legende - fenchel转换技术和特性的经济分析

Ilko Vrankic, Jasminka Šohinger, Mira Krpan
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引用次数: 2

摘要

本文探讨了凸分析与微观经济理论之间的双向关系。本文的动机是观察到消费者行为理论和生产理论在结构上的相似性。变量的行为不是由它们的性质决定的,而是由它们之间的关系决定的,这一事实可以用凸集和凸分析来最好地说明和解释,凸集和凸分析在微观经济理论中占据中心位置。本文是努力使凸分析的复杂结果及其在微观经济理论中的应用更加透明的结果。作者从著名的利润最大化经济现象出发,在凸分析的框架内,以一种新颖的方式推导出一般结果。从这些结果中,直接或间接地得出了消费者和生产者行为理论的结论。作者表明,运用微积分基本定理开辟了一个新的视角,即边际成本曲线可以解释为边际利润曲线。这使得霍特林引理的推导有了新的途径。利用Hotelling引理的新解释,重构了代价函数,证实了legende - fenchel变换的共轭对偶定理。通过将代价函数的图描述为其切线的包络,放宽了可微性的假设,作者重新推导了legende - fenchel变换的性质,并证明了它们一般成立。通过将legende - fenchel变换的共轭对偶定理应用于利润函数,延续了从众所周知的经济事实到凸分析的完全一般结论的路径。利润函数的线性同质性的图形表示说明了利润函数对技术的双重表征的本质。这使得仅使用一阶条件就可以重新推导霍特林引理,从而重构生产函数成为可能。正是这种用于检验勒让德-芬切尔变换的性质及其在消费者和生产者行为理论中的应用的归纳演绎方法,在凸分析和微观经济理论之间建立了双向关系。
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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.
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