Monday, October 18, 2010

Fractal Dimension, (Economic's) Elasticity and Complexity

Update May 2017
This is by far my best idea; I have written it up in a working paper at my academia.edu and vixra, and named it:  Quantum Mechanics, Information and Knowledge, all Aspects of Fractal Geometry and Revealed in an Understanding of Marginal Economics.
I shall post the Abstract, followed by the original post, followed by the paper. I hope to have some collaborate and review my work in time.
Abstract
Fractal geometry is found universally and is said to be one of the best descriptions of our reality – from clouds and trees, to market price behaviour. As a fractal structure emerges – the repeating of a simple rule – it appears to share direct properties familiar to classical economics, including production, consumption, and equilibrium. This paper was an investigation into whether the mathematical principles behind ‘the market’ – known as marginalism – is an aspect or manifestation of a fractal geometry or attractor. Total and marginal areas (assumed to stand for utility) and the cost of production were graphed as the fractal grew and compared to a classical interpretation of diminishing marginal utility theory, and the market supply and demand. PED and PES was also calculated and analysed with respect to (iteration) time and decay.  It was found the fractal attractor demonstrates properties and best models classical economic theory and from this it was deduced the market is a fractal attractor phenomenon where all properties are inextricably linked. The fractal, at equilibrium, appears to be a convergent – zeta function – series, able to be described by Fourier analysis, and involves Pi, i, e, 0, and 1 (of Euler’s identity) in one model. It also demonstrated growth, development, evolution and Say’s Law – production before consumption. Insights from the fractal on knowledge and knowing are also revealed, with implications on the question of what exactly is ‘science’ – and what is ‘art’? A connect between reality and quantum mechanics was identified. It was concluded marginal, classical economics is an aspect of a fractal geometry.  
 Keywords
Marginal, Fractal, Elasticity, Utility, Cost, Production, Price, Growth and Development, Say’s Law


Original Post



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 (?)


Update May 2017
This is by far my best idea; I have written it up in a working paper at my academia.edu and vixra, and named it:  Quantum Mechanics, Information and Knowledge, all Aspects of Fractal Geometry and Revealed in an Understanding of Marginal Economics.
I shall post the Abstract, followed by the original post, followed by the paper. I hope to have some collaborate and review my work in time.
Abstract
Fractal geometry is found universally and is said to be one of the best descriptions of our reality – from clouds and trees, to market price behaviour. As a fractal structure emerges – the repeating of a simple rule – it appears to share direct properties familiar to classical economics, including production, consumption, and equilibrium. This paper was an investigation into whether the mathematical principles behind ‘the market’ – known as marginalism – is an aspect or manifestation of a fractal geometry or attractor. Total and marginal areas (assumed to stand for utility) and the cost of production were graphed as the fractal grew and compared to a classical interpretation of diminishing marginal utility theory, and the market supply and demand. PED and PES was also calculated and analysed with respect to (iteration) time and decay.  It was found the fractal attractor demonstrates properties and best models classical economic theory and from this it was deduced the market is a fractal attractor phenomenon where all properties are inextricably linked. The fractal, at equilibrium, appears to be a convergent – zeta function – series, able to be described by Fourier analysis, and involves Pi, i, e, 0, and 1 (of Euler’s identity) in one model. It also demonstrated growth, development, evolution and Say’s Law – production before consumption. Insights from the fractal on knowledge and knowing are also revealed, with implications on the question of what exactly is ‘science’ – and what is ‘art’? A connect between reality and quantum mechanics was identified. It was concluded marginal, classical economics is an aspect of a fractal geometry.  
 Keywords
Marginal, Fractal, Elasticity, Utility, Cost, Production, Price, Growth and Development, Say’s Law



Original Post
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: May 2017.
Blair D. Macdonald.
2017 October: I see Bitcoin and any crypto current as an ePublic Good - it will fail.

Abstract
Media and entertainment industries are in decline; profitability down due to ‘freer access on computers. Is the internet producing ‘Public goods’ from what were Private goods? With respect to these goods and the Internet, the assumption used to classify ‘Private goods' and Public goods in an economy (the degree of excludability and rivalry) was analysed, and the respective industries tested for being Public Goods. It was concluded these goods within the entertainment/media industries are slowly being repositioned from what are termed 'private' or 'club'/'congestion' goods, to their extreme opposite, Public goods. The ‘free rider problem’ of Public Goods has become the ‘free copy problem’ with respect to these goods. It was hypothesised the Internet was the cause. It was discussed Public Goods – by tradition – failure in the market, and are therefore provided by Government: is this to be the destiny of Internet goods, 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?
Key words: Public Goods, Internet, Copyright, Market Failure, File Sharing

1 Introduction
While there has been much (probably unfounded) concern over the effect new computer technology will have on labour markets, the real problem with computers may lie with the internet, and its effect on markets. Can the current demise the entertainment and media industries be explained by the internet creating ‘Public Goods? Newspapers, TV, movie and music production and the like are – since the advent of the Internet – slowly diminishing and have been scrambling for new models that pay. All of these firms are ‘hit’ by the proliferation of their copyrighted material on the Internet. Not withstanding attempts to extend their respective industries; to a great extend there is market failure. This paper claims this failure is due to the Internet producing an environment of ‘Public Goods. For this to be so, there will have to be evidence of a ‘free rider’ problem, non-excludability and non-rivalry – in the respective industries – all due to the Internet. If true, it may seem trivial on the outside, however on the inside, a problem rises; traditionally, public goods tend to almost always be supported by Government. What does this mean for investigative journalism? Will it lead to support by Government under a merit good rationale.
1.1 Public Goods
While teaching market failure in my economics class recently, and analysing examples of ‘public’ and ‘private’ goods: I noticed newspapers and the like – at least in their digital form – are akin to radio, where radio can be consumed in either pay per listen or free to air. Newspapers have similar problems to that of Public goods. So, what is a public good? The following is taking from a regular economics textbook.
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 - e Public Goods
For the Internet to create an environment for public goods it will have to show evidence of non-excludability and non-rivalry. Non-excludability may be evident in the way the said goods and services are now sold: consumers can view – most, if not all – the product (news articles) online without paying. This is unlike the classic examples of a lighthouse or streetlight, but the effect similar; the firm has lost control of access: if the consumer cannot get from one provider of news, they will (easily) find a substitute, and if an access fee is applied online, this will also lead consumers to likely search for a free substitute. Firms are forced into competitive ‘game play’ to offer material for free. For the music and movie industry, much is available on Youtube; and if not Youtube, a close substitute to it – copies can easily be made by consumers. Entertainment goods are proliferated throughout the Internet, and this is extremely difficult to police; if a copy is found by a firm to be ‘pirated’, this copy may be ‘taken down’ – only to see copies soon available somewhere else, sometime in the future.
Non-rivalry maybe broken down to whether the consumers experience of the good or service is diminished by the use of an other consumer; are the consumers ‘bothered’ by the practice of someone – or themselves – getting something for (next to) no cost? The answer, without any research, is – no. Unlike the use of other traditional ‘Private’ or ‘Common goods’ (figure 1 below) – print newspapers, recorded music – there is no rivalry or bother by others the 'stealing' or copying of the good. The community do not show the concern or fight for the media as they would for the exploitation of ‘Common goods’ – the whales in the oceans or pollution in the atmosphere. As a consequence of this non-excludability and non-rivalry, 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 an opposite domain, public goods – as shown in the table below.
Figure 1. Public Goods and the Internet. All forms of media, once the domain of ‘Private goods’ are now in the domain of ‘Public Goods’.
1.2.1 The Free Rider (free copy) Problem and the Internet
The ‘smoking gun’ evidence of this ‘internet creating public goods’ claim is the presence of the free rider problem. Consumers are not getting a free ride, but rather a – in this case – a free copy: quite literally – when asked to pay by a market provider – consumers simply find a substitute or find a free copy elsewhere (on the internet). And for music and films, copies are made 'free' by file sharing and the like.
2 Discussions and Conclusions
Firms in the arts and entertainment industry demand revenue if they are to survive. While there has been an adaptive response to this problem of market decline: Spotify and the like for music provision, Netflix and the like for movie, and bands concert touring more, it does not seem to totally satisfy the concern of those in the industries – where there is clearly a major decline.
The problem is: public goods tend to fail as a business operation, and are almost always supported by Government, this is of great concern with respect to the future of news media and investigative journalism. Democracy demands
or depends upon a 'free from Government' media – can blogging and other social media meet this problem. It is likely we will see a ‘mass extinction in print and arts; this is not to say (at all) this is the end of the market, but it is a massive change.

The problem of mass-market failure may be extended if we include other flows of knowledge and information with respect to modern tech: the genome, knowledge in general, and anything that can be digitally printed, including solid object 3D printing, and even money in the form of bit-coins.
There is a great paradox: one of the great accomplishments of humanity, the internet, maybe our own, slow undoing. 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.