Simple wealth distribution model causing inequality-induced crisis without external shocks

We address the issue of the dynamics of wealth accumulation and economic crisis triggered by extreme inequality, attempting to stick to most possibly intrinsic assumptions. Our general framework is that of pure or modified multiplicative processes, basically geometric Brownian motions. In contrast with the usual approach of injecting into such stochastic agent models either specific, idiosyncratic internal nonlinear interaction patterns or macroscopic disruptive features, we propose a dynamic inequality model where the attainment of a sizable fraction of the total wealth by very few agents induces a crisis regime with strong intermittency, the explicit coupling between the richest and the rest being a mere normalization mechanism, hence with minimal extrinsic assumptions. The model thus harnesses the recognized lack of ergodicity of geometric Brownian motions. It also provides a statistical intuition to the consequences of Thomas Piketty's recent “$r>g$” (return rate $>$ growth rate) paradigmatic analysis of very-long-term wealth trends. We suggest that the “water-divide” of wealth flow may define effective classes, making an objective entry point to calibrate the model. Consistently, we check that a tax mechanism associated to a few percent relative bias on elementary daily transactions is able to slow or stop the build-up of large wealth. When extreme fluctuations are tamed down to a stationary regime with sizable but steadier inequalities, it should still offer opportunities to study the dynamics of crisis and the inner effective classes induced through external or internal factors.

Figure 1

(a) Gaussian distribution of $w-w_p$ at increasing times as indicated (log-scale), with $\beta =0.06$ and initial distribution concentrated at $w_1$; (b) wealth distribution $P\left(w\right)$ on a linear scale.

Figure 2

Log-log plot of $N$/$k$-iles $x_{N/k}\left(t\right)$ (dashed lines) as a function of time for $N=3600,\beta =0.06$, as defined by Eq. (9), i.e., the point in the Gaussian distribution such that the partial probability above $x_{N/k}\left(t\right)$ is $(k/N)$. The curve $k=1$ is superimposed as a solid dark magenta line. The largest of $N$ wealth lies around and above $x_N\left(t\right)$, plotted with an added solid line. The largest wealth from several simulations are shown at selected time points (green dots), the fraction of points above $x_{N/k}\left(t\right)$ at a given time $t$ showing the expected rarefaction trend with decreasing $k$.

Figure 3

Average wealth of eight simulations (solid lines of different colors) with the same parameters $N=3600,\beta =0.06$. The initial Brownian-like motion around $w_1$ (level indicated by the right-hand side dashed line) is gradually suffering larger and larger fluctuation, essentially associated to the large $x_N\left(t\right)$ at intermediate times ($t\sim 15000–30000$). At larger times, the drift eventually dominates and the average wealth is stuck to the poverty level $w_p$ (indicated by the left-hand side dashed line).

Figure 4

Same log-log plot of maximum wealth vs. time as Fig. 2, in the case of reset average, i.e., forced normalization of the total wealth to $N{w}_1$. After reaching the point of maximum wealth expectation without reset, the permanent regime retains a fluctuation pattern similar to that occurring around the maximum.

Figure 5

Illustration of a typical sample of wealth $w_N\left(t\right)$ under the reset-to-average assumption. (a) Histogram of wealth using a few hundred log-spaced bins, showing intermittency at multiple time scales; (b) the sorted trajectories of $w_j-w_p$ for selected $w_j$'s. Note the correlation among the majority of small wealth, but note that there is rather an anticorrelation among this vast majority and the top three wealths; (c) Gini coefficient of the distribution vs. time, with large random fluctuations that still bear the signatures of intermittency of the few wealthiest agents.

Figure 6

Correlation analysis on ranked wealth. See text and Eq. (14) on the particular correlation-based indicator $C_{i,j}$ used here. We represent as a color a quantity measuring the fluxes and their signs, between the affluent and the poorest of the agents, with a stepwise logarithmic sampling of the $3600\times 3600$ matrix. Note that the anticorrelation is neatly defined, and that it clearly stems in this graph from gains of the few richest. The inset on the bottom left is a zoom on the $\sim 40$ wealthiest agents.

Figure 7

Fate of the richest wealth under different assumptions for the skew parameter $\varepsilon$. The analytical curves provided for the simple case of no skew and no reset in Sec. 2 are left as a guide to the eye.

Figure 8

Gini coefficient under different assumptions for the skew parameter $\varepsilon$ as indicated.

Figure 9

(a–e) For a simulation with a small negative skew parameter $\varepsilon =-0.005$, the evolution is depicted: (a) the histogram of wealth distributions as a color map coded as (d, e); (b) selected sorted wealth from highest to lowest, showing the few occurrences of anticorrelation at times $36000$ and $45000$; (c) the Gini coefficient, with spikes at those moments; (d) and shows the histogram for a case of large positive skew parameter $+0.03$, with extreme intermittency; (e) shows the case of a large negative skew parameter $-0.03$, with a very steady situation.

Figure 10

Map of correlation of wealth variations, $C_{i,j}$, among sorted agent records, as in Fig. 6, for a negative skew coefficient -0.015. The transition from positive to negative fluxes is smooth and lies around $i,j\sim 300$; it is smooth and has no strong values, except close to the diagonal (artifact of correlation when ranked agents “cross”).

Figure 11

(a) Histograms of ($w-w_p$) on log-log scale, for various skew parameters as indicated at a large enough time to escape the initial phase. Note the spikes on the left side of the distributions down to parameter $-0.005$, that are created at the “bust” events of wealthiest agents. Spikes at the right end are the few wealthiest agents; (b) Eigenmodes associated to the largest (unity) eigenvalue for the continuous model without the normalization effect, for negative values of the skew parameter. For too weak values, there is no solution within the simulation space, the distribution is at the verge of unstable negative drift, and hence the eigenmodes explore boundaries.