Sulle code di potenza di Pareto
Toscani, Giuseppe
Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana, Tome 1 (2016), p. 21-30 / Harvested from Biblioteca Digitale Italiana di Matematica
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In questo articolo vengono brevemente presentate le principali analogie tra il problema dell'andamento all'equilibrio delle molecole di un gas rarefatto e la formazione delle code di potenza nella distribuzione della ricchezza in una società di agenti. L'approccio della meccanica statistica al succitato problema di origine economica ha fornito infatti in questi ultimi anni una spiegazione particolarmente convincente sul fenomeno della formazione delle code di potenza di Pareto.

This article briefly introduces the main similarities between the problem of the trend to equilibrium of the molecules of a rarefied gas and the formation of the power tails in the distribution of wealth in a multi-agent society. The approach of statistical mechanics to the above mentioned problem of economic nature in fact provided in recent years a particularly convincing explanation on the phenomenon of formation of Pareto tails.

Publié le : 2016-04-01
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     title = {Sulle code di potenza di Pareto},
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     pages = {21-30},
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Toscani, Giuseppe. Sulle code di potenza di Pareto. Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana, Tome 1 (2016) pp. 21-30. http://gdmltest.u-ga.fr/item/RUMI_2016_1_1_1_21_0/

[1] Amoroso, L., Ricerche intorno alla curva dei redditi. Ann. Mat. Pura Appl. Ser. 4 21, (1925), 123-159. | MR 1553077 | Zbl 51.0405.08

[2] Bassetti, F., Toscani, G., Explicit equilibria in a kinetic model of gambling. Phys. Rev. E81, (2010), 066115. | MR 2736281 | Zbl 1335.91049

[3] Benini, R., Di alcune curve descritte da fenomeni economici aventi relazione colla curva del reddito o con quella del patrimonio. Giornale degli Economisti, Serie II 14, (1897), 177-214.

[4] Bisi, M., Spiga, G., Toscani, G., Kinetic models of conservative economies with wealth redistribution. Commun. Math. Sci.7, (4) (2009), 901-916. | MR 2604625 | Zbl 1188.91115

[5] Bobylev, A.V., The theory of the nonlinear spatially uniform Boltzmann equation for Maxwellian molecules. Sov. Sci. Rev. c7, (1988), 111-233. | MR 1128328 | Zbl 0850.76619

[6] Bobylev, A.V., Carrillo, J.A., Gamba, I., On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Stat. Phys., 98,(2001), 743-773; Erratum on: J. Stat. Phys., 103, (2001), 1137-1138. | MR 1851370 | Zbl 1056.76071

[7] Boltzmann, L., Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzungsberichte der Akademie der Wissenschaften66, (1995), 275-370, in Lectures on Gas Theory. Berkeley: University of California Press (1964) Translated by S.G. Brush. Reprint of the 1896-1898 Edition. Reprinted by Dover Publ.

[8] Carrillo, J.A., Toscani, G., Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma6/7, (2007), 75-198. | MR 2355628 | Zbl 1142.82018

[9] Cercignani, C., The Boltzmann equation and its applications. Springer Series in Applied Mathematical Sciences, Vol. 67, Springer-Verlag, New York1988. | MR 1313028 | Zbl 0646.76001

[10] Cercignani, C., Illner, R., Pulvirenti, M., The mathematical theory of dilute gases. Springer Series in Applied Mathematical Sciences, Vol. 106, Springer-Verlag, New York1994. | MR 1307620 | Zbl 0813.76001

[11] Chakraborti, A., Distributions of money in models of market economy. Int. J. Modern Phys. C13, (2002), 1315-1321.

[12] Chakraborti, A., Chakrabarti, B.K., Statistical mechanics of money: effects of saving propensity. Eur. Phys. J. B17, (2000), 167-170

[13] Chatterjee, A., Chakrabarti, B.K., Manna, S.S., Pareto law in a kinetic model of market with random saving propensity. Physica A335, (2004), 155-163. | MR 2048143

[14] Chatterjee, A., Chakrabarti, B.K., Stinchcombe, R.B., Master equation for a kinetic model of trading market and its analytic solution. Phys. Rev. E72, (2005), 026126.

[15] Cordier, S., Pareschi, L., Toscani, G., On a kinetic model for a simple market economy. J. Stat. Phys.120, (2005), 253-277. | MR 2165531 | Zbl 1133.91474

[16] D'Addario, R., Intorno alla curva dei redditi di Amoroso. Riv. Italiana Statist. Econ. Finanza4, (1) (1932), 723-729.

[17] D'Addario, R., Ricerche sulla curva dei redditi. Giornale degli Economisti e Annali di Economia Nuova Serie, Anno 8, No. 1/2 (1949), 91-114.

[18] Drăgulescu, A., Yakovenko, V.M., Statistical mechanics of money. Eur. Phys. Jour. B17, (2000), 723-729.

[19] Düring, B., Matthes, D., Toscani, G., Kinetic equations modelling wealth redistribution: a comparison of approaches. Phys. Rev. E78, (2008), 056103. | MR 2551376

[20] Düring, B., Matthes, D., Toscani, G., A Boltzmann type approach to the formation of wealth distribution curves. Riv. Mat. Univ. Parma8, (1) (2009), 199-261. | MR 2597795 | Zbl 1189.91097

[21] Ernst, M.H., Brito, R., High energy tails for inelastic Maxwell models. Europhys. Lett.58, (2002), 182-187.

[22] Ernst, M.H., Brito, R., Scaling solutions of inelastic Boltzmann equation with over-populated high energy tails. J. Statist. Phys.109, (2002), 407-432. | MR 1942001 | Zbl 1015.82030

[23] Guala, S., Taxes in a simple wealth distribution model by inelastically scattering particles. Interdisciplinary description of complex systems7, (2009), 1-7.

[24] Majorana, E., Il valore delle leggi statistiche nella fisica e nelle scienze sociali. Scientia36, (1942), 58-66.

[25] Maldarella, D., Pareschi, L., Kinetic models for socioeconomic dynamics of speculative markets. Physica A391, (2012), 715-730.

[26] Matthes, D., Toscani, G., On steady distributions of kinetic models of conservative economies. J. Stat. Phys.130, (2008), 1087-1117. | MR 2379241 | Zbl 1138.91020

[27] G. Naldi, L. Pareschi, G. Toscani eds., Mathematical modelling of collective behavior in socio-economic and life sciences. Birkhauser, Boston 2010. | MR 2761862 | Zbl 1200.91010

[28] Pareschi, L., Toscani, G., Interacting multiagent systems: kinetic equations and Monte Carlo methods. Oxford University Press, Oxford2014. | Zbl 1330.93004

[29] Pareschi, L., Toscani, G., Wealth distribution and collective knowledge. A Boltzmann approach. Phil. Trans. R. Soc. A372, (2014), 20130396. | MR 3268064 | Zbl 1353.91036

[30] Pareto, V., Cours d'èconomie politique. Rouge, Lausanne and Paris, 1897.

[31] Slanina, F., Inelastically scattering particles and wealth distribution in an open economy. Phys. Rev. E69, (2004), 046102.

[32] Stanley, H.E., Afanasyev, V., Amaral, L.A.N., Buldyrev, S.V., Goldberger, A.L., Havlin, S., Leschorn, H., Maass, P., Mantegna, R.N., Peng, C.-K., Prince, P.A., Salinger, M.A., Stanley, M.H.R., Viswanathan, G.M., Anomalous fluctuations in the dynamics of complex systems: from DNA and physiology to econophysics. Physica A224, (1996), 302-321.

[33] Toscani, G., Wealth redistribution in conservative linear kinetic models with taxation. Europhysics Letters88, (1) (2009), 10007.

[34] Toscani, G., Villani, C., Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys.203 (1999), 667-706. | MR 1700142 | Zbl 0944.35066

[35] Villani, C., Cercignani's conjecture is sometimes true and always almost true. Commun. Math. Phys.234, (2003), 455-490. | MR 1964379 | Zbl 1041.82018