The Yule—Simon Model in its Limiting Case as a Pure Migration Process

In this paper the author offers an analytical solution to a problem first raised in 1973, namely, what is the equilibrium distribution of city sizes implied by the Yule–Simon model, when the total population of the urban system in question is stationary? Under the assumption that in-migration and out-migration rates are uncorrelated with city sizes, it is shown that Fisher's log-series distribution is that distribution. Fisher's log-series distribution yields a much less concentrated size distribution than does the Yule distribution, which is the equilibrium distribution associated with a pure growth process. Thus, we might expect a lower level of concentration or city size inequality when the overall urban system is stationary than when it is growing.