# M01. Exponential Growth

It is more or less obligatory to begin by mentioning *Moore's Law*, which, if it has somewhat lost its conviction over the years, nonetheless remains a robust icebreaker. The Law is usually stated as follows: *the density of components in an integrated circuit doubles about once every two years*. 

In truth, the force of the Law comes not so much from it saying that transistor density doubles every *two* years as from it saying that it doubles *every* two years&mdash;it says that the growth of transistor density is an exponential process. It would certainly be a category-error to make this sort of remark about something linear ('I double in age every quarter-century') or logarithmic ('Usain Bolt doubles in speed every decade'). Exponential growth is quite special, and always transient. If you are an optimist, Moore's Law is really a law of human heroism. If you are slightly more disillusioned, or Japanese, it is what we might call a *seishin-ron*, an 'argument from willpower.'

Formally, a continuous signal $x(t)$ exhibits exponential growth if its rate of change is proportional to itself,

$$\frac{dx}{dt} \propto x(t).$$

More conveniently, $x(t)$ is an exponential function of time:

\begin{align}
&\Rightarrow \int \frac{1}{x(t)}dx(t) \propto \int dt\\
&\Rightarrow \ln{|x(t)|} \propto t+c\\
&\ldots\\
&\Rightarrow x(t) = x(0)e^{kt}.
\end{align}

When the exponential process is defined over discrete time, we may write 

$$x(t) =  x(0)(1+\frac{r}{100})^t,$$

where we have written $x(t)$ in the familiar form of an initial value multiplied by a compounding interest rate $r$. 

The doubling time $t_2$ of such an exponential process may be estimated by the so-called Rule of 72,

$$t_2 \approx \frac{72}{r},$$

which follows from 

\begin{align}
&2x(0) = x(0)(1+\frac{r}{100})^{t_2} \\
&\Rightarrow 2 = (1+\frac{r}{100})^{t_2} \\
&\Rightarrow \log_{1+\frac{r}{100}}2 = t_2 \\
&\Rightarrow \frac{\ln2}{\ln{(1+\frac{r}{100})}} = t_2.
\end{align}

With a second-order Taylor-expansion of the denominator,

\begin{align}
t_2 &\approx \frac{\ln2}{\frac{r}{100}-\frac{\frac{r}{100}^2}{2}}\\
&\approx \frac{\ln2}{\frac{r}{100}}\\
&\approx \frac{69}{r}\\
&\approx \frac{72}{r}.
\end{align}

The last approximation may appear slightly cheeky but we do it for the acceptable reason that $69$ has four factors while $72$ has twelve.

An example is maybe in order: a quantity that increases at a rate of $r=8$ percent per unit time (around the average annual return of the S&P 500) will double in roughly $\frac{72}{8} = 9$ time-units.

Finally, we note that the Rule of 72 works best (has minimum error) for *small* values of $r$ (incidentally, around 8 percent). Perhaps what best illustrates the might of Moore's Law is that it corresponds to an annualized rate of *40 percent*.



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