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Distribution of income in big samples

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Pareto law and inequality in income distribution

Benoît Mandelbrot
Mathematician

Views | Duration | ||
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111. Fractals and chaos theory | 282 | 04:04 | |

112. A new alphabet | 187 | 01:19 | |

113. Dimension | 161 | 03:06 | |

114. The inevitability of mathematical development | 212 | 04:35 | |

115. Fractals and chaos theory in mathematical development | 197 | 05:04 | |

116. Economics: Pareto and Bachelier | 244 | 02:22 | |

117. Pareto law and inequality in income distribution | 294 | 04:25 | |

118. Distribution of income in big samples | 141 | 03:17 | |

119. Distribution of price change | 131 | 06:17 | |

120. Inequality in price change distribution | 100 | 03:04 |

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Now what was the finding of Pareto? It was that the distribution of incomes is very long tailed, that the probability of income exceeding value u is roughly of the form u to the power (of) minus some exponent alpha. This in contrast to distributions in physics, which are very often exponential, exponential of minus alpha of u, or even Gaussian, that is exponential of -u^{2}. Therefore, high incomes probability goes down much more slowly than the probability of many of the physical phenomena. What was the fate of Pareto distribution over the long period between its description in, say, 1900, and the time I became involved in the early '60s? I would say that it was a mixture of neglect and scorn. There were a certain number of authors who did test the data, found the distribution to be correct, and followed up a little bit in defending it. But by and large the argument was that it was a rule without any logic behind it, any kind of explanation, and that it did not deserve attention, and that another distribution called lognormal was much preferable. To say that the distribution of income is lognormal is to say the logarithm of income is normal, Gaussian. To say that the distribution of income is power-law, is to say that lognormal is exponentially distributed. Now exponential and the Gaussian are not that terribly different, they're very different but not that terribly different. What is very striking is that once you go back to u from log u enormous differences appear from a viewpoint of this essential feature, which is economic inequality. So perhaps I should follow up first on this matter of economic inequality and contrast three possibilities, which would be a distribution that is power-law; a distribution that is lognormal, distribution that is Gaussian; and perhaps even distribution that is exponential. Imagine, as I like to say as an example, two persons picked in the street, preferably on Wall Street and you have a way of halving their total income, which happens to be two million. Well everybody has a very strong feeling that if the total is two million it is not because each of them had one million plus or minus some factor, but because one had a normal income and the other had all the rest, roughly two million. Therefore, they would have a very strong feeling that economic variables are unequal. If you make a sum of two variables, there is one, which is larger, which predominates, if the sum is large. If you did that for exponential economic variables, you would find no inequality - evenness. For a Gaussian economic variable, each would pick one half and plus or minus a term. The Gaussian being characteristic of physical phenomena, it is very important to observe the physical phenomena, except near the critical points, are not subject to great influence from individual observations. What Pareto law was tackling is the fact that economical phenomena like income are not that uniformly distributed, and was coming close, very close to the core of the problem. Again what would be the case if the distribution of income were lognormal, which many people proclaim to be the case? Well, if two persons together had an income of two million, then indeed one of them would have a normal income and the other a very large income. The lognormal distribution also has the property of being very unequal for small samples like two.

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.

**Title: **Pareto law and inequality in income distribution

**Listeners:**
Bernard Sapoval
Daniel Zajdenweber

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.

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

**Duration:**
4 minutes, 26 seconds

**Date story recorded:**
May 1998

**Date story went live:**
29 September 2010