  A set of data with positive values follows a Pareto distribution if the
log-log plot of value versus rank is approximately a straight line. A Pareto
distribution satisfies Zipf's law if the log-log plot has a slope of -1. Since
many types of ranked data follow Zipf's law, it is considered a form of
universality. We propose a mathematical explanation for this phenomenon based
on Atlas models and first-order models, systems of positive continuous
semimartingales with parameters that depend only on rank. We show that the
stable distribution of an Atlas model will follow Zipf's law if and only if two
natural conditions, conservation and completeness, are satisfied. Since Atlas
models and first-order models can be constructed to approximate systems of
time-dependent rank-based data, our results can explain the universality of
Zipf's law for such systems. However, ranked data generated by other means may
follow non-Zipfian Pareto distributions. Hence, our results explain why Zipf's
law holds for word frequency, firm size, household wealth, and city size, while
it does not hold for earthquake magnitude, cumulative book sales, the intensity
of solar flares, and the intensity of wars, all of which follow non-Zipfian
Pareto distributions.
