Wealth Rheology

Abstract

We study wealth rank correlations in a simple model of macroeconomy. To quantify rank correlations between wealth rankings at different times, we use Kendall's τ and Spearman's ρ, Goodman-Kruskal's γ, and the lists' overlap ratio. We show that the dynamics of wealth flow and the speed of reshuffling in the ranking list depend on parameters of the model controlling the wealth exchange rate and the wealth growth volatility. As an example of the rheology of wealth in real data, we analyze the lists of the richest people in Poland, Germany, the USA and the world.

Keywords: Bouchaud–Mézard model; Gini coefficient; rank correlations; wealth distribution; wealth inequality.

Figure 1

(a) Gini coefficient G (8) plotted as a function of $\alpha$ (solid line) and computed numerically from samples generated in Monte Carlo simulations for $N={10}^4$ (symbols). Different symbols correspond to $\sigma =0.02$, $0.04$ and $0.08$. (b) The auto-correlation time (1) ${\tau }_{\mathrm{ac}}$ and the exponential time (2) ${\tau }_{\exp }$ for the Gini coefficient G measured for consecutive configurations in the stationary state for $\alpha =2$ for different $\sigma$. When $\sigma$ decreases ${\tau }_{\mathrm{ac}}$ grows as ${\sigma }^{-x}$ with $x=2.849\left(50\right)$, and ${\tau }_{\exp }$ grows as ${\sigma }^{-y}$ with $y=1.617\left(15\right)$. (c) Evolution of the Gini coefficient G from the ‘cold’ start, $G=0$, towards the stationary state’s value $G=0.5$ for $\alpha =2.0$. The plots correspond to $\sigma =0.02$, $0.04$ and $0.08$. Please note logarithmic scale on the time axis. (d) Evolution of the Gini coefficient G from the ‘hot’ start. The values fluctuate about the stationary state value $G=0.5$ for $\alpha =2.0$. The plots correspond to $\sigma =0.02$, $0.04$ and $0.08$.

Figure 2

Rank correlations coefficients (a) $\tau$ and (b) $\rho$ for steady state configurations separated by k time steps in the simulated systems for $N={10}^4$ and for various values of $\alpha =2$, 3, 4 and for various values of $\sigma =0.02$, $0.04$ and $0.08$. Please note logarithmic scale on the time axis. The first parameter in the legend is the value of $\alpha$ and the second is the value of $\sigma$.

Figure 3

Dependence of the overlap of top 100 lists at times $k_1$ and $k_2$ on the separation time $k=k_2-k_1$. The overlap is measured as the percentage of people that are on both the lists. The data points are obtained by averaging over pairs of $k_1$, $k_2$ such that $k_2-k_1=k$. They are plotted against the universal argument $x=k{\sigma }^2(\alpha -1)$. The data is obtained by simulations of the model for $N={10}^4$, and for different combinations of $\alpha =2$, 3, 4 and $\sigma =0.02$, $0.04$, $0.08$. The first parameter in the legend is the value of $\alpha$ and the second is the value of $\sigma$. The data is fitted with the Formula (12) with $A=0.7570\left(26\right)$ and $B=0.3341\left(30\right)$. The fit is shown with a solid line.

Figure 4

Time evolution of various rank correlations $\mathsf{\Gamma }$ for top 100 richest people in (a) Poland, data from [11], (b) Germany, data from [12], (c) the USA, data from [13] and (d) the world, data from [13].

Figure 5

Rank coefficients for the richest people in Germany, Poland, the US and the world based on real-world data published in Refs. [11,12,13]. (a) Overlap ratios ${\overline{\mathsf{\Omega }}}_{100}\left(t\right)$ for the four systems. The best fit of the Formula (12), with $A=0.1536\left(42\right)$ and $B=0.0425\left(14\right)$, to the world data is shown with a solid line. (b) Goodman–Kruskal’s $\gamma$ coefficients for the four system.