# A Simple Mechanism Causing Wealth Concentration

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## Abstract

**:**

## 1. Introduction

#### 1.1. Some Stylised Facts Related to Empirical Wealth Distribution

#### 1.2. The General Structure of Kinetic Exchange Models

## 2. Model

- (i)
- agents are equal in the sense that each of them has the same access to the market and the same knowledge about it.
- (ii)
- agent trade only when it is profitable from their perspective.

- The first agent i is chosen randomly with the probability equal to its wealth ${a}_{i}$.
- The second agent j is chosen randomly with the probability equal to its wealth ${a}_{j}$.
- If $i=j$ or agent i has traded with agent j in this cycle, go to point 1. Otherwise, make the trade.

## 3. Results and Discussion

#### 3.1. Wealth Condensation

- (a)
- delta distribution—all agents started with the same amount of money;
- (b)
- uniform distribution—the initial wealth of each agent was uniformly distributed on the interval $[0,2/N)$;
- (c)
- exponential distribution—the initial wealth was drawn according to the exponential distribution of the unit mean and variance;
- (d)
- Gaussian distribution—the initial wealth of each agent was an absolute value of a number drawn according to the normal distribution of the zero mean value and unit variance;
- (e)
- Cauchy distribution—the initial wealth of each agent was an absolute value of a number drawn according to the following probability distribution function$$p\left(x\right)=\frac{1}{\pi ({x}^{2}+1)};$$
- (f)
- $1\%$ of richest agents possessed 100 times more money than the remaining $99\%$ of poorer agents.

- (i)
- linear preferences utility function: $g(i,j)=r\frac{{a}_{i}+{a}_{j}}{2}$—gains from individual transaction depend on assets of both sides of trade process.
- (ii)
- Cobb–Douglas utility function: $g(i,j)=r\sqrt{{a}_{i}{a}_{j}}$—similar as in the above case but gains were much lower when agents assets differed significantly.
- (iii)
- Koopmans and Leontieff utility function: $g(i,j)=r\mathrm{min}({a}_{i},{a}_{j})$—gains are determined by a poorer trader.

#### 3.2. Income and Wealth Tax Influence on the Model

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**The empirical distribution of wealth in the U.S. based on Census Data. The blue line represents one of fitted generalised gamma distribution of the second kind.

**Figure 2.**Histograms of agents assets after 0, 10, 100, and 1000 cycles. Different plots correspond to the different initial distribution of wealth among the agents: ${a}_{i}$ is (

**a**) equal to $1/N$, (

**b**) uniformly distributed in the interval [0, 2/N), (

**c**) exponentially distributed, (

**d**) normally distributed, (

**e**) Cauchy distributed, (

**f**) 1% of richest agents possess 100 times more than the rest 99% of agents.

**Figure 3.**Gini index evolution for all studied agents initial wealth distributions. Inset shows the evolution of the maximal and median wealth in the population of agents. Black line corresponds to equal initial wealth of all agents, red—the uniform distribution of wealth on the interval $[0,2/N)$, blue—the exponential wealth distribution, brown—Gaussian distribution, violet—Cauchy distribution, green—$1\%$ of richest agents have 100 bigger assets than the rest $99\%$ of the population.

**Figure 4.**Histograms of agents assets after 1000 cycles for different size of the population. Because the total wealth changes with population size, we rescaled agents wealth by multiplying it by the number of agents N.

**Figure 5.**Gini index evolution for different payoff function used in the model. The black line corresponds to constant payoffs while the other ones correspond to payoffs depending on trading agents assets: red—payoff proportional to average assets of trading agents, blue—payoff proportional to geometrical mean of trading agents assets, brown—payoff proportional to poorer agent asset. The parameter $r=0.01$ and population size $N={10}^{5}$.

**Figure 6.**Histograms of agents assets after 0, 10, 100, and 1000 cycles. In both plots, the wealth of agents was initially equal (${a}_{i}={10}^{-5}$). The left plot corresponds to income tax (${t}_{I}=0.1$ and ${t}_{W}=0$), and the right one corresponds to wealth tax (${t}_{I}=0$ and ${t}_{W}=0.01$).

**Figure 7.**Gini index evolution for pure wealth (solid black line) and income (dashed red line) tax applied to the model. Initially, the wealth was distributed equally among agents (${a}_{i}={10}^{-5}$). Inset shows the evolution of the maximal and median wealth in the population of agents.

**Figure 8.**Histograms of agents assets after 0, 10, 100, and 1000 cycles. In both plots the initial wealth of agents was taken from the final state of simulation (1000 cycles) presented in a Figure 2. The left plot corresponds to income tax (${t}_{I}=0.1$ and ${t}_{W}=0$), and the right one corresponds to wealth tax (${t}_{I}=0$ and ${t}_{W}=0.01$).

**Figure 9.**Gini index evolution for pure wealth (solid black line) and income (dashed red line) tax applied to the model. The initial wealth of agents was taken from the final state of simulation (1000 cycles) presented in Figure 2a. Inset shows the evolution of the maximal and median wealth in the population of agents.

**Figure 10.**Gini index after 1000 cycles for wealth (solid black line) and income (dashed red line) tax based redistribution. The initial wealth of agents was taken from the final state of simulation (1000 cycles) presented in Figure 2. Inset shows final maximal and median wealth in the population of agents.

Distribution | 2010 | 2018 | ||||
---|---|---|---|---|---|---|

Gini | Aic | Bic | Gini | Aic | Bic | |

Generalised Beta of the Second Kind | 0.4504 | 825,368.9827 | 825,407.7613 | 0.4656 | 914,590.8355 | 914,629.8928 |

Generalised Gamma | 0.4485 | 825,598.8432 | 825,627.9271 | 0.4526 | 915,341.0957 | 915,370.3886 |

Beta of the Second Kind | 0.4545 | 825,501.8130 | 825,530.8969 | 0.4636 | 915,233.3365 | 915,262.6295 |

Dagum | 1,258,915.0143 | 1,258,944.0982 | 0.4693 | 914,642.1348 | 914,671.4277 | |

Singmad | 0.4531 | 827,239.3961 | 827,268.4800 | 0.4600 | 914,833.1633 | 914,862.4562 |

Lognormal | 0.5013 | 832,408.2444 | 832,427.6337 | 0.5206 | 924,094.3485 | 924,113.8772 |

Weibull | 0.4432 | 827,065.1604 | 827,084.5496 | 0.4462 | 916,179.0877 | 916,198.6163 |

Gamma | 0.4409 | 826,112.8345 | 826,132.2238 | 0.4467 | 915,559.4152 | 915,578.9439 |

Doubly lognormal | 1,375,275.6949 | 1,375,295.0841 | 920,090.4281 | 920,109.9568 | ||

Pareto | 0.5047 | 832,191.8408 | 832,211.2301 | 0.5061 | 920,845.1144 | 920,864.6431 |

**Table 2.**Estimated distribution parameters: location $\mu $, scale $\sigma $, skewness $\nu $ and kurtosis $\tau $.

Distribution | 2010 | 2018 | ||||||
---|---|---|---|---|---|---|---|---|

$\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\nu}$ | $\mathit{\tau}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\nu}$ | $\mathit{\tau}$ | |

Generalised Beta of the Second Kind | 108,564.1708 | 1.7786 | 0.7034 | 2.0083 | 113,253.3847 | 2.1917 | 0.5323 | 1.2229 |

Generalised Gamma | 60,663.6500 | 0.9001 | 0.7612 | 81,899.1321 | 0.9057 | 0.8421 | ||

Beta of the Second Kind | 283,044.3149 | 1.5608 | 7.5992 | 372,137.5280 | 1.4877 | 7.2873 | ||

Dagum | 1,012,451,669.9591 | 0.9721 | 0.1021 | 105,486.1708 | 2.4436 | 0.4689 | ||

Singmad | 1,012,451,669.9591 | 1.1348 | 53,697.3940 | 190,335.2758 | 1.3413 | 3.3841 | ||

Lognormal | 10.6958 | 0.9900 | 10.9507 | 1.0373 | ||||

Weibull | 69,527.7650 | 1.1699 | 89,843.3170 | 1.1589 | ||||

Gamma | 6,5806.3882 | 0.8652 | 85,510.9515 | 0.8789 | ||||

Doubly lognormal | 1,012,451,669.9591 | 0.1514 | 60,235.5085 | 1.7064 | ||||

Pareto | 1,549,526.7678 | 24.2996 |

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Cieśla, M.; Snarska, M. A Simple Mechanism Causing Wealth Concentration. *Entropy* **2020**, *22*, 1148.
https://doi.org/10.3390/e22101148

**AMA Style**

Cieśla M, Snarska M. A Simple Mechanism Causing Wealth Concentration. *Entropy*. 2020; 22(10):1148.
https://doi.org/10.3390/e22101148

**Chicago/Turabian Style**

Cieśla, Michał, and Małgorzata Snarska. 2020. "A Simple Mechanism Causing Wealth Concentration" *Entropy* 22, no. 10: 1148.
https://doi.org/10.3390/e22101148