Marginal Utility - derived from the fractal
Marginalism and Marginal Analysis: derived from the fractal.
Here, I perform basic marginal analysis on the Koch Snowflake
Intro
For some time now, I have pondered on how the Koch Snowflake fractal (seen below) demonstrates, uncannily, key features in economic theory.
Is it that the economy - as we experience it - is a fractal? or is the fractal an economy?
After doing the following basic math, fractal geometry is the foundation of the social science of economics.
Towards Equilibrium
From the outset, the concept of Marginalism and fractals have something in common: They are both about the next unit or iteration.
Once a rule, in this case, new triangles, is set in action and allowed to progress (iterate) step by step, it will go on to eventually form a snowflake. After iteration 4, the shape will set and no longer change - at least from the viewers' static perspective.
If one were to continue the iterations beyond the point of equilibrium (4.), the computer generating the pattern would soon crash: the effort (cost) would multiply. At the same time, the extra benefit to the viewer - in terms of the shape – would fall close to, but never, zero.
Take the time to look at this webpage: http://mathforum.org/mathimages/index.php/Koch_Curve, where you can experience the Koch Curve's full workings firsthand.
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
To show that the fractal Koch Snow Flake demonstrates marginal utility theory, I substituted the Economists' utility (satisfaction or benefit gained from another iteration) with the area of the curve, reasoning that after more iterations, one gets more total satisfaction or in this case Total Area (TA). One gets (pleasingly) closer to the true shape of the snowflake.
I have used no units to measure Area, as the area in the diagram corresponds to a curve I used as a reference, http://math.rice.edu/~lanius/frac/koch3.html on the internet.
Results Table
This result table will be central to my analysis of the fractal.
Here, I perform basic marginal analysis on the Koch Snowflake
Intro
For some time now, I have pondered on how the Koch Snowflake fractal (seen below) demonstrates, uncannily, key features in economic theory.
Is it that the economy - as we experience it - is a fractal? or is the fractal an economy?
After doing the following basic math, fractal geometry is the foundation of the social science of economics.
 |
| Koch Snowflake animation: |
Towards Equilibrium
From the outset, the concept of Marginalism and fractals have something in common: They are both about the next unit or iteration.
Once a rule, in this case, new triangles, is set in action and allowed to progress (iterate) step by step, it will go on to eventually form a snowflake. After iteration 4, the shape will set and no longer change - at least from the viewers' static perspective.
 |
| Koch Snowflake development |
If one were to continue the iterations beyond the point of equilibrium (4.), the computer generating the pattern would soon crash: the effort (cost) would multiply. At the same time, the extra benefit to the viewer - in terms of the shape – would fall close to, but never, zero.
Take the time to look at this webpage: http://mathforum.org/mathimages/index.php/Koch_Curve, where you can experience the Koch Curve's full workings firsthand.
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
To show that the fractal Koch Snow Flake demonstrates marginal utility theory, I substituted the Economists' utility (satisfaction or benefit gained from another iteration) with the area of the curve, reasoning that after more iterations, one gets more total satisfaction or in this case Total Area (TA). One gets (pleasingly) closer to the true shape of the snowflake.
I have used no units to measure Area, as the area in the diagram corresponds to a curve I used as a reference, http://math.rice.edu/~lanius/frac/koch3.html on the internet.
Results Table
Hello my name is Rolando and I´m an economics student from El Salvador, I came across with the concept of the fractal numbers this past week and I realized that it has a lot of aplications in the economic sciences, a great example of the aplication is the explanation of the marginal utility with the fractal numbers, another one could be the diminishing returns or using the property of the fractal numbers of the susbsecuent divisions of an original estructure for explain the distribution paterns. I want to ask you how the concept of the fractal numbers is being used in the econmic sciences today? I will apreciate your answer and thanck you very much for your effort
ReplyDeleteHello Rolando,
ReplyDeleteSorry I never replied to your question, I have only now seen it. Have you learn't this through my blog, or have you thought about it yourself?
As I am trying to show, I think economics - and all science's and knowledge - is fractal, and I have a proof, and my blog is it: this is a new frontier. Someday the textbooks will have it in their introductary chapters, it is fundimental.
Thanks for your comment - I have plenty more ideas to come.