Yeah, about that exponential economy...

There's a lot of management and technology writing that refers to "exponential growth," but I think that most of it is a confusion between early life cycle convexity and true exponentials.

Here's a bunch of data points from what looks like exponential growth:

Looks nicely convex, and that red curve is an actual exponential fit to the data,
\[
y = 0.0057 \, \exp(0.0977 \, x)   \qquad  [R^2 = 0.971].
\]
Model explains 97.1% of variance. I mean, what more proof could one want? A board of directors filled with political apparatchiks? A book by [a ghostwriter for] a well-known management speaker? Fourteen years of negative earnings and a CEO that consumes recreational drugs during interviews?

Alas, those data points aren't proof of an exponential process, rather, they are the output of a logistic process with some minor stochastic disturbances thrown in:
\[
y = \frac{1}{1+\exp(-0.1 \, x+5)} + \epsilon_x \qquad \epsilon_x \sim \text{Normal}(0,0.005).
\]
The logistic process is a convenient way to capture growth behavior where there's a limited potential: early on, the limit isn't very important, so the growth appears to be exponential, but later on there's less and less opportunity for growth so the process converges to the potential. This can be seen by plotting the two together:

This difference is important because — and this has been a constant in the management and technology popular press — in the beginning of new industries, new segments in an industry, and new technologies, unit sales look like the data above: growth, growth, growth. So, the same people who declared the previous ten to twenty s-shaped curves "exponential economies" at their start come out of the woodwork once again to tell us how [insert technology name here] is going to revolutionize everything.

Ironically, knowledge is one of the few things that shows a rate of growth that's proportional to the size of the [knowledge] base. Which would make knowing stuff (like the difference between the convex part of an s-shaped curve and an exponential) a true exponential capability.

But that would require those who talk of "exponential economy" to understand what exponential means.