Empirical verification of Zipf's law using the quantile regression method for Brazilian cities

Abstract

This study examines the empirical validity of Zipf's Law in the distribution of Brazilian city sizes between 1950 and 2001, using census data and quantile regression techniques, including simultaneous quantile regression. Zipf's Law posits an inverse proportional relationship between city size and its rank, implying a power-law distribution with a unitary exponent. The results indicate that, for most
of the period—particularly from 1950 to 1980—the estimated Pareto coefficients fall below the theoretical benchmark, reflecting a lack of full conformity with the law, as well as pronounced urban primacy and substantial regional asymmetries.
However, the 1991 and 2001 censuses show stronger convergence in the central quantiles, with coefficients approaching the unitary exponent, suggesting a recent movement toward Zipfian regularity. These findings point to a shift toward greater relative balance within the Brazilian urban system, associated with the structural economic and territorial transformations that occurred during the 1990s.