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Distribution of price change


Distribution of income in big samples
Benoît Mandelbrot Mathematician
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What about big samples, big collections of individuals or of observations? For these it is best to move from income distribution to something that happens to be very similar, which is distribution of city sizes or the distribution of firm sizes. Different individuals observed that the distribution of firms' sizes is very very long tailed, and the number of firms of, say, sales greater than u is also a form proportional to u to some small power like minus one, maybe one point two, or maybe close to one. The same thing for cities. Well two questions occur. First of all how could it be that any law applies to notions as ill-defined as income, firm size or city size? That question I discussed earlier. And the second question is, what are the consequences of this distribution? A third question is how it came to be. The third one I can say nothing about, but the first two - the second question in particular- are very much worth investigating. So imagine that it is true that the distribution of firm sizes is lognormal as some people proclaim. Then if you take the sum of two firms, picked at random, and the sum is very large then one of them is large, therefore there is inequality. You take a sum of ten firms and the largest is going to be a smaller proportion of the sum. A sum of a hundred firms, the largest would be quite a small part of the total. In other words, the lognormal distribution predicts that as you get to larger and larger assemblies, the predominance of the single largest or second largest, third largest of the contributors to the sum becomes smaller and smaller and smaller, rather rapidly. What does the power-law distribution proclaim? It proclaims the sum of many such variables, the contribution of the largest is going to be about the same, independently of how many elements are being added. In one case you have asymptotic concentration; in the other case you have asymptotic lack of concentration. Which is closer to reality? Well, I think, quite clearly, there are many, many industries in which the number of firms is very large, hundreds or even thousands, in which the largest has sales or other measures of size that are very much a non-negligible portion of the whole. That is to say, the idea, the fact of economic concentration is not due to the smallness of the number of participants, but to something more profound. If one dealt with distributions, random variables following the law of exponential distribution or the Gaussian, then the largest would become very rapidly be completely negligible.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Daniel Zajdenweber Bernard Sapoval

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Duration: 3 minutes, 18 seconds

Date story recorded: May 1998

Date story went live: 29 September 2010