Derivation of wealth distributions from biased exchange of money

Figure 1.  Left: Illustration of the $ 3 $ dynamics: at random time, one dollar is passed from a "giver" $ i $ to a "receiver" $ j $. Right: The rate of picking the "giver" $ i $ depends on the wealth $ S_i $

Figure 2.  Left: Distribution of wealth for the three dynamics after $ 50,000 $ steps. The distribution decays for the unbiased dynamics (pink) i.e. poor agents are more frequent than rich agents, whereas in the poor-biased dynamics, the distribution (blue) is centered at the average $ ＄ 10 $. For the rich-biased dynamics, almost all agents have zero dollars except a few with a large amount (more than $ ＄ 30 $). Right: evolution of the Gini index (5) for the three dynamics. The Gini index is lower for the poor-biased dynamics (less inequality) whereas it is approaching $ 1 $ for the rich-biased dynamics

Figure 3.  Schematic illustration of the strategy of proof: The approach of sending $ t\to \infty $ first and then taking $ N \to \infty $ is carried out in [39]. Our strategy is to perform the limit $ N \to \infty $ before investigating the time asymptotic $ t\to \infty $

Figure 4.  Summary of the limit ODE systems obtained in this manuscript. The exact form of the operator $ Q $ will be model-dependent

Figure 5.  Schematic illustration of the coupling strategy. We use an intermediate process $ (\widehat{S}_1, \dots, \widehat{S}_N) $ to decouple the "give" and "receive" parts of the dynamics

Figure 6.  The clocks $ \mathrm{N}_t^{(i, j)} $ used to generate the unbiased dynamics (11) have to be modified to generate the limit dynamics $ (\mkern 1.5mu\overline{\mkern-1.5mu{S}\mkern-1.5mu}\mkern 1.5mu_1(t), \ldots, \mkern 1.5mu\overline{\mkern-1.5mu{S}\mkern-1.5mu}\mkern 1.5mu_k(t)) $ (14). The processes $ \mkern 1.5mu\overline{\mkern-1.5mu{S}\mkern-1.5mu}\mkern 1.5mu_i(t) $ and $ \mkern 1.5mu\overline{\mkern-1.5mu{S}\mkern-1.5mu}\mkern 1.5mu_j(t) $ have to be independent, thus the clocks $ \mathrm{N}_t^{(i, j)} $ for $ 1\leq i, j \leq k $ cannot be used

Figure 7.  Left: comparison between the numerical solution $ {\bf p}(t) $ (36) of the poor-bias model and the equilibrium $ {\bf p}^* $ (48). The two distributions are indistinguishable. Right: decay of the difference $ \|{\bf p}(t)-{\bf p}^*\|_{\mathcal H^0} $ in semilog scale. The decay is exponential as predicted by the theorem 4.5

Figure 8.  Evolution of the wealth distribution $ {\bf p}(t) $ for the rich-biased dynamics (71). The distribution spreads in two parts: a large proportion starts to concentrate at zero ("poor distribution") and while the other part forms a dispersive traveling wave. Parameters: $ \Delta t = 5\cdot10^{-3} $, $ {\bf p}(t)\approx (p_0(t), p_1(t), \dots, p_{1,000}(t)) $. A standard Runge-Kutta of order $ 4 $ has been used to discretize the system

Figure 9.  Left: Estimation of the center $ c(t) $ and standard deviation $ \sigma(t) $ of the dispersive wave along with their parametric (power-law) estimation (91). Right: Comparison of the distribution $ {\bf p}(t) $ (see Figure 8) with the dispersive wave using the standard normal distribution $ \phi $

Figure 10.  Left: Evolution of the corresponding Gini index (93) along with the analytical approximation using the dispersive wave profile (94). Right The Gini index converges to $ 1 $ due to the vanishing dispersive wave transporting all the wealth to infinity