Tuesday, 14 August 2012

Exponentials, % Growth Rate, Doubling Time...

The dynamics of money supply, debt, inflation, population, energy use, digital electronics, green house gas concentrations, bacterial colony growth, radioactivity and many many other phenomena show exponential behaviour.

The science is so simple to comrehend but the consequences can be totally counterintutive, fascinating, baffling and many a times frightening.
And it always starts with a nice slow pace gaining momentum as it goes along and before you know it, it has become a monster.

Let us start with a couple of fun examples; I use them as party games:

1.  Take a big sheet of paper - normal paper is 0.1 mm thick.
Fold it 1 times - the thickness of folded paper will be 0.2 mm.
Fold it again - the thickness is now 0.4 mm.
Fold is again - the thickness is now 0.8 mm.
continue this folding for a total of 50 steps
What do you think the thickness of the folded paper will be??

The height of the folded paper will be more than 100,000 km!

2.   A chess board has 64 squares.  Place
1 grain of rice on the first square,  (Total grains = 1)
2 grains of rice on the second square, (Total grains = 3)
4 grains of rice on the third sqaure, (Total grains = 7)
8 grains of rice on the 4th square,   (Total grains = 15)
16 grains of rice on the 5th square and so on...

Continue doubling the number of grains on the next square
Fill up to the 64th square on the board.
How much rice will you need to fill up the board?

You will need 400 billion tonnes of rice to fill up the board!
This is about 1000 times annual global rice production.

UNBELIEVABLE - Of course one hits reality limits before you can finish.  You cannot fold a piece of paper so many times and there is not enough rice to fill the whole board.
These are examples of Exponential Growth.

There are a few important observations one makes here:

1.  Size of each step is double the size of the previous step
2.  Amount added in a step is greater than the total amount added in all the previous steps by one unit

Let us now consider an example where time is involved:
Look at the next blog for this ...