Abstract :
In this article concentration (i.e., inequality) aspects of
the functions of Zipf and of Lotka are studied. Since both
functions are power laws (i.e., they are mathematically
the same) it suffices to develop one concentration theory
for power laws and apply it twice for the different
interpretations of the laws of Zipf and Lotka. After a brief
repetition of the functional relationships between Zipf’s
law and Lotka’s law, we prove that Price’s law of concentration
is equivalent with Zipf’s law. A major part of
this article is devoted to the development of continuous
concentration theory, based on Lorenz curves. The
Lorenz curve for power functions is calculated and,
based on this, some important concentration measures
such as the ones of Gini, Theil, and the variation coefficient.
Using Lorenz curves, it is shown that the concentration
of a power law increases with its exponent and
this result is interpreted in terms of the functions of Zipf
and Lotka.
