Eisenberg's Duality in Homogeneous Programming,Shephard's Duality and Economic Analysis

This note is to reintroduce to the reader Eisenberg's symmetric duality theorem in homogeneous programming problems as a useful tool in economic analysis, and thereby to pay a due tribute to him for one of his mathematical contributions. His duality result has been almost in oblivion during the development of Shephard's duality theory between cost and production, and has seldom been mentioned in the literature about the dualities concerning Shephard's distance function, Luenberger's benefit function and directional distance functions proposed by many authors. We show that from Eisenberg's duality it is possible to derive in a systematic way these dualities so far obtained. We also present a further extension of the duality for generalized directional distance functions. In addition, we explain the relationships between the duality theorem of linear programming and that of homogeneous programming, and show how to apply the latter in those economic models in which linear programming has been utilized.