher and Pry made three important assumptions, which constitute rules for their method, assumptions that today are integrated into the thinking of forecasters (Fisher & Pry, 1971):

 Many technological advances are primarily competitive substitutions of one method of satisfying a need with another.

 If substitutions progressed to “a few percent”, then they will progress to completion.

 That fractional rate of (fractional) substitution of new for old is proportional to the remaining amount of the old not yet substituted (a special form of a law developed by Raymond Pearl (1925)).

Fisher and Pry used the linear form of the logistic function described by Equation (29):

𝑓 = 1

2[1 + 𝑡𝑎𝑛ℎ⍺(𝑡 − 𝑡0)]

(29)

Where: f is the fraction of takeover by the new technology, tanh is the hyperbolic tangent function, ⍺= ½ the initial annual exponential takeover rate, and T0 is the year in which the

new technology captures 50 percent of the usage or f = ½.

Their method involved transforming the data via a log-linear transform to a reasonably straight line, although Griliches and others did similar transforms. The functional form is in Equation (30):

𝑓 1 − 𝑓= 𝑒

2⍺(𝑡− 𝑡₀) (30)

Fisher and Pry pointed out that population growth and per capita consumption were not included directly in the model and that understanding the unit of substitution required significant thought for the model to be calibrated correctly. Their observations are significant because almost all subsequent models require the same assumptions, although it is often implicit. The area of time dependent market size has been addressed by a range of authors (Centrone, Goia, & Salinelli, 2007; Kalish, 1985; Mahajan & Peterson, 1978; Sharif & Ramanathan, 1981), but models with adjustable market size capability have not come into common use.

Fisher and Pry demonstrated the model’s potential effectiveness at predicting the future with historical data for 17 substitution situations, for example: synthetic versus natural fibres, plastic versus leather, and synthetic versus natural rubber. They noted that progress of the substitution to a few percent market share penetration was a critical requirement. Mathematically, this was required to deal with their function, which starts at some infinite time in the past. From a pragmatic perspective, it probably allows the data of substitution to become viably visible. This viably visible test is important because you need a viable trend in order to recognise that diffusion is indeed following the trajectory of a successful technology. Moreover, you need to get above the noise floor to the extent that the data of interest is at a stronger level than measurement and stochastic noise levels in the data. In their case, they believed that between diffusion = 10 percent share and diffusion = 90 percent share their model functioned well. Much earlier, Griliches similarly trimmed the two

asymptotes but to share = five percent and share = 95 percent. Fisher and Pry (1971) felt in their method, that their forecasts could be relied on if used after an initial 10 percent market share diffusion (effectively left truncating the data).

The trimming of the upper limit (right truncating) is rarely if ever mentioned, and is important because beyond 90 percent, a typical S-curve function takes a very long time to approximate 100 percent and data can vary extensively from pattern as it moves asymptotic to that limit. Hence including the last 10 percent of growth in your forecast can often make the forecast errors appear very large in the time domain. Despite the limited discussion in the literature, the practice of trimming out a portion of the top and bottom data has become common practice. This is discussed in detail section 6.4.5.

3.3.8.

Substitutions over successive generations

A significant body of the literature on modelling successive generations of technology (one substituting for the other over time) builds on Frank Bass’ model of growth for consumer durables (Bass, 1969), in particular on Norton and Bass (1987). (Norton & Bass) sought to provide a predictive model for initial diffusion and subsequent waves of substitution of products into a market. It is a model focusing on the growth of the technologies in a category over time. As in the earlier Bass diffusion model (Bass, 1969), at their model’s foundation is the impact of innovators which diminished with time and the role of the imitators, who rely on promotion and word-of-mouth for confirmation of the suitability of their purchase and whose impact on diffusion rises over time to become dominant. Additionally, the Norton and Bass model also incorporates substitution theory, where those later generations compete with earlier generations. Their model does this by using a concept of generations of innovation, each generation modelled as gaining from substitution of the earlier generation and losing to the later generation by its substitution.

Three of their generations are expressed in Equations (31), (32), and (33) below. Modelled for three generations it is:

1st Generation. _𝑆 1(𝑡)= 𝐹(𝑡1)𝑙1[1 − 𝐹(𝑡−𝑡2)] (31) 2nd Generation. _𝑆 2(𝑡) = 𝐹(𝑡−𝑡2)[𝑙2+ 𝐹(𝑡1)𝑙1][1 − 𝐹(𝑡−𝑡3)] (32) 3rd Generation. 𝑆3(𝑡) = 𝐹(𝑡−𝑡3)[𝑙3+ 𝐹(𝑡−𝑡2)[𝑙2+ 𝐹(𝑡)𝑙(1)]] [1 − 𝐹(𝑡−𝑡4)] (33)

Where: 𝑆𝑖(𝑡)is the shipments (sales) of generation i at time (t-𝑡𝑖) being the time since

introduction of generation i,𝑙𝑖 i = the incremental potential served by the ith generation, not

capable of being served by any earlier generation (Norton & Bass, 1987).

Where: _𝐹

(𝑡) =

1 − 𝑒−(𝑝+𝑞)𝑡

1 +𝑞_{𝑝 𝑒}−(𝑝+𝑞)𝑡

(34)

Equation (34) is the cumulative distribution from the original Bass model and assumes that

p and q are the same value across the generations (Bass, 2004).

In a more generalised variant of the Norton and Bass model (Norton & Bass, 1987), p and q

could be varied by generation. However, Bass (2004) said of that model, that it did not appear to be much of an improvement, consequently, providing some support for the use of the simple version. While there are many substitution models that operate with share of the market as the unit of diffusion, the Norton and Bass model gives us insight into the size of the market. Norton and Bass demonstrated the usefulness of this model through the fitting of their model to early data for the diffusion and subsequent substitution of dynamic RAM and static RAM memory, and microprocessor and microcontroller technologies. They predicted the growth of that technology from this early data and compared it with the data they held for later years, in what is now the tradition of technology forecasting model testing. Norton and Bass noted two behaviours not covered by their model. First, the identification of those that were late adopters and opted for a later generation technology rather than the first generation (the leapfrogging phenomenon). Second, those adopters that switched or substituted their earlier choice for a later generation (another form of leapfrogging but perhaps also characterised as switching). Furthermore, they felt there was potential for adapting the model to deal with the situation where only partial substitution takes place even after extended periods. Most recently, Jiang and Jain (2012) have addressed both

generational leapfrogging and switching while still retaining a model that simplifies to the original Norton and Bass model. This model has yet to be demonstrated as a superior forecasting tool compared to the original Norton and Bass models by anyone else.