The analysis of city land area, population size and urban density explain a multidimensional, spatial core and periphery model of municipal and metropolitan districts. The two-dimensional relationships describe the log rank size rule, a combination of the log of population size of a central core city and place ranking among the top 100 cities in population size. A second, two-dimensional estimation of spatial elasticity, provides a description of municipal boundary expansion (or contraction) by land area. A third, two-dimensional model relates spatial elasticity to the compactness of land area by estimation of the number of townships contained within city land area. Forth, urban density is considered a single dimensional analysis, estimated by a constant even though urban density measurement is derived from the ratio of population size to land area. In summary, this planning and development model is by design multidimensional in measurement, specification and precision estimation of core area and population size. Given these basic two and three-dimensional results, the construction of a model of city land area, population size and urban density explains variation during city time series in boundary changes that are strongly significant, fixed city effects by location and any imperfect competition by local jurisdiction. In the absence of metropolitan districts, by either county amalgamation or city-county consolidation decisions, inter-local coordination sustains the evolving major cities.
Dr. Allen Brierly