  We develop a mean-field theory of the growth, exchange and distribution (GED)
model introduced by Kang et al. (preceding paper) 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 was 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, critical slowing down, 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 $N \to \infty$. 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.
