A theoretical relationship between the rank-size rule and Clark's law of urban population distribution: Duality in the rank-size rule

This paper theoretically shows the necessary and sufficient conditions for the robust rank-size rule: the rank-size rule that robustly holds for any choice of threshold urban population density by which populations of cities are determined. First, it is shown that when urban population distribution follows Clark's law, the robust rank-size rule holds if and only if the gradient parameter of the negative exponential distribution of the ith ranked city as √i times as large as that of the first ranked city and the population density at the center is the same for all cities. Second, when urban population distribution follows a certain general class of urban population distribution functions, the robust rank-size rule (of population) holds if and only if the boundary condition is satisfied and the rank-size rule holds with respect to urban areas. These two rank-size rules, the rank-size rule of population and the rank-size rule of urban areas, form the primal-dual relation. Third, if the robust rank-size rule holds, then the constant density rule holds, that is, the average population density of a city over its urban area is the same for all cities.