On the existence of immigration proof partition into countries in multidimensional space

Abstract

The existence of immigration proof partition for communities (countries) in a multidimensional space is studied. This is a Tiebout type equilibrium its existence previously was stated only in onedimensional setting. The migration stability means that the inhabitants of a frontier have no incentives to change jurisdiction (an inhabitant at every frontier point has equal costs for all possible adjoining jurisdictions). It means that inter-country boundary is represented by a continuous curve (surface). Provided that the population density is measurable two approaches are suggested: the first one applies an one-dimensional approximation, for which a fixed point (via Kakutani theorem) can be found after that passing to limits gives the result; the second one employs a new generalization of Krasnosel’skii fixed point theorem for polytopes. This approach develops [8] and extends the result to an arbitrary number of countries, arbitrary dimension, possibly continuous dependence on additional parameters and so on.