Iteration and Sensitivity for a Spatial Equilibrium Problem with Linear Supply and Demand Functions

We investigate a set of spatial equilibrium conditions which apply, for example, to a multiregional, multicommodity market or to a physical system of interconnected sources and sinks. Under additional hypotheses, the equilibrium conditions are the Kuhn-Tucker conditions for a concave, quadratic programming problem. For the case in which the equilibrium conditions do not correspond to a programming problem, an iterative process for obtaining a solution is discussed. This iteration is motivated by the Project Independence Evaluation System (PIES) energy model algorithm. An algorithmic method to determine the sensitivity of an equilibrium solution vector to data perturbations is described. Both the sensitivity analysis and the convergence of the iterative solution method are based upon a solvability lemma, which guarantees that a solution of the equilibrium conditions may be expressed in terms of the problem data. Nondegeneracy assumptions for the inter-regional flows are unnecessary for this discussion. An example of the equilibrium problem which arose in modeling the supply and demand of eastern coal is briefly described.