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Growth Curves

The difference between linear and exponential growth, and why it matters.

The Frogs And The Lily Pad

This example is taken from Dennis Sherwood's book 'Seeing The Forest For The Trees' and is a great way to visualise the difference between linear and exponential growth:

"A colony of frogs is living happily on one side of a large pond.  On the other side is a lily pad.  One day a chemical pollutant flows into the pond and stimulates the growth of the lily pad - it starts to double in size every 24 hours.

Question 1: If the lily pad will cover the entire pond in 50 days, on what day is the pond half-covered?

Question 2: The frogs have a method of stopping the growth of the lily pad, but it will take 10 days to put into effect.  What proportion of the pond is covered at the latest possible time to take action and save themselves?"

The answer to question 1 is Day 49.  The easiest way to visualise this: if the lily pad doubles in size every 24 hours and will cover the entire pond on the 50th day, then if you start from the 50th day and work back, the lily-pad will halve in size, hence day 49 will cover 1/2 the pond, day 48 will cover 1/4 of the pond and so on...  Many people incorrectly answer Day 25 - that would be true if it were linear growth.

The answer to question 2 is 0.00098 - the frogs need to identify the problem and take action by the time the lily pad covers less than one thousandth of the pond.   Would the frogs even have noticed anything was different yet?  Compare this to linear growth where, on the 40th day, the lily pad would cover 80% of the pond - 'Houston, we have a problem'.  This is a huge difference, as shown on the graph below:  (Red line is Linear growth, Blue line is Exponential growth.)

It can be difficult to spot (and even harder to accept) the early warning signs of an exponential growth curve, making it difficult to act before it is too late.  One of the biggest mistakes can be to misinterpret too small a range of data.  As you can see on the chart, it takes a long time for an exponential growth curve to become visible, but once it does, it gets big very, very quickly.  If you take a small segment of any part of that exponential growth curve, what will it look like?  Linear!   If you use a logarithmic scale, what will it look like?  Linear!  (A useful trick for those wanting to hide an exponential growth curve...)

Join The Dots

Why does this matter?  Increasingly we are discovering that many aspects of the real world are not as linear as we had thought (or hoped) they were.  In systems thinking, every system identified as having a reinforcing loop will experience exponential growth (or loss - it goes both ways) unless an external force is applied to balance the loop.  This applies to everything - business models, technology adoption, media hype, finances, physics, biology, weather systems, health epidemics (worried about bird flu yet?)... everything!  Ever wondered why some new phenomenon appears to have appeared suddenly from nowhere?  Yep, you guessed it, exponential growth at work.

The most obvious example in the Information Technology world has been ' Moore's Law', which predicted that the performance of computer chips will double every 18 months.  It is the Internet that has made exponential growth visible to us all.  The 'dot com' boom and bust that took place from 1997 to 2002 was an example of exponential growth followed by extreme loss.   The current hype about Web 2.0 is an exponential growth curve starting to become visible.  Ever read the books 'Tipping Point' and/or 'Good To Great'?  They are both talking about exponential growth at work.

Historically, we have tended to assume linear growth for many systems - dating right back to the days of Adam Smith.   The assumption of finite resources and finite demand, when building economic models, leads to well-behaved predictable markets.   It's what we've tended to do in the past and, like learning to write with your left hand, it's difficult to unlearn what you are used to and switch to a new method... 

Looking at the bigger picture, next time you hear politicians stating that the sea level, average temperatures, number of catastrophic tsunamis, hurricanes and earthquakes have all only increased marginally over the past 100 years, ask yourself if that increase is the first visible sign of an exponential growth curve...

The Equations

The basic equations, where x is the initial value and n is the multiplier, are:

• Linear growth = nx
• Exponential growth = xⁿ

If n = 3, then linear growth = 3x = x + x + x, and exponential growth = x³ = x * x * x. 

For example, if you were to invest 4 gold coins (x=4) over 6 years (n = 6), this is the growth you would see over the period.

Period:	Linear Growth:	Exponential Growth:
Year 1	4	4
Year 2	4 + 4 = 8	4 * 4 = 16
Year 3	4 + 4 + 4 = 12	4 * 4 * 4 = 64
Year 4	4 + 4 + 4 + 4 = 16	4 * 4 * 4 * 4 = 256
Year 5	4 + 4 + 4 + 4 + 4 = 20	4 * 4 * 4 * 4 * 4 = 1,024
Year 6	4 + 4 + 4 + 4 + 4 + 4 = 24	4 * 4 * 4 * 4 * 4 * 4 = 4,096

So if you had linear growth, you would receive 24 gold coins back, if you had exponential growth, you would receive 4,096 back after the same period of years.  Quite a difference!

Notes

The sign '*' is used to represent multiplication.  The letters x and n are used to the describe the formulas, other letters can be used too (you will often see e^x, for example)