Kinetic equations modelling wealth redistribution: A comparison of approaches

Kinetic equations modelling the redistribution of wealth in simple market economies is one of the major topics in the field of econophysics. We present a unifying approach to the qualitative study for a large variety of such models, which is based on a moment analysis in the related homogeneous Boltzmann equation, and on the use of suitable metrics for probability measures. In consequence, we are able to classify the most important feature of the steady wealth distribution, namely the fatness of the Pareto tail, and the dynamical stability of the latter in terms of the model parameters. Our results apply, e.g., to the market model with risky investments [S. Cordier, L. Pareschi, and G. Toscani, J. Stat. Phys. 120, 253 (2005)], and to the model with quenched saving propensities [A. Chatterjee, B. K. Chakrabarti, and S. S. Manna, Physica A 335, 155 (2004)]. Also, we present results from numerical experiments that confirm the theoretical predictions.

Figure 1

Regimes for the formation of Pareto tails.

Figure 2

CPT model: Decay of the Wasserstein distance to the steady state in zones II (left-hand side) and III (right-hand side).

Figure 3

CCM model: Decay of the averaged Wasserstein distance to the steady states for $\epsilon \equiv 1∕2$ and for $\epsilon ∊(0.4,0.6)$ uniformly distributed.

Figure 4

CCM model: Decay of the Wasserstein distance to the averaged steady state for $\epsilon \equiv 1∕2$ (left-hand side) and for $\epsilon ∊(0.4,0.6)$ uniformly distributed (right-hand side).

Figure 5

“Winner takes all” model: Evolution of the fraction of agents with zero wealth (left-hand side) and blow up (right-hand side).