Invariant-Distributional Regularities and the Markov Property in Urban Models: An Extension of Schinnar's Result

Abstract

This paper presents a formal derivation of the condition identified by Schinnar (1978) in which the prediction of nonbasic employment in models of the Garin–Lowry type can be independent of the spatial distribution of basic employment. The distributional invariance which determines this result is also a feature of the series-expansion form of model, which is characterised by the Markov property. It is shown that, in the limit, the invariant distribution is equivalent to the steady state of a finite Markov chain, and convergence to this limit is then traced by use of the spectral decomposition of the distribution. This leads to a new interpretation of the model based on spatially dependent and independent components, and statistics are suggested which measure the degree of invariance in any application. Finally these ideas are given substance by recomputing Schinnar's (1978) example in terms of its spectral decomposition, and this leads to several suggestions for future research which are drawn together in the conclusion.