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

TL;DR

This paper presents a simple generational model explaining city size distributions as a result of stay-or-leave decisions, linking family ties and risk-avoidance to the power law behavior observed in urban hierarchies.

Contribution

It introduces a minimal model with two parameters that accounts for city size distributions and connects them to genealogical stay-or-leave data, emphasizing social factors over economic optimization.

Findings

Power law exponent b1 2.2 matches genealogical data.

Model explains city size distributions with only two parameters.

Zipf's Law emerges as a limiting case when stay probability approaches one.

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.