City size distributions are driven by each generation's stay-vs-leave decision

Abstract

Throughout history most young adults have chosen to live where their parents did while a smaller number moved away. This is sufficient, by proof and simulation, to account for the well-known power law distributions of city sizes. The model needs only two parameters, $r$ = the probability that a child stays, and the maximum number of cities (which models the observed saturation at high city rank). The power law exponent follows directly as $\alpha = 1 + 1/r$, with Zipf's Law simply the limiting case as $r \rightarrow 1$. Observed exponents $(\alpha = 2.2 \pm 0.4, n = 158)$ are consistent with stay-or-leave data from large genealogic studies. This model is self-initializing and could have applied from the time of the earliest stable settlements. The driving narrative behind city-size distributions is fundamentally about family ties, familiarity, and risk-avoidance, rather than economic optimization.