Equilibrium Models of Cities: 2. Single-Species Cities

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Abstract
This second part of a study of a city as an ‘urban gravitational plasma’ investigates in detail the case where the city consists of only one species of civic matter, and is circularly symmetric. To increase the relevance of the theory to actual urban situations, this civic matter is assumed throughout to be a citizen population, though the theory would apply just as well if other illustrations, such as floor space or traffic flows, etc., were to be chosen instead. The population is assumed to attract itself in a way which tends to increase its density in high density regions and to decrease it in low density regions. This ‘clumping’ effect is offset by another inducement on the population to relocate itself in places where some ‘dissatisfaction potential’ is less. Again, for illustration, it is assumed throughout that the dissatisfaction has the form of a housing rental, that is, the price of the composite bundle of ‘housing’ commodities and utilities. It is shown that the competition between the two civic forces of attraction and dispersal can lead to equilibrium distributions of the population in which the forces are everywhere in balance. The forms of these distributions depend greatly on the extent to which the housing rental is proportional to the local population density. Different degrees of this dependence are shown to give rise to many different forms of the equilibrium configurations available to a city. These are classified according to a regular scheme, and their properties explored and illustrated in detail. The manner in which one equilibrium configuration may grow into another, with or without any change in the total population in the city, leads to the idea of an ‘equilibrium growth’ in a city. Again their different possible types are examined in detail. Finally, certain classes of the equilibrium configurations are shown to resemble closely the familiar negative exponential and gaussian distributions of population density. The resemblance can be so close as to make it extremely likely that many actual cities, that have been shown elsewhere to exhibit population density distributions of those forms, may in fact be exhibiting equilibrium distributions of the kind deduced in this study.

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