Convexity and duality properties of a quadratic intraregional location model

• Published in: Regional science and urban economics 1978-02, Vol.8 (1), p.5-19
• Main Author: Snickars, Folke
• Format: Article
• Language: English
• ISSN: 0166-0462; 1879-2308
• DOI: 10.1016/0166-0462(78)90009-1

This paper has two main purposes. The first one is to analyse the convexity and duality properties of a quadratic intraregional location model that has been developed for long-term indicative planning in the Stockholm region. The second one is to review the results of Koopmans and Beckmann (1957) about the inadequacy of a linear price system in sustaining an optimal assignment of plants to locations when the costs of transporting intermediary commodities are taken into consideration. At the outset a model is formulated which is a transposition of a continuous Koopmans-Beckmann model into the urban scene. It is shown that this quadratic programming model is non-convex in all practical cases of interest, due to the simple fact that transportation costs increase with distance. A modification of the model is proposed in which the centralising transportation cost criterion is traded of against a decentralising so called congestion cost which penalizes over-exploitation of urban space. It is shown that the modified model tends to be convex. In the light of these results Kuhn- Tucker theory is used to derive a set of conditions that will ensure that the optimal solution is stable relative to all potential moves by individual decision-makers. This result forms the basis for the conclusion that the failure of the price system in the Koopmans-Beckmann model is rather due to properties of the quadratic criterion function than the integral restrictions on the variables.