Mean-field theory of an asset exchange model with economic growth and wealth distribution

We develop a mean-field theory of the growth, exchange, and distribution (GED) model introduced by Liu et al. [K. K. L. Liu et al., preceding paper, Phys. Rev. E 104, 014150 (2021)] that accurately describes the phase transition in the limit that the number of agents N approaches infinity. The GED model is a generalization of the yard-sale model in which the additional wealth added by economic growth is nonuniformly distributed to the agents according to their wealth in a way determined by the parameter lambda. The model is shown numerically to have a phase transition at lambda = 1 and be characterized by critical exponents and critical slowing down. Our mean-field treatment of the GED model correctly predicts the existence of the phase transition, a critical slowing down, and the values of the critical exponents and introduces an energy whose probability satisfies the Boltzmann distribution for lambda < 1, implying that the system is in thermodynamic equilibrium in the limit that N -> infinity. We show that the values of the critical exponents obtained by varying lambda for a fixed value of N do not satisfy the usual scaling laws, but do satisfy scaling if a combination of parameters, which we refer to as the Ginzburg parameter, is much greater than one and is held constant. We discuss possible implications of our results for understanding economic systems and the subtle nature of the mean-field limit in systems with both additive and multiplicative noise.