Inequality measures in wealth exchange models based on saving propensity

Wealth inequality has gradually become a widely discussed and researched topic. Building upon existing researches related to saving propensity and kinetic models of wealth distribution, we propose the implicit saving propensity (ISP) model and the explicit saving propensity (ESP) model. We obtain the steady state wealth distributions as well as the Gini and $k$-indices of the ISP and ESP models through simulations. We find that the ISP model produces very limited types of wealth distribution shapes and yields relatively large Gini indices, whereas the ESP model, after incorporating saving behavior, is able to generate more diverse results, including producing power-law tails, and the Pareto exponent exhibits robustness for some specific parameters. The Gini index of the ESP model is also compared with empirical data from countries around the world and can effectively cover them. This indicates that, after incorporating saving behavior, the explanatory power and realistic value of our model have been significantly enhanced. Moreover, the mixture of Gamma distribution is used to fit the wealth distributions with different parameters, which shows the excellent agreement between simulation and theoretical results. Based on the mechanism of saving propensity, the ISP and ESP models are effective to reveal the wealth inequality qualitatively and quantitatively, which have flexible applicability in modeling the wealth dynamics in the real-world.