Equilibrium Models of Cities: 1. An Axiomatic Theory

Abstract

A study is undertaken of the concept of a city as an ‘urban gravitational plasma’ consisting of one or more species of civic matter (populations, activity rates, and so on) interacting on themselves and each other, and, at the same time, responding to relocation coercions induced by satisfaction potentials of various kinds (housing rentals, amenity levels, and so on). The latter are assumed to be coupled to the territorial densities of the individual species of civic matter through equations of state, for which the housing rental-population density relation in market equilibrium theory is a prototype. The study is divided into four parts. The first part (presented here) approaches the problem from a formal axiomatic viewpoint, and the axioms and definitions are discussed in relation to the real urban situations from which they are abstracted. The notion of equilibrium configurations for a city is introduced, and the general equilibrium equations necessary for their existence are developed. Three particular illustrations of these equations are offered: that of a single species city, and of a two species city—both with an ideal (polytropic) state equation—and that of a single species city with an imperfect (van der Waals) state equation. These illustrations will be examined in detail in the subsequent three parts of this study.

Experimental Analysis of Eleven Models of Urban Population Density

“…In Amson's (1972) theory the distribution of housing costs determines the distribution of population, so why should parks be left out when the surrounding built-up area (with its generally elevated housing costs) is left in? Equally, if March's (1970) and Bussiere and Snickars's (1970) derivations of models from assumptions about transport and locational costs are noted, perhaps the exclusion of areas that have an effect on these costs without the concomitant exclusion of the affected areas might involve the introduction of bias?…”

“…Over the years most of the attention has been devoted to the negative-exponential distribution; only the occasional paper has appeared concerning the others. The quadratic gamma and normal gamma distributions were only mentioned in passing by Amson (1972) as logical parents for the family (he called them both "quadratic gamma distributions"-the name 'normal gamma' is adopted here for reasons of clarity and symmetry).…”

“…Models proposed to relate distance from a city centre and local density can be divided into two main groups-the quadratic gamma family and the equilibrium models proposed by Amson (1972). Table 1 gives a chronology of recent model-building activities related to urban population densities.…”

Urban population density models

“…This model was first mentioned by Amson [14] and has the form D(r)=Aexp(-br 2 )r c (3.14) where A, b and c are constants. For c > 0, the maximum density occurs at r = yj(c/2b).…”

“…These models have been derived by Amson [14][15][16][17][18] in a series of papers by using two equations, a 'civic state' equation, relating population distribution and housing costs, and an 'equilibrium' equation.…”