# Equivalence of Price and Quantity Formulations of Spatial Equilibrium: Purified Duality in Quadratic and Concave Programming

https://doi.org/0012-9682(197011)38:6<889:EOPAQF>2.0.CO;2-I
p. 889-906

A. D. Woodland, T. Takayama

In this paper, a first attempt is made to formulate a spatial equilibrium quadratic programming problem in its primal and "purified" dual forms. This formulation is in sharp contrast to the quadratic primal and dual forms formulated by Dorn [1] and Hanson [2]. Equivalence of the quantity and purified price formulations is formally proved. The proof is important because it permits the formulation of spatial equilibrium problems in terms of either the quantity domain or the price domain. The equivalence is then used to establish the mutually dual quadratic problems in quantify and price separately. The notion of a purified dual is extended to concave programming with linear inequality constraints to deal with spatial equilibrium involving nonlinear demand and supply functions.