There has been a
very long discussion here:
new Chris Martenson
July 4 - Bubbles, August 5 - Fuzzy Numbers
http://www.greenenergyinvestors.com/index....mp;hl=martensonabout the exponential function. I do not want that repeated here.
I find this quite ironic considering this:
Prof. Albert Bartlett
The Most IMPORTANT Video You'll Ever See (parts 1 of 8) - great videos on basic economics, like what exponential growth is, 70 / % = years to double
Posted by:
http://www.youtube.com/user/wonderingmind42Part 1:
http://www.youtube.com/watch?v=F-QA2rkpBSYPart 2:
http://www.youtube.com/watch?v=Pb3JI8F9LQQPart 3:
http://www.youtube.com/watch?v=CFyOw9IgtjYPart 4:
http://www.youtube.com/watch?v=yQd-VGYX3-EPart 5:
http://www.youtube.com/watch?v=qHuwgxrTKPoPart 6:
http://www.youtube.com/watch?v=-3y7UlHdhAUPart 7:
http://www.youtube.com/watch?v=RyseLQVpJEIPart 8:
http://www.youtube.com/watch?v=VoiiVnQadwE ----------------
Re-reading Bobsta's post:
QUOTE
Now if you take *any* exponential graph and you're at the end of it (i.e. the right hand side) you will *always* be in the vertical phase!
I now see what was meant.
Yes, if you draw any exponential function from the start of time to now, you will be at the 'vertical' phase of the curve. I agree.
But, I don't think that is what Chris means at all.
I think this is what Chris means.
1. First you have to estimate the total resources available. Yes it's an estimate, but that's what you do.
2. Then you draw the exponential function from the start of time
until the resources are exhausted. This is very important. It
is not drawn just to 'now'.
The result is that most of the time 'now' will be somewhere on the curve.
I think the point Chris is making is this. We are getting close to the end time of a number of curves drawn as in (2). So we are on the vertical phases of them.
I do not want to get back into an excessively long discussion about the maths of the exponential function.
It is explained simply here:
Exponential Growth
http://www.math.cornell.edu/~numb3rs/kostyuk/num109.htmQUOTE
In plain words, this means that x grows exponentially if it increases proportionally to its own value. Most often exponential growth occurs in situations where "x creates more x", typical examples being population growth and compound interest. Exponential growth can occur both when the time intervals are discrete, for example in annual or monthly interest compounding, and when the time variable is continuous, as in continuous compounding of interest or modelling of large populations. The discrete time case is more often encountered in practice and is easier to analyse mathematically, since we don't need to resort to the exponential function.
Any suggestion that 3% growth per year is not a simple function is just plain wrong.
Now having re-watched Prof Albert Bartlett's videos, I realise that using the "doubling time" makes thinking about this even easier.
As he explains, the doubling time ~= 70 / growth rate.
eg if the growth rate is 7%/year, the doubling time = 70 / 7 = 10 years.
It is then easier to imagine doubling the value every n years.
That gives this type of curve, with each doubling point marked:
It is no more complex than this.
A tree has 15 apples.
Day 1: Eat 1 apple.
Day 2: Eat twice as many = 2. Total consumed = 3.
Day 3: Eat twice as many = 4. Total consumed = 7.
Day 3: Eat twice as many = 8. Total consumed = 15.
There are no more apples.
Please note: On the last day, ~ half the apples were consumed. It took 3x days to consume the ~ first half.
This simply demonstrates how growth rates result in a vertical phase, which are relatively short.
Can you answer this question from Part 3 of 8 of Prof. Albert Bartlett's video series ?
If not, please watch it until you can.
I repeat:
IMO it's vital this is understood.
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