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Economic Analysis of Technology and Properties of Legendre-Fenchel Transformations

Author

Listed:
  • Ilko Vranki?

    (Faculty of Economics and Business Zagreb)

  • Mira Krpan

    (Faculty of Economics and Business Zagreb)

  • Jasminka ?ohinger

    (Faculty of Economics and Business Zagreb)

Abstract
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.

Suggested Citation

  • Ilko Vranki? & Mira Krpan & Jasminka ?ohinger, 2019. "Economic Analysis of Technology and Properties of Legendre-Fenchel Transformations," International Journal of Economic Sciences, International Institute of Social and Economic Sciences, vol. 8(2), pages 159-183, December.
    • Handle: RePEc:sek:jijoes:v:8:y:2019:i:2:p:159-183
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    References listed on IDEAS

    as
    1. McFadden, Daniel, 1978. "Cost, Revenue, and Profit Functions," Histoy of Economic Thought Chapters, in: Fuss, Melvyn & McFadden, Daniel (ed.),Production Economics: A Dual Approach to Theory and Applications, volume 1, chapter 1, McMaster University Archive for the History of Economic Thought.
    2. Fuss, Melvyn & McFadden, Daniel, 1978. "Production Economics: A Dual Approach to Theory and Applications (II): Applications of the Theory of Production," History of Economic Thought Books, McMaster University Archive for the History of Economic Thought, volume 2, number fuss1978a.
    3. Rosas, Juan Francisco & Lence, Sergio H., 2015. "Duality theory econometrics: How reliable is it with real-world data?," 2015 AAEA & WAEA Joint Annual Meeting, July 26-28, San Francisco, California 205294, Agricultural and Applied Economics Association.
    4. Cornes,Richard, 1992. "Duality and Modern Economics," Cambridge Books, Cambridge University Press, number 9780521336017, September.
    5. Burgess, David F., 1975. "Duality theory and pitfalls in the specification of technologies," Journal of Econometrics, Elsevier, vol. 3(2), pages 105-121, May.
    6. Appelbaum, Elie, 1978. "Testing neoclassical production theory," Journal of Econometrics, Elsevier, vol. 7(1), pages 87-102, February.
    7. Fuss, Melvyn & McFadden, Daniel (ed.), 1978. "Production Economics: A Dual Approach to Theory and Applications," Elsevier Monographs, Elsevier, edition 1, number 9780444850133.
    8. Diewert, W E, 1971. "An Application of the Shephard Duality Theorem: A Generalized Leontief Production Function," Journal of Political Economy, University of Chicago Press, vol. 79(3), pages 481-507, May-June.
    9. Shumway, C. Richard, 1995. "Recent Duality Contributions In Production Economics," Journal of Agricultural and Resource Economics, Western Agricultural Economics Association, vol. 20(1), pages 1-17, July.
    10. Fuss, Melvyn & McFadden, Daniel, 1978. "Production Economics: A Dual Approach to Theory and Applications (I): The Theory of Production," History of Economic Thought Books, McMaster University Archive for the History of Economic Thought, volume 1, number fuss1978.
    11. Harold Hotelling, 1932. "Edgeworth's Taxation Paradox and the Nature of Demand and Supply Functions," Journal of Political Economy, University of Chicago Press, vol. 40(5), pages 577-577.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Mira Krpan, 2023. "Duality in the analysis of monopsony in the labor market," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 31(3), pages 975-990, September.

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    More about this item

    Keywords

    Hotelling?s lemma; Legendre-Fenchel transformations; convexity; duality; inversion; sub-differentiability; monotonicity; cyclical monotonicity.;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • D01 - Microeconomics - - General - - - Microeconomic Behavior: Underlying Principles
    • D21 - Microeconomics - - Production and Organizations - - - Firm Behavior: Theory

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