Comptes Rendus
no. 4
From statistical physics to social sciences / De la physique statistique aux sciences sociales
Dynamics of wealth inequality
[Dynamique de l'inégalité de richesse]
Zdzislaw Burda1  ; Pawel Wojcieszak1  ; Konrad Zuchniak1
1 Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, 30059 Kraków, Poland
Comptes Rendus. Physique, Volume 20 (2019) no. 4, pp. 349-363.

Nous étudions un modèle d'évolution de la répartition de la richesse dans un système macro-économique basé sur les agents. L'évolution est mue par des fluctuations stochastiques multiplicatives régies par la loi de la croissance proportionnelle et des interactions entre agents. Nous nous intéressons principalement aux interactions qui accroissent les inégalités de richesse, c'est-à-dire à une mise en œuvre locale du principe de l'avantage accumulé. De telles interactions déstabilisent le système. Elles sont confrontées dans le modèle à un mécanisme réglementaire mondial qui réduit les inégalités de richesse. Différents scénarios se dessinent, comme un effet net de ces deux mécanismes concurrents. Lorsque l'effet de la régulation globale (interventionnisme économique) est trop faible, le système est instable et n'atteint jamais l'équilibre. Lorsque l'effet est suffisamment fort, le système évolue vers une distribution stationnaire limitante avec une queue de Pareto. Entre les deux, il y a une phase critique. Dans cette phase, le système peut évoluer vers un état stable avec une répartition multimodale des richesses. La fonction de densité cumulative correspondante suit un motif d'escalier caractéristique qui reflète l'effet de la stratification économique. Les escaliers représentent les niveaux de richesse des classes économiques séparées par des écarts de richesse. Comme nous le montrons, le schéma est typique des systèmes macro-économiques avec une liberté économique limitée. On peut trouver un tel modèle multimodal dans les données empiriques, par exemple, dans le percentile le plus élevé de la répartition des richesses pour la population des zones urbaines de la Chine.

We study an agent-based model of evolution of wealth distribution in a macroeconomic system. The evolution is driven by multiplicative stochastic fluctuations governed by the law of proportionate growth and interactions between agents. We are mainly interested in interactions increasing wealth inequality, that is, in a local implementation of the accumulated advantage principle. Such interactions destabilise the system. They are confronted in the model with a global regulatory mechanism that reduces wealth inequality. There are different scenarios emerging as a net effect of these two competing mechanisms. When the effect of the global regulation (economic interventionism) is too weak, the system is unstable and it never reaches equilibrium. When the effect is sufficiently strong, the system evolves towards a limiting stationary distribution with a Pareto tail. In between there is a critical phase. In this phase, the system may evolve towards a steady state with a multimodal wealth distribution. The corresponding cumulative density function has a characteristic stairway pattern that reflects the effect of economic stratification. The stairs represent wealth levels of economic classes separated by wealth gaps. As we show, the pattern is typical for macroeconomic systems with a limited economic freedom. One can find such a multimodal pattern in empirical data, for instance, in the highest percentile of wealth distribution for the population in urban areas of China.

Publié le :
DOI : 10.1016/j.crhy.2019.05.011
Keywords: Wealth inequality, Wealth distribution, Population dynamics, Stochastic evolution, Agent-based modelling, Monte Carlo methods
Mot clés : Inégalité de richesse, Répartition de la richesse, Dynamique de la population, Évolution stochastique, Modélisation basée sur les agents, Méthodes de Monte-Carlo
Zdzislaw Burda 1 ; Pawel Wojcieszak 1 ; Konrad Zuchniak 1

1 Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, 30059 Kraków, Poland 
@article{CRPHYS_2019__20_4_349_0,
     author = {Zdzislaw Burda and Pawel Wojcieszak and Konrad Zuchniak},
     title = {Dynamics of wealth inequality},
     journal = {Comptes Rendus. Physique},
     pages = {349--363},
     publisher = {Elsevier},
     volume = {20},
     number = {4},
     year = {2019},
     doi = {10.1016/j.crhy.2019.05.011},
     language = {en},
}
TY  - JOUR
AU  - Zdzislaw Burda
AU  - Pawel Wojcieszak
AU  - Konrad Zuchniak
TI  - Dynamics of wealth inequality
JO  - Comptes Rendus. Physique
PY  - 2019
SP  - 349
EP  - 363
VL  - 20
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crhy.2019.05.011
LA  - en
ID  - CRPHYS_2019__20_4_349_0
ER  - 
%0 Journal Article
%A Zdzislaw Burda
%A Pawel Wojcieszak
%A Konrad Zuchniak
%T Dynamics of wealth inequality
%J Comptes Rendus. Physique
%D 2019
%P 349-363
%V 20
%N 4
%I Elsevier
%R 10.1016/j.crhy.2019.05.011
%G en
%F CRPHYS_2019__20_4_349_0
Zdzislaw Burda; Pawel Wojcieszak; Konrad Zuchniak. Dynamics of wealth inequality. Comptes Rendus. Physique, Volume 20 (2019) no. 4, pp. 349-363. doi : 10.1016/j.crhy.2019.05.011. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2019.05.011/

[1] T. Piketty Capital in the XXI Century, Harvard University Press, 2014

[2] E. Saez; G. Zucman Wealth inequality in the United States since 1913: evidence from capitalized income tax data, 2014 http://www.nber.org/papers/w20625 (NBER Working Paper Series, Working Paper 20625)

[3] T. Piketty About capital in the twenty-first century, Am. Econ. Rev. Pap. Proc., Volume 105 (2015) no. 5, pp. 48-53 | DOI

[4] Oxfam Briefing Paper An economy for the 99%, January 2017 https://www.oxfam.org/

[5] Credit Suisse Research Institute, Global Wealth Databook 2016, November 2016.

[6] J.B. Davis; A.E. Shorrocks The distribution of wealth (A.B. Atkinson; F. Bourguignon, eds.), Handbook of Income Distribution, Elsevier, 2000 (Chapter 11)

[7] J. Benhabib; A. Bisin Skewed wealth distributions: theory and empirics, 2016 http://www.nber.org/papers/w21924 (NBER Working Paper Series, Working Paper 21924)

[8] J.D. Ostry; A. Berg; C.G. Tsangarides Redistribution, Inequality and Growth, 2014 (IMF Discussion Note, SDN 14/02)

[9] J.-P. Bouchaud J. Stat. Mech. (2015)

[10] V. Pareto Cours d'Économie Politique, II, F. Rouge, Lausanne, 1897

[11] P.A. Samuelson A fallacy in the interpretation of the Pareto's law of alleged constancy of income distribution, Riv. Int. Sci. Econ. Commer., Volume 12 (1965), pp. 246-250

[12] R. Gibrat Les Inégalités économiques, Paris, 1931

[13] T. Piketty; E. Saez A theory of optimal capital taxation, 2012 http://www.nber.org/papers/w17989 (NBER Working Paper Series, Working Paper 17989)

[14] R. Cont; J.P. Bouchaud Herd behavior and aggregate fluctuation in financial market, Macroecon. Dyn., Volume 4 (2000)

[15] J. Angle The surplus theory of social stratification and the size distribution of personal wealth, Soc. Forces, Volume 65 (1986), pp. 293-326

[16] E. Samanidou; E. Zschischang; D. Stauffer; T. Lux Agent-based models of financial markets, Rep. Prog. Phys., Volume 70 (2007), pp. 409-450

[17] G.U. Yule A mathematical theory of evolution, based on the conclusions of Dr. J.C. Willis, Philos. Trans. R. Soc. B, Volume 213 (1925), p. 21

[18] H.A. Simon On a class of skew distribution functions, Biometrika, Volume 42 (1955), p. 425

[19] A.L. Barabasi; R. Albert Emergence of scaling in random networks, Science, Volume 286 (1999), p. 509

[20] J.-P. Bouchaud; M. Mézard Wealth condensation in a simple model of economy, Physica A, Volume 282 (2000), p. 536

[21] F.J. Dyson A Brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys., Volume 3 (1962), p. 1191

[22] P.J. Forrester Log-Gases and Random Matrices, Princeton University Press, 2010

[23] P. Bialas; Z. Burda; D. Johnston Condensation in the Backgammon model, Nucl. Phys. B, Volume 493 (1997), p. 505

[24] P. Bialas; L. Bogacz; Z. Burda; D. Johnston Finite size scaling of the balls in boxes model, Nucl. Phys. B, Volume 575 (2000), p. 599

[25] X. Gabaix Power laws in economics and finance, Annu. Rev. Econ., Volume 1 (2009), p. 255

[26] M. De Nardi Models of Wealth Inequality: A Survey, NBER, 2015 (Working Paper 21106)

[27] A. Adamou; P. Ole Dynamics of inequality, Significance, Volume 13 (2016), p. 32

[28] Y. Berman; O. Peters; A. Adamou Far from equilibrium: wealth reallocation in the United States, 2016 | arXiv

[29] Y. Berman; E. Ben-Jacob; Y. Shapira The dynamics of wealth inequality and the effect of income distribution, PLoS ONE, Volume 11 (2016)

[30] T. Blanchet, J. Fournier, T. Piketty, Generalized Pareto curves: theory and applications, WID.world working paper series 2017/3.

[31] M. Levy; S. Solomon Power laws are logarithmic Boltzmann laws, Int. J. Mod. Phys. C, Volume 7 (1996), p. 595

[32] S. Ispolatov; P.L. Krapivsky; S. Redner Wealth distributions in models of capital exchange, Eur. Phys. J. B, Volume 2 (1998), p. 267

[33] V.M. Yakovenko; J.B. Rosser Colloquium: statistical mechanics of money, wealth, and income, Rev. Mod. Phys., Volume 81 (2009), p. 1703

[34] Z. Burda; J. Jurkiewicz; M.A. Nowak Is econophysics a solid science?, Acta Phys. Pol. B, Volume 34 (2003), p. 87

[35] B.K. Chakrabarti; A. Chakraborti; S.R. Chakravarty; A. Chatterjee Econophysics of Income and Wealth Distributions, Cambridge University Press, 2013

[36] T. Ma; J.G. Holden; R.A. Serota Distribution of wealth in a network model of the economy, Physica A, Volume 392 (2013), p. 2434

[37] Z. Liu; R.A. Serota Correlation and relaxation times for a stochastic process with a fat-tailed steady-state distribution, Physica A, Volume 474 (2017), p. 301

[38] Y. Berman; Y. Shapira; M. Schwartz A Fokker-Planck model for wealth inequality dynamics, Europhys. Lett., Volume 118 (2017), p. 3

[39] Z. Liu; R.A. Serota On absence of steady state in the Bouchaud–Mézard network model, Physica A, Volume 491 (2018), p. 391

[40] B.V. Gnedenko; A.N. Kolmogorov Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, 1954

[41] K. Itô On Stochastic Differential Equations, Memoirs, American Mathematical Society, vol. 4, 1944

[42] N.G. Van Kampen Stochastic Processes in Physics and Chemistry, Elsevier, 2007

[43] H. Kesten Random difference equations and renewal theory for products of random matrices, Acta Math., Volume 131 (1973), p. 207

[44] D.G. Champernowne A model of income distribution, Econ. J., Volume 63 (1953), p. 318

[45] H.O.A. Wold; P. Whittle A model explaining the Pareto distribution of wealth, Econometrica, Volume 25 (1957), p. 591

[46] I.S. Gradshteyn; I.M. Ryzhik Table of Integrals, Series, and Products, Academic Press, 2007

[47] T. Piketty; L. Yang; G. Zucman Capital Accumulation, Private Property and Rising Inequality in China, 1978-2015, 2017 (WID.world working paper series, No. 2017/6)

[48] Z. Burda et al. Wealth condensation in Pareto macroeconomies, Phys. Rev. E, Volume 65 (2002)

[49] P. Ball Wealth spawns corruption, physicists are explaining how politics can create the super-rich, Nature (2002) (News020121-14) | DOI

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Risk premium and fair option prices under stochastic volatility: the HARA solution

Srdjan D. Stojanovic

C. R. Math (2005)

Structuration optimale de produits financiers et diversification en présence de sources de risque non-négociables

Pauline Barrieu; Nicole El Karoui

C. R. Math (2003)