Mathematical Sciences Department Financial Math Seminar
Fei Cao, UMass Amherst
Monday, March 31st
2:00pm - 3:00 pm
Title: Derivation of wealth distributions from biased exchange of money --- Models and results
Abstract: In this talk, we will illustrate the use of kinetic theory to better understand the time evolution of wealth distribution and their large-scale behavior such as the limiting money distribution and the evolution of wealth inequality (e.g. Gini index). We investigate three types of dynamics denoted unbiased, poor-biased and rich-biased exchange models. At the individual level, one agent is picked randomly based on its wealth and one of its dollars is redistributed among the population. Proving the so-called propagation of chaos, we identify the limit of each dynamics as the number of individual approaches infinity using both coupling techniques and martingale-based approaches. Equipped with the limit equation, we identify and prove the convergence to specific equilibrium for both the unbiased and poor-biased dynamics. In the rich-biased dynamics however, we observe a more complex behavior where a dispersive wave emerges. Although the dispersive wave is vanishing in time, it's also accumulates all the wealth leading to a Gini approaching 1 (its maximum value). We characterize numerically the behavior of dispersive waves but further analytic investigation is needed to derive such dispersive waves directly from the dynamics. If time allows, we also talk about the possibility of allowing agents to go into debt where a central bank is introduced.