Reallocating geometric Brownian motion
Geometric Brownian Motion is a simple random multiplicative growth process, used widely in finance. It is non-ergodic, in that the time-average growth rate of a single trajectory differs from the growth rate of the ensemble average. A finite sample of such motions is a toy model of a human economy, in which individual wealths evolve without interacting. Only recently have the statistical properties of the sample mean been understood, in the context of the Random Energy Model (Derrida 1980). In particular, a short-time regime exists where the sample is self-averaging. We add reallocation to this model. Individuals contribute proportionally to their wealth and receive an equal share of the amount collected. Sufficiently fast reallocation extends the self-averaging regime indefinitely. This maximises the time-average growth rate of the sample mean and yields a stationary wealth distribution. Slower reallocation results in a sample dominated by large deviations. Negative reallocation is qualitatively different: the sample splits into positive and negative wealths, whose distribution never equilibrates. Finally, we fit the model parameters to historical wealth inequality in the United States. We find that the effective reallocation rate is currently negative, i.e. from poorer to richer. Mainstream economic analyses miss such findings by assuming equilibrium.