Monday, October 18, 2010

Fractal Dimension, (Economic's) Elasticity and Complexity


Fractal Dimension and Elasticity - is there a connection?
I have found - after analysing the Koch Snowflake fractal with standard economic analysis - that there maybe a connection between Economics 'Elasticity coefficent' (which is a universal measure of the change in one variable to a change in another) - and the fractal dimension. That is to say, Price Elasticity of Demand (PED) is possibly the same as , or at least related to, the fractal dimension (D) - a measure of complexity. This may help explain the logarithmic nature of the demand curve.
Here's my reasoning:
Below you can also listen to a video blog of me on this topic too.

 Recap: Marginal Analysis of the fractal

In my last blog I established that the blue (MA) curve  in Fig. 1 -showing the change in area in the 'Koch Snowflake' after an other iteration- is mimicking or demonstrating something central to traditional marginal analysis and utility theory: that it maybe the origin of the consumer demand curve as we know it.   The next question begs - what's the MA's elasticity?




Sunday, October 17, 2010

The Passing of Benoit Mandelbrot

Last night I was camping out with my wife  in the forests in Sweden and slept under the stars and this very wonderful scotch pine above.
Apart from the romance of the setting, I went to sleep thinking over fractal explanations of (mostly) inflation of money, and it's connection to economic growth, development, and sustainability.
They are really on my mind at the moment and I will write on my findings in the future in the near future.

I heard this morning that Benoit Mandelbrot - the father of Fractals has died.
I don't know what to think: I thought I might get to meet him someday.
He curtainly hadn't made it into the mainstream: when I ask, hardly anyone knew of him or his fractals - outside maths of course.
Yet he to me made what I believe the greatest discovery in human history (and I mean it) cause I am sure everything is fractral - evolution and all.  We just don't really know it yet how to use it.
His name and input will only grow now - in  the very same way (funny enough) as the fractals he founded.


If I (and I am sure many other thinkers) have anything to do with it - we will put his fractals up front.

Wednesday, October 13, 2010

Fractals - Marginal Analysis of the Koch Snowflake

Marginalism, and Marginal Analysis : derived from the fractal (?)

Intro
For some time now I have pondered on the fractal, particularly the development the Koch Snowflake fractal, (seen below), I have noticed how it uncannily demonstrates many of the key features of economic theory.
Could it be that the economy - as we experience it - is a fractal? Is it that the fractal can be understood as an economy?
I am now confident, after doing the following analysis of the (Koch Snowflake) fractal that fractal geometry is the foundation of neoclassical economics and our reality. 
The following is the first in a series of insights I have found within the fractal:

The marginal utility (or benefit) of additional triangle area - after each iteration

To show that the fractal demonstrates marginal utility (or benefit) I substituted area for utility, and then analysed the change in area over the development of the Koch Snowflake fractal - from a triangle (iteration 1 below), to the complete shape of the snowflake at iteration 4 (below). Shape is usually forms at and around 7 plus or minus 2 iterations: it is at iteration 4 (below) because of its size and relative thickness of line.

Towards  (Fractal) Equilibrium



Koch Snowflake development: iteration 1 to 3
Once a rule -  in this case new triangles - is set in action, and allowed to iterate (repeat) step by step, it will go on to eventually form a snowflake.  After iteration 4, the shape will set at what I term fractal equilibrium: no more iterations will change the shape after this iteration - assuming a static viewers perspective, ie no zooming.


Koch Snowflake development animation: iteration 1 to 6

If one were to continue the iterations beyond the point of fractal equilibrium (iteration 4 in the figure above and iteration 6 in the animation above) the computer generating the pattern would soon crash: the effort (cost) would multiply while the extra benefit to the viewer - in terms of the shape – would fall close to, but never equaling, zero.
Take the time to look at this webpage http://mathforum.org/mathimages/index.php/Koch_Curve where you can experience the fill working of the Koch Curve first hand.

Marginal Analysis: Method.
I analysed the growth of the Koch Snowflake using marginal analysis (a method found in any basic economic textbook) which looks at (in this case) the extra 'area' added - iteration by iteration.

Working from internet sources I constructed a table showing the area progression of the curve - iteration after iteration:

Taking s as the side length, the original triangle area is . The side length of each successive small triangle is 1/3 of those in the previous iteration; because the area of the added triangles is proportional to the square of its side length, the area of each triangle added in the nth step is 1/9th of that in the (n-1)th step. In each iteration after the first, 4 times as many triangles are added as in the previous iteration; because the first iteration adds 3 triangles, the nth iteration will add triangles. Combining these two formulae gives the iteration formula:

Area (A) of each triangle (as in column 2) = (s^2 √(3))/4

Results
Results Table

The Fractal Total and Marginal Area Curves
Fig 1. shows the total area (TA) and marginal area (MA) at each iteration.
Notice, from the start, that Fig.1 resembles any marginal utility diagram found in any elementary economic text books.


Fig 1. is derived from the fractal - not from any (fictitious) made up numbers, as in text books. All fractals - whether computer generated or not - will chart as Fig 1.

Diminishing Marginal Utility.



fig 4-1
Figure 4-1 above shows how the fractal analysis is similar if not the same as a typical economics text book diagram: marginal utility reference.
The MA curve is a 'Power law' function, and that it demonstrates Pareto distribution - the 80/20 rule.
As the total area of the Koch Snowflake increases, after each iteration, the marginal or extra area (also) diminishes.
The fractal is the perfect demonstration of diminishing marginal utility; it could be interpreted that diminishing marginal utility is a perfect demonstration of the (universal) fractal - that both are the same thing?
Here are some internet references on marginal utility:
http://www.investopedia.com/university/economics/economics5.asp
http://www.google.se/imgres?


A window into reality:
My fractal-marginal analysis diagram is a snap-shot, taken somewhere (anywhere) in the set: this insight in turn has opened a window of enormous insight into reality, and has led me to the world quantum mechanics. The fractal (above) shows no scale - the numbers assigned are purely arbitrary and serve merely to show the relationship between the total area, 160, and the first triangle area 100. In fact triangles range in size from small, (but never reaching 0), to the infinitely large.
From this early insight (in this blog) I have gone on to unlock the quantum features of the fractal and have recently suggested a direct link between the demand curve and the de Broglie wave function.


negative marginal utility misattributed
Next: Marginal cost, Fractal production.

Saturday, October 9, 2010

The creation of new public goods: the internet (or information age) paradox



First published:October 9, 2010

Update:March 22, 2015.
Blair D. Macdonald.

Abstract
Based on the economic model used to classify 'goods' in an economy; private goods such as found in the entertainment/media industries, or any item on the internet subject to file sharing or digital copying in any form including – at the extreme – the human genome, solid object 3D printing, and even money in the form of bit-coins, are being slowly repositioned from what are termed 'private' or 'club'/'congestion' goods, to the extreme opposite, public goods. The ‘free rider problem’ of Public Goods has become the ‘free copy problem’.  Public Goods failure in the market, and are therefore provided by Government.  Is this the destiny of internet goods?

Key words: Public Goods, Internet, Copyright, Market Failure


1        Introduction

One of the great questions of our time is what is to be the fate of the entertainment and media industries - newspapers, movies, music production, TV and the like - all of which since the advent of the internet, have been scrambling for new models that pay? Will they survive in this age of Internet and file sharing? I have found something that suggests why it is so difficult for them.
It came to me while teaching the market failure economics and analysing examples of  ‘public’ and ‘private’ goods: I was looking at the (market) differences between 'Cable TV' - which is known as a ‘club good' or 'natural monopoly', – and 'free to air TV' – which is known as a ‘public good:  I began to see other examples or repetition’s of the subject at different scales and contexts, and after doing so, noticed something interesting, if not astounding, newspapers and the like are now – at least in their digital form – public goods.

1.1      Public Goods

The following is a typical textbook definition of a Public Good.
 Public goods are goods that are both non excludable and non-rival. Consumers cannot be prevented from consuming them once they are provided and additional consumers do not reduce the amount left for other people, e.g. national defence. Once a country is defended all of its inhabitants benefit automatically. Many public goods such as lighthouses could in theory be provided by the market mechanism but are not; these are called 'quasi public goods', rather than 'pure public goods '. Public goods suffer from the free rider problem. If asked whether they would pay for them, households would lie and say no because, once provided, they could benefit for free anyway. Because no one is willing to pay for these goods (because they hope someone else will) they will not be provided in a free market. Therefore, the Government must provide them. (Note: for a 'private good' if one unit is consumed by one person it cannot be consumed by another. (This is not the case for public goods.) [1]

1.2      Public Goods and the Internet

The entertainment, media industries or anything subject to file sharing or digital copying in any form, have slowly been re-positioned from being private or club/congestion goods, to their now extreme opposite, public goods – as shown below in figure. 


Figure 1. Public Goods including effects of the Internet.

1.3      The free rider (free copy) problem

The proof of this 'public good' claim is the presence of the free rider. Quite literally – when asked to pay by a market provider – consumers simply find a substitute or find a free copy elsewhere (on the internet). We quite literally get a free ride – or in this case, a free copy. And for music and films, copies are made 'free' by file sharing and the like.

1.4      Lack of Rivalry

Unlike the use of other traditional Private or Common goods – for example print newspapers – there is no rivalry: there is little bothered by the 'stealing' or copying of the good, and they do not show real concern – or fight – for the media – as they would for the exploitation of Common's goods, e.g. the of ocean's or atmosphere. 

2        Discussions

The problem is public goods tend to fail as a business operation, and are almost always supported by Government. Traditional examples are lighthouses, and streetlights. So this leaves a great paradox: we have the Internet, and do well for it, but on the other hand maybe our own, slow undoing. Democracy demands or depends on a 'free from Government' media; and firms in the arts and entertainment industry demand revenue if they are to survive.
It is as if the Internet has formed some kind of an economic black hole. Is there any escaping? Our own privacy on the Internet may well fall under this public goods paradox also: including – in the future – ones genome.

References

1.         Gillespie A. Advanced Economics Through Diagrams. Oxford University Press; 2001.