Continuum and thermodynamic limits for a wealth-distribution model

We discuss a simple random exchange model for the distribution of wealth. There are N agents, each one endowed with a fraction of the total wealth; indebtedness is not possible, so wealth fractions are positive random variables. At each step, two agents are randomly selected, their wealths are first merged and then randomly split into two parts. We start from a discrete state space, discrete time version of this model and, under suitable scaling, we present its functional convergence to a continuous space, discrete time model. Then, we discuss how a continuous time version of the one-point marginal Markov chain functionally converges to a kinetic equation of Boltzmann type. Solutions to this equation are presented and they coincide with the appropriate limits of the invariant measure for the marginal Markov chain. In this way, in this simple case, we complete Boltzmann’s programme of deriving kinetic equations from random dynamics.