Platos poor city and rich city

Philippe WoloszynChargé de Recherche CNRS

ESO UMR CNRS 6590, Université de Haute Bretagne Rennes2,Maison de la Recherche en Sciences Sociales, 35043 Rennes, France

philippe.woloszyn@univ-rennes2.fr

10th international Conference of Territorial Intelligence Liege 2011

From economical to social entropy measure.Towards an inequality multidimensional evaluation.

poor city

An inequality measure has been inspired by Plato: The Plato inequality refers to Plato's "two cities", a city devided into at least two quantiles:

20% of the citizens control a share of 80% of all resources

and rich city

Pareto « 80-20 » law, according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is α = log45.

Fractal distribution: segments of the curves are self-similar

…Pareto distribution

Probability density function for various α with xm = 1.

Cumulative distribution functions for various α with xm = 1

Pareto-Zipf LawPareto-Zipf Law

[Mandelbrot (1965)]

As Pareto distributions are continuous probability distributions, Zipf law is the discrete counterpart of the Pareto distribution.

ksCf x loglog

Informational impedance adaptation level between economical and social dimensions

After Plato's two-quantile-societies and Pareto-Zipf law, Time series indicators has been constructed, such as Robin Hood Index (Hoover inequality) Gini inequality and Entropy measures

Inequality Issuization

Measure of redundancy, lack of diversity, isolation, segregation, inequality, non-randomness, and compressibility, describes the maximum possible entropy of the data minus the observed entropy, through generalized entropy index computation

It measures an entropic "distance" between population and an hypothetic "ideal" egalitarian state. Derived from Shannon’s entropy mesurement, entropy is a measure of economical, social and environmental « order »

.)(

1log)()(log)()()(

xpxpxpxpXIXH

XxXx

Econogical weight

Socioecological weight

Socioeconomical weight EconomicalEntropy

SocialEntropy

EnvironmentalEntropy

Vulnerability

Resi

lienc

e

DEVELOPMENT

NEEDS

RESOURCE

Entropy as a sustainability measure

Applications of entropy statistics

The parameter α in the GE class represents the weight given to distances between incomes at different parts of the income distribution

GE(1) is Theil’s T index :

Organisation theory Saviotti (1988)

Income inequality (Theil 1967: 91-134 and 1972: 99-109)

Technological evolution (innovation studies Frenken et al. 1999) Regional industrial diversification (Hackbart and Anderson 1975; Attaran 1985)

Corporate diversification and profitability (Jacquemin and Berry 1979; Palepu 1985; Hoskisson et al. 1993)

Industrial concentration (Hildenbrand and Paschen 1964; Finkelstein and Friedberg 1967; Theil 1967: 290-291)

World Inequality: Population and Income Shares of the 54 Richest and 54Poorest Countries in the World in 1970.

Simulation of the Evolution of Inequality Measures as the Shares of IncomeChange.

Theil IndexDeconstruction

Source: Conceição, P. and Ferreira, P. (2000) : The Young Person’s Guide to the Theil Index: Suggesting Intuitive Interpretations and Exploring Analytical Applications, UTIP Working Paper N°14, 54p

Time series spatialization: REGIONAL PATTERNS OF INEQUALITY

Fractal behavior of the Theil Index (attractor ?)

Vertical cut, with origin in individual i, intercepts the boundary of individual i (itssource point), but also the boundaries of group l and group a.

SocialEntropy

Econogical weight

Socioecological weight

Socioeconomical weight EconomicalEntropy

EnvironmentalEntropy

Vulnerability

Resi

lienc

e

DEVELOPMENT

NEEDS

RESOURCE

Example classification using numerical techniques. The top row shows how the system is clustered at several levels, parameterized by taxonomic level h. The classification is summarized in a taxonomic tree, or dendrogram (bottom).

Social entropy

social entropy (computed using Shannon’s information entropy formulation(Shannon, 1949)) as an appropriate measure of diversity

TUCKER BALCH: Hierarchic Social Entropy: An Information Theoretic Measureof Robot Group Diversity, Autonomous Robots 8, 209–237, 2000

branching structure of the dendrograms for these two societies is the same. However, the more compact distribution of elementsin the system on the upper right is reflected in the branches being compressed towards the bottom of the corresponding dendrogram (lower right).

Hierarchic Social Entropy Metric

Because hierarchic entropy is scale invariant it can distinguishbetween the two societies regardless of the value of x.

Groups of countries derived from the classification according to their inequalitylevel

Application: Inequality trends comparison among European countries

Dendrogram of the countries’ common space based inequality indexEuropean Community Household Panel (ECHP)

KRZANOWSKI, W.J. (1979) “Between-groups comparison of principal components”.Journal of the American Statistical Association, 74, 703-707. Correction note(1981), 76, 1022.KRZANOWSKI, W.J. (1982) “Between-groups comparison of principal components –some sampling results”. Journal of Statistical Computation and Simulation, 15,141-154.

From economics to social: the « social capital » ecosocial combination. (equitable/social justice)

Relationship Between Bridging Social Capital and Governance Source: Narayan (1999)

Ecosocial indexing

Econogical weight

Socioecological weight

Socioeconomical weight Economical Social

Environmental

Vulnerability

Resi

lienc

e

DEVELOPMENT

NEEDS

RESOURCE

Ecosocial entropy

The case α = ∞ corresponds to perfectly equal distribution (G = 0) and the α = 1 line corresponds to complete inequality (G = 1)

Gini coefficient a measure of the deviation of the Lorenz curve from the equidistribution plot of the fraction of total income held by a given fraction of the population (Heaps et al., 1998, Aaberge 2005).

Income patterns for Pareto distributionLorenz Curve & Gini Coefficient

Gini coefficient for a Pareto distributionLorenz curves for Pareto distributions.

Measure of the maximum vertical distance between the Lorenzcurve and the line of perfect equality (45° line)

Foster, Greer, and Thorbecke (1984) class of poverty measures

Three dimensions of poverty—headcount, shortfall, and inequality among the poor—

α the parameterz the distance to the hunger Linei-> n the number of individuals

Pareto distribution of income patterns: a powerty measurement? Robin Hood Index, hunger line and Hoover inequality

Hunger Lines vs. Mean Income

Hunger Line: the social risk perception

Sources: Current hunger levels from FAO (1997a); Incomes from WRI (1996b), Heaps, et al. (1998)

Hunger line tend to rise as average incomes do (Ravallion et al., 1991; World Bank, 1990)

From Plato inequality To redistributive aggression?

Tresspassing the hunger Line

Conclusion: Inequality is leading to battlefields

Unfair wealth distribution leads to social aggression

Ecosocial entropy measure is a risk indicator

Pareto, Vilfredo, Cours d’Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino, Librairie Droz, Geneva, 1964, pages 299–345.

Seal, H. (1980). Survival probabilities based on Pareto claim distributions. ASTIN Bulletin, 11, 61–71.

M. E. J. Newman (2005). "Power laws, Pareto distributions and Zipf's law". Contemporary Physics 46 (5): 323–351

Aaberge, R. (2000): Characterizations of Lorenz curves and income distributions, Social Choice and Welfare 17, 639-653.

Aaberge, R. (2001): Axiomatic characterization of the Gini coefficient and Lorenz curveorderings, Journal of Economic Theory, 101, 115-132.

Reed, W.J., 2000 a. The Pareto law of incomes – an explanation and an extension. Submitted toJournal of Business and Economic Statistics.

Reed, W.J., 2000 b. On the rank-size distribution for human settlements. Submitted to RegionalScience and Urban Economics.

Gabaix, X., 1999. Zipf’s law for cities: an explanation. Quarterly Journal of Economics, 114739-767.

Brakman, S, H.Garretsen, C. Van Marrewijk and M. van den Berg, 1999. The return of Zipf:Towards a further understanding of the rank-size distribution. Journal of Regional Science,39:739-767.