HIST363: Global Perspectives on Industrialization

The literature on the diffusion of innovations has generally found that the diffusion process follows an S-shaped or sigmoid pattern. The diffusion process takes off rather slowly, then, after a while, it accelerates, and finally it slows down until a certain saturation level is reached (for two insightful overviews of the literature on the diffusion of inventions, see Lissoni and Metcalfe and Stoneman). Also in our case, the dynamics of the cumulative number of engines installed seems to follow, in most counties, an S-shaped profile. Following the pioneering contribution of Griliches, it has been common to estimate a simple symmetric logistic growth equation such as in order to provide a characterization of the diffusion process ¹⁴:

$N_{t} = {\frac{K}{{1 + {\text{e}}^{( - a(t - b))} }}}$

In this perspective, the analysis of data on innovation diffusion using a generalized version of the simple logistic growth equation originally suggested by Richards in the study of growth processes in the field of botany seems particularly promising. The Richards equation is given by:

$N_{t} = {\frac{K}{{\left( {1 + T \cdot {\text{e}}^{( - a(t - b))} } \right)^{(1/T)} }}}$

In Eq. 2 the definition of the variables and parameters is the same of Eq. 1. In comparison to Eqs. 1, 2 contains a fourth shape-parameter, $T$. It can be shown that the Richards equation encompasses the logistic and the Gompertz as special cases: when the parameter $T = 1$ we are in the case of the simple logistic equation, when $T → 0$ the Richards equation tends towards the Gompertz growth curve. The chief advantage of the Richards equation is clearly that by containing the additional shape-parameter $T$, it can be used to characterize a wide variety of sigmoid patterns with different positions of the inflection point. In general, if $T < 1$ the inflection point of the curve will be located when $N_t  < K/2$, instead if $T > 1$ the inflection point will be situated in correspondence of a value of $N t  > K/2$.

Using non linear least squares, we have estimated both the simple logistic equation ("symmetric") and the Richards equation ("generalized") for each county and adopted as preferred model the one that provided a better fit of the data (we have done this exercise for counties with at least 10 engines installed in the period 1700–1800). In order to characterize the relative speed of the diffusion process in each county we have calculated the parameter $Δt$, which is the number of years needed to grow from the 10 to the 90% of the estimated saturation level $K$.¹⁶ The results are reported in Table 2. The full estimates are instead reported in Table 5. The estimated diffusion paths for Newcomen and Watt engines are represented in Figs. 2 and 3 together with the cumulative number of engines (represented by the dots).

$Δt$ is the number of years necessary for moving from 10 to 90% of the estimated saturation level. G = Generalized, S=Symmetric.

Fig. 2

Estimated diffusion paths for Newcomen engines

Fig. 3

Estimated diffusion paths for Watt engines

In the case of Newcomen engines the generalized specification is preferred in 11 counties whereas the standard symmetric logistic in 10 counties. In the case of Watt engines, instead, the generalized specification is preferred in 2 counties whereas the standard symmetric logistic in 9. Note that in the case of Newcomen engines, for ten counties the estimates of the parameter $T$ are greater than 1. This means that the S curves display an asymmetric growth pattern where the point of inflection is situated to the right of the point $N_t = K/2$. This finding may perhaps be interpreted as an effect of the competition of Watt engines in the last 25 years of our time window causing a progressive slowing down of the rate of growth of Newcomen engines.

Figure 2 and Table 2 reveal some other interesting aspects of the spread of Newcomen engines. There is a group of "pioneer" counties characterized by relatively fast diffusion rates $(Δt$. These locations are Cornwall where steam engines were employed in copper and tin mines, some typical "textile" counties like Lancashire (cotton) and West Riding and Gloucester (wool) and the Scottish counties (Ayr, Lanark, Fife and Stirling). It is worth noting that in Scotland the diffusion begins only in the second half of the eighteenth century (around 1760). Presumably, the establishment of the Carron ironworks (which made use of the cylinder boring machine designed by John Smeaton) in Stirling in 1760 spurred the rapid adoption of steam power in several Scottish counties from the early 1760s.¹⁷ Coal mining areas of the Midlands (Derby) and of the North East (Northumberland and Durham) together with the London and Nottingham, Stafford and Leicester are characterized by intermediate rates of diffusion $(60 < Δt < 100)$. Finally there is a group of "laggard" counties with relatively slow diffusion rates $(Δt > 100)$: Warwick, Shropshire, Cumberland and Worcester.

Table 2 also shows that, in general, Boulton and Watt engines were characterized by a faster diffusion rate. The average rate of diffusion $(Δt)$ for Newcomen engines is equal to 68 years, whereas for Watt engines it is equal to 18 years.

As in the case of Newcomen engines, the adoption of the Watt engine in various locations also appears to be characterized by a sequential order. Also in the case of Watt engines, we can identify a group of "pioneer" counties such as Cornwall, Shropshire, Northumberland, Durham and Lancashire where the diffusion process is particularly fast $(Δt < 20)$. We have regions with intermediate rates of diffusion such as Cheshire and West Riding $(Δt = 20)$. Finally, we have "laggard" counties with relatively slow diffusion rates such as London, Stafford and Warwick $(Δt > 24)$.

This general exploration of the patterns of diffusion confirms that steam engine technology by the end of the eighteenth century had already become a source of power capable of being used in a wide variety of production processes and in different local contexts. Of course, this exercise is simply meant to provide a broad characterization of the diffusion process in different locations. As noted by Dosi:

In other words, our reconstruction of the patterns of diffusion needs to be integrated by further research on the "microbiology" of the diffusion process.