Le Châtelier Principle and the Flow Sensitivity of Spatial Commodity Models

Abstract

One of the major tools of spatial interaction analyses is, needless to say, the spatial equilibrium model developed from (1) the linear programming transportation problem by Hitchcock (1941), Kantorovich (1942) and Koopmans (1949), and (2) an analogue of Kirchhoff’s law of electric circuits by Enke (1951). Samuelson (1952) showed that the Enke problem could be converted into a mathematical programming model. In 1964 Takayama and Judge reformulated it into an operationally efficient concave quadratic programming model. Since then, there has been a proliferation of theoretical advances that improve and extend the basic linear programming transportation and spatial equilibrium models. These include their transformation into a new algorithm by Liew and Shim (1978) and Nagurney (1986); Thore’s formulation with income (1982); sensitivity analyses of model performance by Yang and Labys (1981, 1982), Irwin and Yang (1982), Chao and Friesz (1984), Daffermos and Nagurney (1984), and Tobin (1984, 1987); computational evaluations by Meister, Chen and Heady (1978), and Nagurney (1987b); iterative solution methods by Pang and Chan (1982), and Irwin and Yang (1982); a linear complementarity formulation by Uri (1976) and Takayama and Uri; sensitivity analysis of the complementarity problem by Tobin (1984) and Yang and Labys (1985); a sufficient condition for the complementarity problem by Smith (1984); a flow dependent spatial equilibrium problem by Smith and Friesz (1985); nonlinear complementarity models by Irwin and Yang (1983) and Friesz, et al. (1983); variational inequality solutions by Pang and Chang (1982), Daffermos (1983), Harker (1984), Tobin (1986) and Nagurney (1987a); a path dependent spatial equilibrium model by Harker (1986); transhipment and location selection problem by Tobin and Friesz (1983, 1984); and evaluations of the spatial equilibrium models and Maxwell-Boltzmann entropy models by Yang (1990). For the detailed description of the advances in the spatial equilibrium models, readers are referred to Labys, Takayama and Uri (1989) and Labys and Yang (1991).