Testing the Zipf's law under heterogeneous urban-rural hierarchies with spatial quantile regressions

Despite the widespread recognition of Zipf's law in urban economics, empirical evidence reveals deviations from the strict adherence to its postulated parameters. In our contribution to this ongoing debate, we applied Zipf's law to population density of a metropolitan hierarchy in Greece (1033 municipalities) over a substantial timeframe (1961-2021). To overcome the specific limitations of global Ordinary Least Squares (OLS) models, cross-section spatial quantile regressions had been performed. The outcomes unveiled heterogeneous rank-size relationships for urban municipalities corresponding with the first and the second quartile of density distribution. These relationships became more homogeneous in rural municipalities (third and fourth quartiles). Consistent with the predictions of Zipf's law, slope coefficients exhibited a notable increase over time, approaching 1 for mixed urban-rural municipalities. Conversely, they remained slightly below 1 for strictly urban municipalities and slightly above 1 for strictly rural ones. Space was found to play a significant role during late urbanization (1980s-1990s), suggesting an impact of agglomeration economies on metropolitan hierarchy. Suburbanization exerted a reverse effect resulting in more heterogeneous hierarchies since the late-1990s. Our findings underscore the intricate nexus between deviations from Zipfian patterns in urban-rural population distribution and the spatially explicit dynamics of city life cycles.