Economic Analysis of Technology and Properties of Legendre-Fenchel Transformations

This paper examines a two-way relationship between convex analysis and microeconomic theory. Motivation for this paper are the observed similarities in the structure of the theory of consumer behavior and production theory. The fact that the behavior of variables is not determined by their nature but, rather, by their relationships is best illustrated and explained by using convex sets and convex analysis, which occupy central place in microeconomic theory. This paper is the result of efforts to make complex results of convex analysis and its application in microeconomic theory more transparent. Starting with the well-known economic phenomenon of profit maximization the authors derive in a novel way general results within the framework of convex analysis. From those results follow, directly and indirectly, the conclusions of the theory of consumer and producer behavior. The authors show that applying the Fundamental Theorems of Calculus opens up a new perspective in which the marginal cost curve can be interpreted as the marginal profit curve. This enables the derivation of Hotelling's lemma in a new way. Using the new interpretation of Hotelling's lemma, the authors reconstruct the cost function and confirm the Conjugate Duality Theorem of Legendre-Fenchel transformations. Relaxing the assumption of differentiability by 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 path from the well-known economic facts to completely general conclusions of convex analysis is continued by applying the Conjugate Duality Theorem of Legendre-Fenchel transformations to the profit function. The essence of the dual characterization of technology by the profit function is illustrated by the graphical representation of linear homogeneity of the profit function. It results in the possibility to reconstruct the production function while using only the First Order Conditions to rederive Hotelling's lemma. It is this inductive-deductive approach used to examine the properties of Legendre-Fenchel trasformations and their application in the theory of consumer and producer behavior that establishes a two-way relationship between convex analysis and microeconomic theory.