ECONOMIC ANALYSIS OF TECHNOLOGY AND PROPERTIES OF LEGENDRE-FENCHEL TRANSFORMATIONS

Abstract

This paper examines the two-way relationship between convex analysis and microeconomic theory. The motivation for the paper stems from observed similarities in the structure of consumer behavior theory and production theory. The fact that the behavior of variables is not determined by their nature but rather by their relationships is best illustrated using convex sets and convex analysis, which occupy a central place in microeconomic theory. This paper represents an effort to make complex results of convex analysis and their application in microeconomic theory more transparent.  Starting from the well-known economic phenomenon of profit maximization, the authors derive general results within the framework of convex analysis in a novel way. From these results follow, directly and indirectly, the conclusions of consumer and producer theory. The authors show that applying the Fundamental Theorems of Calculus opens a new perspective in which the marginal cost curve can be interpreted as the marginal profit curve, enabling a new derivation of Hotelling’s lemma. Using this interpretation, the authors reconstruct the cost function and confirm the Conjugate Duality Theorem of Legendre–Fenchel transformations. By relaxing the assumption of differentiability and describing the graph of the cost function as the envelope of its tangents, the authors rederive the properties of Legendre–Fenchel transformations and show that they hold in general. The analysis is further extended by applying the Conjugate Duality Theorem to the profit function. The dual characterization of technology through the profit function is illustrated graphically via its linear homogeneity, which allows reconstruction of the production function using only first-order conditions to rederive Hotelling’s lemma. This inductive–deductive approach to examining Legendre–Fenchel transformations and their application in consumer and producer theory establishes a two-way relationship between convex analysis and microeconomic theory.

Data:
Received: 8 Oct 2019
Revised: 19 Nov 2019
Accepted: 6 Dec 2019
Published: 20 Dec 2019