Production Quotas and Land Values: Importance of the Dual in a Spatial Linear Programming Problem

Abstract

HE primary purpose of this presentation is to illustrate the use of a neoclassical economic interpretation of the dual solution to a spatial equilibrium linear programming model. Past discussions of the HeadyEgbert type models have devoted their primary attention to production allocation schemes. That is, the forward solution has been emphasized. However, a wealth of economic information can be obtained from the dual of these types of models. The models which prompted the development of this paper, and the type to which we shall refer, were designed to observe certain characteristics of agricultural supply control programs. The models feature spatially separated producing and consuming regions. Each production region has the potential for producing one or more of several crop enterprises. Production within any region is limited by the total amount of land available and acreage quotas on individual crop enterprises. Each consumption region has a prespecified requirement for each of the products considered. The product requirements can be met by production from within the region of consumption or imported from outlying consumption regions. The solution to the spatial linear programming problem is attained by minimizing the production and transportation costs of satisfying the stated requirements. The dual solution provides equilibrium product prices and imputed values to the limiting resources. The acreage allotment shadow prices, which we shall call quota rents, are an estimate of the marginal cost of the allotment. In reality, these quota rents may be capitalized into cropland market values if the quotas are fixed to the land. The quota rents are a guide for estimating the governmental costs of reducing crop production in certain regions below that shown to exist in the programming solutions. We also learn which regions may have the lowest costs associated with output reduction-that is, imputed values of zero on their crop allotment restraints. The information derived regarding equilibrium product prices and regional land rents is equally useful for a priori evaluation of specific supply control programs. To demonstrate the derivation of the dual solution we will initially employ an interregional competition model with n producing regions and m