# A Wealth Distribution Agent Model Based on a Few Universal Assumptions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Outline of the Model

#### 2.1. Fundamental Characteristics

#### Basic Assumptions

**Assumption 1**(First basic assumption)

**.**

**Assumption 2**(Second basic assumption)

**.**

#### 2.2. Taxation

**Assumption 3**(Third basic assumption)

**.**

#### 2.2.1. Taxation on Wealth

#### 2.2.2. Taxation on Income or Capital Gain

#### 2.3. Simulation Setup

**five Monte Carlo steps**constitute a

**stage**, and every step is synchronous: the state of an agent (increase/decrease in wealth) is only updated when the Monte Carlo step is completed (all agents have been updated). Tax collection and redistribution occur only once at the end of the stage. Therefore, Monte Carlo steps can be interpreted as the passage of months, while a stage as the passage of an entire year (annual tax).

- Agents with ${w}_{i}\ge 1$, which are shown in the distributions;
- Agents with ${w}_{i}<1$, which are taken as the poverty rate and only appear as a percentage.

#### 2.4. First Scenario: Raw Model and Taxation on Wealth

#### 2.5. Second Scenario: The Wealth–Connection Model with Wealth Taxation

#### 2.6. Third Scenario: Favoring the Rich on Transactions and Wealth Taxation

#### 2.7. Fourth Scenario: Favoring the Rich in Transactions and Taxation on Annual Income (Capital Gains)

## 3. Results

#### 3.1. Raw Model

#### Statistics

#### 3.2. Wealth–Trade Link

#### Statistics

#### 3.3. Favoring the Rich on Transactions

#### 3.3.1. Distributions

- The problem of poverty is not simply solved with higher tax rates. How to redistribute the tax collected is also an essential point. Here, the tax has been redistributed equally among agents. A redistribution of tax that favors the poor is likely to decrease the level of poverty. However, this has not been considered in this work, and it would be interesting to analyze this issue in a future work.
- A strong tax system does not necessarily mean lower poverty rates. As said before, how to redistribute taxes is also a key point.
- It is not necessary to eliminate inequality in order to end poverty. If the rich are taxed properly—on wealth—and redistribution favors the poor, poverty can be virtually eliminated.

#### 3.3.2. Statistics

#### 3.4. Annual Income Tax Model

#### 3.4.1. Distributions

#### 3.4.2. Statistics

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Chakrabarti, B.; Chakraborti, A.; Chakravarty, S.; Chatterjee, A. Econophysics of Income and Wealth Distributions, 1st ed.; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Drăgulescu, A.; Yakovenko, V.M. Exponential and power-law probability distributions of wealth and income in the united kingdom and the united states. Physica A
**2001**, 299, 213. [Google Scholar] [CrossRef] - Banerjee, A.; Yakovenko, V.M. Universal patterns of inequality. New J. Phys.
**2010**, 12, 075032. [Google Scholar] [CrossRef] - Parsson, J.O. Dying of Money: Lessons of the Great German and American Inflations, 1st ed.; Dog Ear Publishing: Indianapolis, IN, USA, 2011. [Google Scholar]
- Piketty, T. Capital in the Twenty-First Century, 1st ed.; The Belknap Press of Harvard University Press: Cambridge, MA, USA, 2014. [Google Scholar]
- Lorenz, M.O. Methods of Measuring the Concentration of Wealth; American Statistical Association: Alexandria, VA, USA, 1905; Volume 9, pp. 209–219. [Google Scholar] [CrossRef]
- Ceriani, L.; Verme, P. The Origins of the Gini Index: Extracts from Variabilità e Mutabilità (1912) by Corrado Gini. J. Econ. Inequal.
**2012**, 10, 421. [Google Scholar] [CrossRef] - Chatterjee, A.; Chakrabarti, B.K. Kinetic exchange models for income and wealth distributions. Eur. Phys. J. B
**2007**, 60, 135. [Google Scholar] [CrossRef] - Chatterjee, A.; Chakrabarti, B.K. Kinetic market models with single commodity having price fluctuations. Eur. Phys. J. B
**2006**, 54, 399. [Google Scholar] [CrossRef] - Chakraborti, A.; Patriarca, M. Variational principle for the pareto power law. Phys. Rev. Lett.
**2009**, 103, 228701. [Google Scholar] [CrossRef] [PubMed] - Braunstein, L.A.; Macri, P.A.; Iglesias, J.R. Study of a market model with conservative exchanges on complex networks. Physica A
**2013**, 392, 1788. [Google Scholar] [CrossRef] - Drăgulescu, A. Applications of physics to economics and finance: Money, income, wealth, and the stock market. arXiv
**2003**, arXiv:0307341v2. [Google Scholar] - Queiros, S.M.D.; Anteneodo, C.; Tsallis, C. Power-law distributions in economics: A nonextensive statistical approach. In Noise and Fluctuations in Econophysics and Finance; SPIE: Bellingham, WA, USA, 2005; Volume 5848. [Google Scholar]
- Aoyama, H.; Nagahara, Y.; Okazaki, M.; Souma, W.; Takayasu, H.; Takayasu, M. Pareto’s law for income of individuals and debt of bankrupt companies. Fractals
**2000**, 8, 293. [Google Scholar] [CrossRef] - Clementi, F.; Gallegati, M. Pareto’s Law of Income Distribution: Evidence for Germany, the United Kingdom, and the United States. In Econophysics of Wealth Distributions, 1st ed.; Springer: Milan, Italy, 2005. [Google Scholar]
- Siciliani, I.D.; Tragtenberg, M. Kinetic theory and brazilian income distribution. Physica A
**2019**, 513, 166. [Google Scholar] [CrossRef] - Saez, E.; Zucman, G. The Triumph of Injustice: How the Rich Dodge Taxes and How to Make Them Pay, 1st ed.; W. W. Norton: New York, NY, USA, 2019. [Google Scholar]
- Steinbaum, M. Effective Progressive Tax Rates in the 1950s; Roosevelt Institute: New York, NY, USA, 2017; Available online: https://rooseveltinstitute.org/2017/08/08/effective-progressive-tax-rates-in-the-1950s/ (accessed on 12 February 2021).
- Iglesias, J.R.; de Almeida, R.M.C. Entropy and equilibrium state of free market models. Eur. Phys. J. B
**2012**, 85, 85. [Google Scholar] [CrossRef] - Tesfatsion, L.; Judd, K. Handbook of Computational Economics, Vol. 2: Agent-Based Computational Economics, 1st ed.; Staff General Research Papers; Iowa State University, Department of Economics: Ames, IA, USA, 2006. [Google Scholar]
- Moran, J.; Bouchaud, J.-P. May’s instability in large economies. Phys. Rev. E
**2019**, 100, 032307. [Google Scholar] [CrossRef] [PubMed] - de Oliveira, P. Investment/taxation model: Investors in groups. Physica A
**2020**, 537, 122588. [Google Scholar] [CrossRef] - de Oliveira, P. Investment/taxation/redistribution model criticality. Eur. Phys. J. B
**2020**, 93, 196. [Google Scholar] [CrossRef] - Iglesias, J.R.; Cardoso, B.-H.F.; Gonçalves, S. Inequality, a scourge of the XXI century. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 95, 105646. [Google Scholar] [CrossRef] - Cardoso, B.-H.F.; Iglesias, J.R.; Gonçalves, S. Wealth concentration in systems with unbiased binary exchanges. Physica A
**2021**, 579, 126123. [Google Scholar] [CrossRef] - Cardoso, B.-H.F.; Gonçalves, S.; Iglesias, J.R. Wealth distribution models with regulations: Dynamics and equilibria. Physica A
**2020**, 551, 124201. [Google Scholar] [CrossRef] - de Oliveira, P.M.C. Rich or poor: Who should pay higher tax rates? Eur. Phys. Lett.
**2017**, 119, 40007. [Google Scholar] [CrossRef] - Fujiwara, Y.; Souma, W.; Aoyama, H.; Kaizoji, T.; Aoki, M. Growth and fluctuations of personal income. Physica A
**2003**, 321, 598. [Google Scholar] [CrossRef]

**Figure 1.**The cumulative probability distribution of net wealth in the US (

**left**, 1997) and UK (

**right**, 1996) shown in log–log scales. Points represent data from the IRS/HMRC, and solid lines are the fitted lines to the exponential (Boltzmann–Gibbs) and power-law (Pareto) [1].

**Figure 2.**Probability function, Equation (11).

**Figure 3.**Raw model with taxation on wealth: $\gamma ={10}^{-3}$ and $\tau =0.4$. In the figure on the left, we can see the average wealth held by the 90 and 99 quantiles, i.e., the $10\%$ and $1\%$ richest agents, respectively, compared with the standard deviation. On the right, the fraction of wealth held by the $10\%$ and $1\%$ richest agents is shown. The time evolution of the Gini index is also shown, stabilizing slightly above $0.3$.

**Figure 4.**Statistics for the wealth–connection linked model and taxation on wealth: $\gamma ={10}^{-3}$ and $\tau =0.4$. In the figure on the left, we can see the average wealth held by the 90 and 99 quantiles, i.e., the $10\%$ and $1\%$ richest agents, respectively, compared with the standard deviation. Note that these values are larger than in the raw case, Figure 5. On the right, the fraction of wealth held by the $10\%$ and $1\%$ richest agents is shown. The increase in wealth concentration is evident. Consequently, the Gini index also increases. The time evolution of the Gini index is also shown, stabilizing just below $0.4$.

**Figure 5.**Evolution of the distributions for the model that favors the rich: $\gamma ={10}^{-4}$ and $\tau =0.4$. At each stage, the figure on the left is the distribution of income, and the figure on the right is the distribution of the number of connections. The orange line is the poverty rate.

**Figure 6.**Stage 23 of Figure 5. Pareto tail (dotted red line) is clear, with Pareto exponent $\alpha =5.63$. $\gamma ={10}^{-4}$ and $\tau =0.4$.

**Figure 7.**Evolution of distributions for the model that favors the rich: $\gamma ={10}^{-3}$ and $\tau =0.4$. The orange line is the poverty rate.

**Figure 8.**Evolution of total tax revenue and total taxed agents for different values of $\gamma $ (tax growth rate according to wealth).

**Figure 9.**Evolution of the top $10\%$ of agents for different values of $\gamma $ (tax growth rate according to wealth).

**Figure 10.**Evolution of the top $1\%$ of agents for different values of $\gamma $ (tax growth rate according to wealth).

**Figure 11.**Evolution of the standard deviation ($\sigma $) and the Gini coefficient for different values of $\gamma $ (tax growth rate according to wealth).

**Figure 12.**Evolution of probability distributions for the model with capital gain taxation: $\gamma =0.1$ and $\tau =0.4$. The orange line is the poverty rate.

**Figure 14.**Stage 41 of Figure 12, scenario of annual income taxation. A second Pareto tail appears, with $\alpha =1.06$ (dotted red line). $\gamma =1/10$ and $\tau =0.4$.

**Figure 15.**Evolution of total tax revenue and total taxed agents for different values of $\tau $ (tax limit). $\gamma =0.1$.

**Figure 16.**Evolution of the top $10\%$ of agents for different values of $\tau $ (tax limit) and $\gamma =0.1$.

**Figure 17.**Evolution of the top $1\%$ of agents for different values of $\tau $ (tax limit) and $\gamma =0.1$.

**Figure 18.**Evolution of the standard deviation ($\sigma $) and the Gini coefficient for different values of $\tau $ (tax limit) and for $\gamma =0.1$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Calvelli, M.; Curado, E.M.F.
A Wealth Distribution Agent Model Based on a Few Universal Assumptions. *Entropy* **2023**, *25*, 1236.
https://doi.org/10.3390/e25081236

**AMA Style**

Calvelli M, Curado EMF.
A Wealth Distribution Agent Model Based on a Few Universal Assumptions. *Entropy*. 2023; 25(8):1236.
https://doi.org/10.3390/e25081236

**Chicago/Turabian Style**

Calvelli, Matheus, and Evaldo M. F. Curado.
2023. "A Wealth Distribution Agent Model Based on a Few Universal Assumptions" *Entropy* 25, no. 8: 1236.
https://doi.org/10.3390/e25081236