Single‐ and Multi‐Commodity Models of Spatial Equilibrium in a Linear Programming Framework
Authors: Willett Keith 
Published: July 2010

This paper uses linear programming to construct a set of single‐ and multi‐commodity models of spatial equilibrium within a competitive market. The models are based on grid linearization techniques which appear in the agricultural economics literature. First, a single‐commodity spatial equilibrium model is constructed and is shown to satisfy both the pricing and quantity conditions required to obtain a spatial competitive market equilibrium solution. Next, the model is extended to include multiple commodities. This extension lakes two forms. The first form slates the objective function as the sum of consumers' and producers' surplus, and requires the assumption that the cross‐price elasticity coefficients of supply and demand are symmetric. Finally, a model with asymmetric cross‐price elasticities is staled as a primal‐dual problem in which the constraints are the conditions necessary for a spatial competitive market equilibrium.

Cet exposé présente un ensemble de modèles ďéquilibre spatial qui appartiennent à des denrées individuelles el multiples au marché de concours et qui sont construits dans le cadre de la programmation linéaire. La construction des modèles de programmation linéaire est fondée sur des techniques du quadrillage linéaire qui se montrent dans les revues ďéconomie rurale. Ďabord, un modèle ďéquilibre spatial ďune denrée individuelle est construit et on fait preuve que ce modèle remplit les conditions de prix el de quantité qui sont exigées pour qu ‘on peut se procurer des résultats. Ensuite le modèle de la denrée individuelle est étendu pour comprendre des denrées multiples. Ľauteur présente deu x formes du modèle étendu. Ďabord. un modèle est développé qui affirme la fonction économique sous la forme de la somme du surplus du producteur et celui du consommateur. Cette conformation exige que les élasticités prix‐croisées de ľoffre et de la demande soient symétriques. Enfin, un modèle dont les élasticités sont asymétriques est exprimé en fonction ďun problème primal‐dual, un problème oú les contraintes sont les conditions nécéssaires pour un équilibre spatial au marché de concours.

Beginning with Samuelson's work, mathematical programming methods to solve spatial competitive market equilibria problems have appeared extensively throughout the agricultural economics literature. The vast majority of these mathematical programming models have been developed in terms of a quadratic programming framework. Duloy and Norton have shown how a quadratic objective function can be approximated as a linear objective function through the use of grid linearization techniques.

A set of spatial competitive market equilibrium models for single and multiple commodities in a linear programming framework were presented in this paper. The linearization techniques used by Duloy and Norton provided a basis for constructing the linear programming whole. First, a single‐commodity spatial equilibrium model was constructed and shown to satisfy both the pricing and quantity conditions required for a competitive spatial equilibrium solution to be obtained.

The single‐commodity model was extended to include multiple commodities. This extension took two forms. First, if the model objective function is stated in the form of Samuelson's “net social payoff‘ function, the commodity demand and supply functions must be assumed to be integrable. This assumption is consistent with the neoclassical assumptions of the theory of production, but creates a number of problems on the demand side. It is implied here that the demand cross‐price effects will be equal across all commodity prices. However, this will be true only for very restrictive cases.

The second form of the multiple commodity model was designed to circumvent the integrability assumption. This was accomplished by specifying a primal‐dual problem in which the constraints are the necessary conditions for a spatial competitive market equilibria.

Implementation of the models set out in this paper require a set of numerical initial conditions. However, these do not necessarily need to be market‐clearing conditions.