Modelling EV adoption with Fisher-Pry models

Electric vehicles (EVs) are an emerging technology with the potential to replace combustion engine vehicles.

This model begins with the assumption that the rate of change in the fraction of EVs (denoted $f$) is proportional to the fraction of EVs and the remaining fraction of the market yet to be substituted. This describes the ordinary differential equation

$$ \begin{align} \frac{df}{dt} &= b f(1 - f) \end{align} $$

Noting that this ODE is separable, with the help of an integral table it can be solved as follows

$$ \begin{align*} \int \frac{1}{f(1-f)} \,df &= \int b \,dt \cr \ln\frac{f}{1-f} &= a + bt \tag{2} \cr \frac{f}{1-f} &= e^{(a + bt)} \cr f &= e^{(a + bt)} - fe^{(a + bt)} \cr f + fe^{(a + bt)} &= e^{(a + bt)} \cr f(1 + e^{(a + bt)}) &= e^{(a + bt)} \cr f &= \frac{e^{(a + bt)}}{1+e^{(a + bt)}} \cr &= \frac{1}{1 + e^{-(a + bt)}} \end{align*} $$

This model of technological substitution was introduced by Fisher and Pry in their accessible A Simple Substitution Model of Technological Change.

To apply this model we can use the equation at $(2)$ to determine the parameters $(a, b)$. In particular, collect data measuring $f$ and transform that according to the left hand side of the equation, then as indicated by the right hand side we need to fit a straight line to this.

Using the proportion of EVs (Electric and plug-in hybrid) in the Australian Capital Territory as an example (data at this link). The semilog plot of $f/(1-f)$ is

Observe that from mid-2019 it appears that EV adoption is taking effect. Fitting a straight line to this region is plotted below and returned the parameters $(a,b)=(-6.291, 0.054)$.

These parameters give the model plotted below. The data have been plotted as circles too and it is evident that the model is a good fit. The model suggests that EVs might make up $50%$ of cars in the second half of 2029.