# Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange

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## Abstract

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## 1. Introduction

## 2. Simulations for a Fixed Value of the Initial Total Income and Gini Index

#### 2.1. Langevin Equation. Deterministic Solutions

#### 2.2. Stochastic Time-Series

## 3. Dependence of the Correlations on the Total Income and on G

## 4. Langevin Equation with Ornstein–Uhlenbeck Noise and Dependence of the Correlations on $\Gamma $ and $\tau $

## 5. Fluctuations of the Populations in Dependence from $\Gamma $

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Patriarca, M.; Chakraborti, A. Kinetic exchange models: From molecular physics to social science. Am. J. Phys.
**2013**, 81, 618. [Google Scholar] [CrossRef] - Pareschi, L.; Toscani, G. Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods; Oxford University Press: Oxford, UK, 2013. [Google Scholar]
- Dragulescu, A.; Yakovenko, V.M. Statistical mechanics of money. Eur. Phys. J. B
**2000**, 17, 723–729. [Google Scholar] [CrossRef] - Schinckus, C. Between complexity of modelling and modelling of complexity: An essay of econophysics. Physica A
**2013**, 392, 3654–3665. [Google Scholar] [CrossRef] - Gallegati, M.; Kirman, A. Reconstructing Economics: Agent Based Models and Complexity. Complex. Econ.
**2012**, 1, 5–31. [Google Scholar] [CrossRef] - Tesfatsion, L.; Judd, K.L. Handbook of Computational Economics, Volume 2: Agent Based Computational Economics; North-Holland: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Bertotti, M.L. Modelling taxation and redistribution: A discrete active particle kinetic approach. Appl. Math Comput.
**2010**, 217, 752–762. [Google Scholar] [CrossRef] - Bertotti, M.L.; Modanese, G. Micro to macro models for income distribution in the absence and in the presence of tax evasion. Appl. Math. Comput.
**2014**, 244, 836–846. [Google Scholar] [CrossRef] - Bertotti, M.L.; Modanese, G. Discretized kinetic theory on scale-free networks. Eur. Phys. J. Spec. Top.
**2016**, 225, 1879–1891. [Google Scholar] [CrossRef] - Bertotti, M.L.; Modanese, G. Microscopic models for the study of taxpayer audit effects. Int. J. Mod. Phys. C
**2016**, 27. [Google Scholar] [CrossRef] - Bertotti, M.L.; Chattopadhyay, A.K.; Modanese, G. Stochastic effects in a discretized kinetic model of economic exchange. Physica A
**2017**, 471, 724–732. [Google Scholar] [CrossRef] - Bertotti, M.L.; Chattopadhyay, A.K.; Modanese, G. Economic inequality and mobility for stochastic models with multiplicative noise. arXiv, 2017; arXiv:1702.08391. [Google Scholar]
- Stiglitz, J.E. The Price of Inequality: How Today’s Divided Society Endangers Our Future; W.W. Norton & Company: New York, NY, USA, 2012. [Google Scholar]
- Atkinson, A.B. Inequality: What Can Be Done? Harvard University Press: Cambridge, UK, 2015. [Google Scholar]
- Corak, M. Income inequality, equality of opportunity, and intergenerational mobility. J. Econ. Perspect.
**2013**, 27, 79–102. [Google Scholar] [CrossRef] - Andrews, D.; Leigh, A. More inequality, less social mobility. Appl. Econ. Lett.
**2009**, 16, 1489–1492. [Google Scholar] [CrossRef] - Bertotti, M.L.; Modanese, G. Economic inequality and mobility in kinetic models for social sciences. Eur. Phys. J. Spec. Top.
**2016**, 225, 1945–1958. [Google Scholar] [CrossRef] - Spannagel, D.; Broschinski, S. Reichtum in Deutschland Wächst Weiter; WSI Report; Wirtschafts- und Sozialwissenschaftliches Institut: Duesseldorf, Germany, 17 September 2014. [Google Scholar]
- Jerrim, J.; Macmillan, L. Income inequality, intergenerational mobility, and the Great Gatsby curve: Is education the key? Soc. Forces
**2015**, 94, 505–533. [Google Scholar] [CrossRef] - Aghion, P.; Caroli, E.; Garcia-Penalosa, C. Inequality and economic growth: The perspective of the new growth theories. Appl. Econ. Lett.
**1999**, 37, 1615–1660. [Google Scholar] [CrossRef][Green Version] - Bertotti, M.L.; Chattopadhyay, A.K.; Modanese, G. Uncertainty dynamics in a model of economic inequality. Int. J. Des. Nat. Ecodyn.
**2017**, 13, 16–22. [Google Scholar] [CrossRef]

**Figure 1.**Dependence of the correlation ${R}_{GM}$ on the Gini index with mixed noise in a range which extends far below the usual values of G. Each dot represents the average of 80 realizations with 2500 steps. There are 480 dots in total.

**Figure 2.**Histogram of a typical deterministic equilibrium configuration, with 10 income classes (see details in the text). The bars represent the populations ${x}_{1},\dots ,{x}_{10}$.

**Figure 3.**Behavior of the Gini index G and the social mobility M as functions of the total income $\mu $ for the deterministic model, at equilibrium. The asymptotic equilibrium of the deterministic equations does not depend on the detailed initial conditions ${x}_{i}(0)$ (class populations), but only on the total income $\mu =\mu (0)={\sum}_{i}{r}_{i}{x}_{i}(0)$ ($\mu $ is conserved in the deterministic evolution). This plot is obtained by taking several values of $\mu (0)$ as explained in Section 2.1, letting the system evolve deterministically and computing G and M in the equilibrium state. The relation between G and $\mu $ allows to determine a range of values of $\mu $ which corresponds to a range of realistic values of G.

**Figure 4.**Example of time-series of G and $\mu $ in the canonical case, with multiplicative white noise of amplitude $\Gamma =0.01$. The time steps in the series are 40,000, with a sampling each 100 steps.

**Figure 5.**Time auto-correlation of G in the time-series of Figure 4. T denotes the number of time steps.

**Figure 6.**Histogram of the correlation ${R}_{GM}$ in 6000 realizations of 5000 steps, with multiplicative noise of amplitude $\Gamma =0.001$.

**Figure 8.**Correlation ${R}_{GM}$ for Ornstein-Uhlenbeck noise in dependence on noise amplitude $\Gamma $ and memory time $\tau $. Each dot is the average of 50 realizations with 5000 steps.

**Figure 11.**Average deviation ${\delta}_{av}$ from deterministic equilibrium in dependence on the noise amplitude $\Gamma $, for multiplicative white noise. Each dot is the average of 500 realizations with 5000 steps.

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**MDPI and ACS Style**

Bertotti, M.L.; Chattopadhyay, A.K.; Modanese, G. Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange. *Entropy* **2018**, *20*, 166.
https://doi.org/10.3390/e20030166

**AMA Style**

Bertotti ML, Chattopadhyay AK, Modanese G. Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange. *Entropy*. 2018; 20(3):166.
https://doi.org/10.3390/e20030166

**Chicago/Turabian Style**

Bertotti, Maria Letizia, Amit K. Chattopadhyay, and Giovanni Modanese. 2018. "Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange" *Entropy* 20, no. 3: 166.
https://doi.org/10.3390/e20030166