Do US city sizes obey Zipf's law?

The claim, stated as a near-law in urban economics since Zipf (1949) and reconfirmed for the US by Gabaix (1999): a country’s city sizes follow a rank-size rule — the Nth-largest city has roughly 1/N the population of the largest. Regress log(rank) on log(population) and the slope should land close to −1. Tested here against every one of the 19,479 populated, legally incorporated US places the Census Bureau’s 2024 Population Estimates Program tracks — cities, towns, villages and boroughs, reconstructed from the Bureau’s own county-part rows so that places split across county lines (New York city is five rows; Chicago and Houston are two and four) are summed to a single whole-place total, not silently dropped. 4 places with a 2024 estimate of exactly zero — incorporated municipalities with no counted residents — are excluded from the log-log fit and listed on the page, not folded in as a guess. 19,475 places remain.

The exponent is not one number — it moves, smoothly and by a lot, depending how far down the list you fit. Restricted to the 100 biggest cities, the slope is -1.39, 95% CI [-1.41, -1.37] — visibly steeper than Zipf’s −1, and because rank within one fixed sample is a deterministic function of population (not iid noise around it), the plain-OLS interval understates the real uncertainty. A case-resampling bootstrap that redraws the full 19,475-place national list with replacement and refits at each cutoff gives a properly wide interval instead — [-1.77, -1.13] for the top 100 — and it still excludes −1. The top 1,000 tells the same story, tighter: -1.37, bootstrap CI [-1.48, -1.26]. America’s biggest cities are more concentrated at the very top than the textbook rule predicts.

The exponent only touches −1 in one narrow band, not a stable regime. Widening the cutoff to the top 4,300 places (population ≥ 6,286, Horace city, North Dakota, sits right at the boundary) lands the slope at -0.9997, 95% CI [-1.0050, -0.9945] — the one cutoff on this ladder whose interval contains −1, and the bootstrap agrees, [-1.0279, -0.9731]. It does not hold as a plateau: one step wider, at the top 5,000, the slope has already drifted to -0.9519 and the interval no longer reaches −1. Zipf’s exact exponent is a crossing point on a moving curve here, not a regime the data settles into.

Across the full national list the slope keeps flattening, to -0.48 (95% CI [-0.4871, -0.4818], bootstrap [-0.4893, -0.4795]) — roughly half the textbook value, and R²=0.870 even here, so the flattening is not the fit degrading, it is the exponent itself changing. The pull comes from sheer count: thousands of incorporated towns with a few hundred residents or fewer, down to Warm River city, Idaho (population 1) and Monowi village, Nebraska (population 2), sit at the bottom of a national list the same way NYC sits at the top — and there are vastly more of them than a single power law fit to the big end would predict.