AN ALGORITHM FOR THE SPATIAL PRICE EQUILIBRIUM PROBLEM ON A GENERAL NETWORK IN THE SPACE OF THE PATH FLOWS*

ABSTRACT. The spatial price equilibrium on a general network may be formulated as a nonlinear-cost mathematical programming problem with simple constraints, when the decision variables are the path flows. The solution of this problem is difficult due to the very large number of variables (paths) and the impracticality of generating all the paths from all the origins to all the destinations. In this paper, we develop a Gauss-Seidel-Newton Projection algorithm and combine it with a restriction strategy. That makes it unnecessary to generate a priori all the paths. This algorithm may be further improved by exploiting the equivalence between the spatial price equilibrium on a general network and the network equilibrium. Computational results that we present in this paper demonstrate the efficiency of the proposed solution algorithms.     

THE CORE OF A SPATIAL PRICE EQUILIBRIUM GAME*

ABSTRACT This paper presents a critical appraisal of the use of the core as a solution concept for games involving spatially separated producers. Starting from the classical Samuelson/Takayama-Judge spatial price equilibrium model, the core of a game between the producers of commodities in this economy is defined, the conditions ensuring the nonemptiness of the core are stated, and the problems surrounding the definition and computation of the characteristic function are addressed. An application to the eastern United States’coal market is then presented in order to illustrate the usefulness of the theoretical and algorithmic results of this paper.     

COMPUTATIONAL COMPARISONS OF SPATIAL PRICE EQUILIBRIUM METHODS*

ABSTRACT. In this paper we propose new algorithms for the solution of both general and standard spatial price equilibrium problems, and test their performance with existing algorithms on randomly generated problems. For the standard problem, we propose decomposition schemes based on the concept of “equilibration operator” and compare their performance with the Frank-Wolfe method. For the general problem, we present alternative variational inequality formulations defined over Cartesian products of sets and then exploit these formulations to construct Gauss-Seidel-type serial decomposition methods. We then compare their performance with the projection method. Our computational tests suggest that the new schemes are substantially more efficient than earlier ones.     

SENSITIVITY ANALYSIS FOR GENERAL SPATIAL PRICE EQUILIBRIA

ABSTRACT. General spatial price equilibrium models are formulated as variational inequalities. Sensitivity analysis results recently developed for variational inequalities are reviewed which give conditions for existence and equations for calculating the derivatives of solution variables with respect to perturbation parameters. These results are applied to the variational inequality formulations of general spatial price equilibria, and derivatives of supplies, demands, flows, and prices with respect to perturbations of supply functions, demand functions, and transportation-cost functions are calculated.     

GENERAL EQUILIBRIUM OF SPATIAL PRODUCT AND LABOR MARKETS*

ABSTRACT A simple general equilibrium model relates spatial product markets and spatial labor markets. The firm is treated as being a spatial monopolist or as a Löschian competitor in the output market and as a spatial monopsonist in the labor market. Derived free spatial demand and free regional labor supply are defined, and their properties examined. The model provides the framework for analyzing the impact of a technological improvement in labor productivity on the structure of the spatial markets. The impact of entry on spatial labor supply is an important determinant of whether or not entry lowers wages and raises output prices. Unlike the spaceless competitive paradigm, zero-profit long-run equilibrium can occur in a space economy under conditions of increasing returns to scale.     

GENERAL EQUILIBRIUM ANALYSIS OF URBAN SPATIAL STRUCTURE. THE PORT-CITY MODEL RECONSIDERED*

In this paper, I consider a general equilibrium urban spatial structure in the setting of a port city by adopting Cobb-Douglas type functional forms for production and utility functions. Unlike previous port-city models, this one can generate four land-use patterns, that is, (1) monocentric, (2) completely integrated, (3) partially integrated (with integration at the center), and (4) partially integrated (with integration at the urban fringe) land-use patterns, and the last pattern suggests a new explanation of suburbanization of employment. The existence and uniqueness of each land-use pattern is proven in a simple constructive way.     

RECURSIVE ALGORITHMS FOR THE ELIMINATION OF REDUNDANT PATHS IN SPATIAL LAG OPERATORS*

ABSTRACT. Recursive algorithms for the elimination of redundant paths in spatial lag operators are introduced. It is shown that these algorithms have superior computational properties in comparison with the cumbersome procedure proposed by Ross and Harary (1952). A rigorous definition of spatial lag operators is given, while a number of mathematical results and properties are derived. Theoretical and empirical results regarding the performance of the proposed algorithms are presented.     

THE OUTPUT EFFECTS OF SPATIAL PRICE DISCRIMINATION REVISITED*

ABSTRACT Some prior findings on spatial price discrimination are extended. Under certain spatially defined market conditions, discriminatory pricing is known to yield greater output than does nondiscriminatory f.o.b. mill pricing. However, this seemingly surprising result is based on the form of the basic demand function which is relatively less convex than a certain standard curve. The present paper makes this prior result more general by permitting the assumed basic demand to be relatively more convex.     

THE EFFECT OF RETAIL FIRM OWNERSHIP ON PRICE EQUILIBRIUM*

ABSTRACT. This paper examines the effect of retail firm ownership on price equilibrium using a simple linear-city model. It is shown that price divergence emerges due to the differences in retail firm ownership, because retail firms under different ownership internalize shopping externalities differently. It is also shown that if a commercial center has two specialized retail firms, these stores charge the same markup for different goods at the equilibrium.     

MODELING SPATIAL STRUCTURES THROUGH EQUILIBRIUM STATES FOR TRANSITION MATRICES*

ABSTRACT. In this paper we present a model for spatial interaction within a network of towns. This interaction is modeled through equilibrium states for certain Markov chains where, in particular, explicit formulas for these states are given. Our model exploits and intertwines ideas from gravity models, the competing destinations model and the intervening opportunities model. The central idea in the paper is to capture the effect of spatial structure in a framework where interaction is determined by the global network configuration.     

SPATIAL PRICE POLICY AND THE DEMAND FOR TRANSPORTATION

For a spatial market of given radius transportation cost per unit sold is less under mill pricing than under any uniform pricing. This is true whether profit-maximizing monopoly prices or welfare-maximizing prices or any other prices are chosen.