Explicit equilibria in bilinear kinetic models for socio-economic interactions

Bilinear kinetic models of Maxwell-Boltzmann type are often used to model socio-economic systems composed by agents that undergo binary interactions, which in general obey to some conservation law. Then, the details of the microscopic interaction are such that an equilibrium solution emerges. At difference with the classical Boltzmann equation for elastic gas particles, the analytic form of the equilibria is known only in some particular case. In the present note, we present and discuss the situations in which this equilibrium profile can be explicitly found. Applications to dissipative gases and to wealth redistribution models are presented.