Is the World Economy Zipf-Distributed? Exact Pareto Tail Estimates for National GDP and Population, 1990–2024, With the Rank-Minus-One-Half Correction and an Exact Chi-Square Interval

Zipf's law says that the second-largest unit in a size distribution is half the largest, the third a third, and so on, which is a Pareto tail with exponent α = 1. I test it for the sizes of national economies (GDP in current US dollars) and national populations, using World Bank data for 1990, 2000, 2010 and 2024 (vintage 8 October 2026). I prove that, when the cut-off is the smallest observation in the tail, the m larger values give an exactly unbiased estimator (m − 1)/S of α, with S the sum of log excesses, and an exact interval from 2αS ~ χ² with 2m degrees of freedom. A simulation confirms the unbiasedness (mean 0.9989 for α = 1, n = 50) and the coverage (0.9497 for a nominal 0.95). The rank-minus-one-half regression of Gabaix and Ibragimov removes most of the bias of the usual log-rank regression (mean 1.0104 against 0.9228) but is less precise than the Hill estimator. For GDP the tail chosen by the Clauset–Shalizi–Newman method covers the 44 to 62 largest economies, and the exponent is 0.628 in 1990 (95% interval 0.488–0.808), 0.763 in 2000, 0.760 in 2010 and 0.807 in 2024 (0.616–1.059). The tail of national GDP has become thinner but remains heavier than Zipf. For population the exponent rose from 0.771 in 1990 to 1.060 in 2024 (0.798–1.409), consistent with Zipf. India moved from 12th to 5th place by GDP between 1990 and 2024.