Abstract
This chapter provides a theoretical framework of technology diffusion, which is defined as a dynamic and time-attributed process involving the transfer of information, knowledge and innovations, and standing for a continuous and gradual spread of new ideas throughout large-scale and heterogeneous societies. First, it extensively discusses theoretical technology diffusion concepts and models, explaining the technology diffusion trajectories by the use of S-shaped curves. Second, it presents the fundamental ideas and models standing behind the idea of technological substitution. Third, there is demonstrated a novel approach to identification of the ‘technological take-off’ and ‘critical mass’ effects with respect to the dynamics of the technology diffusion process and its prerequisites. Finally, based on theoretical frameworks derived from economic growth theories, it shows conceptualizations of technology convergence and technology convergence clubs.
(…) diffusion concerns issues that are among the more difficult to analyze adequately. Time is involved. Uncertainty is inherent. Change is the major topic. Imperfect markets abound
Paul Stoneman (2002)
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Notes
- 1.
In his work ‘Diffusion of Innovation’, E. Rogers presents 508 different case studies explaining the diffusion of different innovations adopted by both companies and individuals in rural areas (see Rogers and Havens 1962).
- 2.
Agents.
- 3.
The rate of diffusion is additionally associated with the concept of ‘critical mass’ and it reveals ‘network effects’—explained in Sect. 3.3.
- 4.
The unique characteristics and basic mathematics related to sigmoid curves are explained in Sect. 3.2.
- 5.
Goeffrey Moore, in his book ‘Crossing the Chasm’ (1991), proposes a modified version of Roger’s bell-curve. He emphasizes the role of ‘disruptive innovations’ that generate the chasm (gap, discontinuities) between the group of innovators and the early adopters and the group of the early majority, the late majority and the laggards.
- 6.
The concepts and mathematics underlying logistic growth are explained in Sect. 3.2.2.
- 7.
‘Word of mouth’ models are also labelled ‘contact’ or ‘disease’ models.
- 8.
The benefits from the adoption of new technology are mainly associated with introducing ‘process innovation’ that underlies company performance. This can be conditioned, inter alia, by prospective profitability, expected risk, organizational structure and other factors which may impact outcomes for a company.
- 9.
The models, are labelled ‘stock’, as diffusion in time (t + 1) depends on the stock (number) of given technology users in period ‘t’.
- 10.
The logistic equation is also recognized as the Verhulst-Pearl equation, as Pearl and Reed (1922), in the early 1920s already adopted similar formulas in the biological sciences.
- 11.
Referring to Benjamin Gompertz (1825) and his ‘law of mortality’, which is a mathematical specification to model time-series (Gompertz model, Gompertz growth).
- 12.
Base of naatural logarithms.
- 13.
Following Meyer et al. (1999), we define \( \left(1 - \frac{Y(t)}{\kappa}\right) \) as a ‘slowing term’ (‘negative feedback’), which is close to 1 as \( Y(t)\ll \kappa \), but if \( Y(t)\to \kappa \) then \( \left(1 - \frac{Y(t)}{\kappa}\right)\to 0. \)
- 14.
For estimates of the asymmetric responses 5-parameter logistic functions (5PL) are applied. A standard 5PL is as follows (Gottschalk and Dunn 2005): \( y=f\left(x;p\right)=d+\frac{\left(a-d\right)}{{\left[1+{\left[\frac{x}{c}\right]}^b\right]}^g} \), where \( p=\left(a,b,c,d,g\right) \), \( c>0 \) and \( g>0 \). If we restrict \( g=1 \), a 4-parameter logistic function is generated.
- 15.
The parameters in Eqs. (3.14 and 3.15) can be estimated by applying ordinary least squares (OLS), maximum likelihood (MLE), algebraic estimation (AE), or nonlinear least squares (NLS). As Satoh and Yamada (2002) suggests, NLS returns the relatively best predictions, as the estimates of standard errors (of κ, β, α) are more valid than those returned from estimation using other methods. Adoption of NLS allows avoiding time-interval biases, which are revealed in the case of OLS estimates (Srinivasan and Mason 1986). However, the main disadvantage of the NLS procedure is that estimates of the parameters may be sensitive to the initial values in the time-series adopted.
- 16.
Also labelled ‘width’.
- 17.
The parameter α as such, is not economically interpretable, thus it is exclusively estimated to calculate the ‘specific duration’.
- 18.
Also labelled ‘characteristic duration’ or ‘specific time’.
- 19.
If a Fisher-Pry transform is applied for normalization, then the logistic curves become linear, which additionally facilitates further analysis of growth sub-phases.
- 20.
Conceptually, technological substitution models refer to the seminal works of Alfred Lotka (1920) and Vito Volterra (1926), who were the first to introduce a generalized version of the logistic growth equation. They developed a model of competition among different species in biological systems (Voltera) and chemical chain reactions (Lotka). Today, the Volterra-Lotka competition equation is widely adopted for qualitative analysis of technological substitution if at least two competing technologies are involved.
- 21.
Relaxing the assumption of a fixed total number of users would allow the system to grow infinitely, which is not the case in real-data based empirical studies.
- 22.
Theodore Modis (2003) distinguishes six ways that two competitors can affect the growth rate in a competitive system. These are: (1) pure competition (competitors need to fight to survive in the same environment, as they use the same resources, which are limited); (2) predator-prey competition (one competitor is labelled prey and the second the predator—the ‘predator’ population grows as there are abundant ‘preys’; this kind of competition generates cyclical growths and declines in populations of ‘predators’ and ‘preys’. Lotka-Volterra equations are applied to describe this kind of competition; (3) symbiosis (competitors are interrelated as the existence of the first is totally dependent on the existence of the second); (4) parasitic (the first competitor benefits from the second, but is does not affect the latter’s existence, also labelled ‘win-impervious’ competition); (5) symbiotic (the first competitor benefits from the second, but the latter is negatively affected by the competition but remains indifferent to the loses, also labelled ‘loss-indifferent’); (6) no competition (the two competitors are not overlapping each other as they use different resources to survive.
- 23.
Oliver et al. (1985) recall that the critical mass effect is also known as the ‘snob and bandwagon effect’, the ‘free rider problem’ or the ‘tragedy of commons’.
- 24.
Self-sustaining.
- 25.
Many claim (see, e.g. Bonacich et al. 1976; Frohlich et al. 1971; or Hardin 1982) that Olson’s concept of critical mass was too general and unconditional and so it did not allow for any mathematical formalization. Additionally, their experiments have proved that Olson’s concepts was not correct, as in many cases people’s real behaviour does not confirm Olson’s assumptions.
- 26.
The notion of critical mass is also known as ‘installed base’ (Grajek and Kretschmer 2012).
- 27.
In fact, they precondition the value of critical mass on prices, arguing that lower prices require lower critical mass, to assure sustainability of the diffusion process.
- 28.
For the formal specification, see Sect. 3.2.
- 29.
- 30.
Baraldi (2012) specifies the network effects as: \( {X}_{i,t}=f\left[{\left(\frac{GDP}{population}\right)}_{i,t},\ {p}_{i,t},\ g\left({X}_{i,t-1}\right)\right] \), where i denotes country, and t the time period. X i,t is thus the installed base, p i,t is price, \( {\left(\frac{GDP}{population}\right)}_{i,t} \) is GDP per capita, and \( g\left({X}_{i,t-1}\right) \) reveals network externalities in country i at time t. To control for concavity, \( g\left({X}_{i,t-1}\right) \) includes a squared term for the lagged installed base. To estimate the size of the critical mass, Baraldi (2012) follows Rohlfs (1974), Katz and Shapiro (1985) and Economides and Himmelberg (1995a) and formalizes the inverse demand function as \( {p}_{i,t} = \alpha +{\beta}_1bas{e}_{i,t}+{\beta}_2 \ln \left(bas{e}_{i,\ t-1}\right)+{\beta}_3{X}_{i,t}+{\varepsilon}_{i,t} \), where X i,t captures control variables. To assure concavity, \( {\beta}_2>0 \), and \( {\beta}_1<0 \) must be satisfied. If \( {\beta}_2>0 \) and \( {\beta}_2>{\beta}_1 \), the network externalities are revealed and the upward slope of the demand curve emerges. The higher β 2, the sooner the critical mass point is reached.
- 31.
The condition follows: \( U = a+b\left({n}^e\right)>P \), where U is the utility function and P is the product price.
- 32.
In our case, expressed as number of users per 100 inhabitants.
- 33.
In the literature discussing ‘technological catching-up’, the term is often confused with ‘technology convergence’. In effect, it is misleading to use these two terms alternatively. Technological catching-up is the process through which countries benefit from the stock of knowledge available in the rest of the developed world, and goes far beyond simple technology convergence (Rogers 2010). The technological catching-up theories instead seek to answer how technologically backward countries may benefit from their underdevelopment and by diminishing the relative gap in Total Factor Productivity (TFP) experience economic growth (Soete and Turner 1984). The idea of incorporating different aspects of ‘technology’ into growth models traces back to pioneering works by Veblen (1915), Nurkse (1955), Gerschenkron (1962), Rostow (1971), Schumpeter (1984). Nelson and Phelps (1966) were the first to formalize the Veblen-Gerschenkron ‘relative backwardness’ idea and they introduced the idea of the function of technological catching-up depending on human capital and its absorptive capabilities (also argued by Abramovitz (1986)): \( \frac{dA}{dt}/\ A = \varnothing (.)\left(\frac{T-A}{A}\right) \), where T stands for the level of the best practice technology, A is the level of technology in a backward country, and \( \varnothing (.) \) is the function of absorptive capacities. Recently the literature treating international technological catching-up, and technology diffusion and transfer as factors contributing to rapid economic growth is pervasive. The most prominent evidence can be found in works by, inter alia, Fagerberg (1987, 1994), Perez and Soete (1988), Verspagen (1994), Dowrick (1992), Ben-David (1993), Coe and Helpman (1995), Barro and Sala-i-Martin (1990), Keller (1996), Bassanini et al. (2000), Dowrick and Rogers (2002), Castellacci (2002, 2006a, b, 2008, 2011), Liebig (2012), Stokey (2012), Shin (2013) and Serranito (2013).
- 34.
- 35.
- 36.
- 37.
Gerschenkron’s ‘relative backwardness’ idea (1962) was formalized in a model by Nelson-Phelps (1966), who argued that the growth of technology in an economically backward country is proportional to the gap between the backward country and the country using the most advanced technological solutions (located close to the Technology Frontier Area) (Gomulka 2006).
- 38.
Productivity convergence.
- 39.
Findlay (1978a), however, argues that the gap to the world technology frontier cannot be too large, and countries located below a threshold value of the gap will not be able to catch-up economically.
- 40.
The coefficient of variation is highly useful in σ-convergence testing if two or more country groups are compared in terms of their internal convergence.
- 41.
If σ-convergence is tested with regard to the coefficient of variation, then the coefficient of variation is \( \frac{\sigma_{i,t}}{\theta_{i,t}} \), where θ i,t is the mean of the tested variable over the whole sample.
- 42.
The σ-convergence hypothesis was tested, inter-alia, in works by de la Fuente (2003), Canaleta et al. (2002), Rey and Dev (2006), Young et al. (2008), Egger and Pfaffermayr (2009), Garrido-Yserte and Mancha-Navarro (2010), Schmitt and Starke (2011), Smetkowski and Wójcik (2012), Delgado (2013) and Thirlwall (2013).
- 43.
Conventionally, Eq. (3.41) is estimated applying OLS. However, if we relax the assumption that the variables are normally distributed, the estimated coefficients might be biased and inefficient. Koenker and Bassett (1978) suggest the adoption of non-parametric quantile regression to avoid the problem. The quantile regression approach is highly useful when the original variable distribution is highly skewed (asymmetric). Standard β-convergence estimates allow for assessment of variable behaviour but are based on the conditional mean, while quantile regression (q-regression, q-convergence) introduces estimates in non-central locations (Koenker 2004; Hao and Naiman 2007). Using the quantile regression approach, it is possible to determine any number of quantiles for estimation, which allows modelling of variable behaviour in any pre-defined location of variable distribution.
- 44.
Also explaining the partial correlation between a variable growth rate and its initial level.
- 45.
The body of evidence on conditional convergence is massive. Seminal contributions in the field were made by, inter alia, Dowrick and Nguyen (1989), Barro and Sala-i-Martin (1990); Mankiw et al. (1992), Quah (1993, 1999), Pritchett (1997), Del Bo et al. (2010), Schmitt and Starke (2011), Barro (2012), and Yorucu and Mehmet (2014).
- 46.
Also labelled ‘stochastic convergence’ (see i.e. McGuinness and Sheehan 1998).
- 47.
The approach for convergence testing using a time-series has been applied in a multitude of studies, e.g. using empirical evidence on inter-regional stochastic convergence, by inter alia, Johnson (2000), Drennan et al. (2004), Alexiadis and Tomkins (2004), Herrerías and Monfort (2013), Lin et al. (2013); or inter-country stochastic convergence as in the works of Datta (2003), Bentzen (2005), and Canarella et al. (2010).
- 48.
The possibility of applying the formula in Eq. (3.46) to use it for absolute convergence testing, however, is determined by specific econometric tests. The most commonly used for this purpose is the Augmented Dickey Fuller test (1979, 1981), which introduces cointegration and unit root procedures to the empirical analysis of time-series.
- 49.
Generally in terms of β-convergence.
- 50.
Quah (1997, 1999) argues that countries may form ‘coalitions’ and behave non-linearly in their convergence patterns for three main reasons: countries’ behaviour along their development paths are heavily preconditioned by other counties (e.g. by trade flows, human labour flows); countries tend to specialize to boost economies of scale; and human capital, culture, social and absorptive capabilities matter for development (see also Abramovitz 1989).
- 51.
The evidence on convergence club identification, mainly with respect to per capita income, can be found in works by, inter alia, Ben-David (1994, 1998), Armstrong (1995, 2002), Dewhurst and Mutis-Gaitan (1995), Fagerberg and Verspagen (1996), Verspagen (1997), Desdoigts (1999), Baumont et al. (2003), Durlauf (2003), Su (2003), Canova (2004), Fischer and Stirböck (2006), Le Gallo and Dall’Erba (2006), Alexiadis (2013b), Lechman (2012c), Song et al. (2013), Brida et al. (2014) and Fischer and LeSage (2014)
- 52.
It is important to note that Baumol (1986) approach to convergence club identification is heavily pre-conditioned by the initial level of per capita income.
- 53.
Cubic specification.
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Lechman, E. (2015). Technology Diffusion. In: ICT Diffusion in Developing Countries. Springer, Cham. https://doi.org/10.1007/978-3-319-18254-4_3
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