Expanding the expanding equilibrium algorithm for spatial price equilibria: Multiple commodities, nonlinearities, and congestion.

摘要

This dissertation concerns spatial price equilibrium models: models that use the equilibrium conditions of a multiple region economy to describe and predict commodity prices and trade quantities. The network-based expanding equilibrium algorithm is one technique available for solving the single commodity spatial price equilibrium problem with linear regional excess demand functions. We review the algorithm and implement it on the IBM-compatible microcomputer. We demonstrate its ability to solve large-scale problems more efficiently than two alternative algorithms proposed for this problem.; We discuss ways to modify the expanding equilibrium algorithm so that linear, multiple commodity problems and nonlinear, single commodity problems may be solved. We present alternative diagonalization schemes that solve the multiple commodity case via a sequence of single commodity subproblems. We recommend the use of our partially-iterative scheme; network characteristics of the problem are exploited so that direct solutions can be provided for problems of very large size. For the nonlinear case, we present two strategies for integrating Newton's method into the expanding equilibrium algorithm. The first strategy is an outer approach in which the nonlinear problem is solved via an approximating sequence of linearized problems. The second is an inner approach where Newton's method is embedded directly into the ratio test of the algorithm. We recommend the use of our outer-Newton strategy, which we show to be relatively invariant to the form of the nonlinear excess demand functions used.; Setting aside our preoccupation with solution techniques, we develop three versions of a spatial price equilibrium model that includes competitive markets for transportation services. These models represent the cases where (1) a local market for transportation services exists between each pair of regions, (2) there is a single market for transportation services, global to the entire carrier network, and (3) some combination of the two previous market structures is present. We demonstrate the mathematical equivalence of these models to spatial price equilibrium models including the phenomena of congestion. We close with a conjecture regarding the application of the expanding equilibrium algorithm to the transportation market/congested spatial price equilibrium problem.