An Alternative Formulation for Spatial-Interaction Modeling

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Abstract
Elementary geometric assumptions are used to derive a spatial-interaction model in which places are related to each other through a set of simultaneous linear equations. The system has simple properties with respect to aggregation and turnover, yet incorporates spatial competition, adjacency, and effects of geographic shadowing. The objective function satisfied by the model reduces congestion and minimizes the per capita work; solves a quadratic transportation-problem and fulfils in-sum and out-sum constraints. The associated Lagrangians can be interpreted as pushes and pulls, or as shadow prices. A spatially continuous version of the model consists of coupled elliptical partial differential equations with Neumann boundary conditions, solvable by numerical methods. With migration data from the United States of America the model yields an amazingly good fit, better than existing models and with fewer free parameters. Inversion of the model yields an estimate of distances between regions. In-movement rates are predicted from out-movement rates in the model.

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