Peter Cochrane's Blog: Tech growth and decline - why people just can't do the maths

Too many stumble over the basics...

Written at Grenoble airport and dispatched to from my car via 3G later the same day at 2.1Mbps.

Exponential functions are very useful in describing many natural and unnatural phenomena such as the diffusion of a gas or the growth of a technology.

But the word 'exponential' is often used out of context and the real meaning bastardised by erroneous context and misunderstanding.

I was recently at a presentation given by an economist who proudly declared that he did not believe in exponentials. These are functions of the form ab and very commonly expressed in the natural form ex or EXP[x].

His thesis was that markets and other activities such as technology growth do not follow the exponential curve. They are governed entirely by the logistic or S curve.

He also postulated that nothing progresses much beyond the 50 per cent point of the S curve before deviating, flipping and gong into decline.

Graph showing the exponential and logistic curves commonly used to describe growth

The exponential and logistic curves commonly used to describe growthImage: Peter Cochrane/

There was nothing new or novel in this revelation. He was correct in a way but reality is much less kind to the theory. Many activities follow exponential growth or decline or follow the S curve, in an approximate manner - much in the same way that data aligns with the normal or bell curve in probability and statistics.

To get a good fit to any mathematical or theoretical curve we often have to think in terms of a sufficiently large sample over a sufficiently long time.

Even detailed and accurate scientific results suffer from scatter due to measurement uncertainties and the stability of the environment and materials.

Little surprise then that we see the growth of computing power, memory, networks and mobile phones as being about the exponential line rather than following it with great precision.

Graph showing real product growth and collapse with time

This graph shows real product growth and collapse with timeImage: Peter Cochrane/

In textbooks and academic papers the representation of the product birth-to-death process tends to be somewhat idealised and we have to remember to colour all these models with a degree of practical experience and judgement. Even getting accurate and timely sales information in the real world can be a challenge.

Textbook representation of the product birth-to-death processImage: Peter Cochrane/

When a market becomes saturated, or a gas has completely diffused, all activity will grind to a halt, and in the ideal case it will be defined by the logistic curve. There really are no surprises here.

However, what this economist was not getting is that technology does something rather magical. It...