# Duality Theory of Non-Convex Technologies Timo Kuosmanen Wageningen University EEA-ESEM 2003, Stockholm 20-24 August 2003.

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Duality Theory of Non-Convex Technologies Timo Kuosmanen Wageningen University EEA-ESEM 2003, Stockholm 20-24 August 2003.

What is duality theory? Equivalence between direct models expressed in terms of physical quantities and indirect models expressed in monetary terms. Consider e.g. utility functions vs. expenditure functions. In the context of production: Shephard (1953): Cost and Production Functions, Princeton. Färe and Primont (1995): Multi-Output Production and Duality: Theory and Applications, Kluwer.

Duality of profit and production possibilities Example: production possibility set: profit function: If T is a non-trivial (non-empty, closed, …) production set that satisfies free disposability and convexity, then

What if convexity fails to hold?

Objectives 1) Understanding of the asymmetric role of convexity in Duality Theory, and its practical implications. 2) Generalizing Duality Theory to account for non- convexities rather than assume them away

Models considered

Asymmetry of assumptions Model A Model B

Conclusion 1 All models contain information of production possibilities and profit possibilities. Models differ in terms of their information content. Assuming convexity and monotonicity simplifies the setting so that the information content of all models is the same.

Conclusion 2: We can always recover a model that contains less information from a model that contains more information. No convexity is needed! Example: We can always recover the profit function is the production possibility set T is known. This does not depend on convexity or non- convexity of T.

Conclusion 3: In principle, the information content of the model is independent of the orientation of the model as production, economic, or indirect model. Example: Constrained profit function and the production possibility set are both complete, equivalent representations of technology.

Conclusion 4: In practice, the information content of production models is usually higher than the information content of economic or indirect models. Example: Standard profit function enables us to recover an outer approximation - the convex monotonic hull - of the production possibility set, but not necessarily the exact set.

Illustration u T x cm(T)

Conclusion 5: Economic models retain their usual regularity properties (monotonicity, convexity/concavity) even if the underlying technology is non- convex. Example: Cost function is concave, and continuous in input prices even if the production possibility set is non-convex.

Implications to Testing hypotheses empirically indirect / dual approach for testing convexity Model specification Found duality between the indirect cost/revenue functions and the Petersen technology (non-convex technology with convex input isoquants and concave output isoquants.

Paper availability To appear in Journal of Productivity Analysis Preprint downloadable from my homepage http://www.sls.wau.nl/enr/staff/kuosmanen/ or send e-mail to Timo.Kuosmanen@wur.nl

Constrained profit function Definition: Can model profit maximization under exogenous quantity / budget constraints. This function contains as its special cases (among others) profit function cost function revenue function cost indirect revenue fnction revenue indirect cost function resticted profit function (McFadden, 1978)

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