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Duality Theory of Non-convex Technologies Timo Kuosmanen

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Motivation Stems from my earlier interest in non-convex technologies, e.g. –Kuosmanen (2001): DEA with Efficiency Classification Preserving Conditional Convexity, European Journal of Operational Research 132(2), –Cherchye, Kuosmanen and Post (2000a): What is the Economic Meaning of FDH? A Reply to Thrall, Journal of Productivity Analysis 2000, 13(3), –Cherchye, Kuosmanen and Post (2001): Why Convexify? A Critical Assesment of Convexity Assumption in DEA, Helsinki School of Economics and Business Administration, Working Paper W-270.

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Motivation DEA model specification often justified by duality arguments: We use the convex DEA model, because the duality theory requires convexity. A relevant argument or not? Desire to understand the role of convexity and free disposability in the duality theory in a more profound fashion. Why duality fails without convexity? Could it be repaired? How?

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What duality theory? Shephard (1953): Cost and Production Functions, Princeton. Equivalence between a production model and an economic model. Example: If T is a non-trivial (non-empty, closed, …) production set that satisfies free disposability and convexity, then

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Idea #1 The assumptions of convexity and free disposability can be by-passed by deriving an inexact (outer bound) representation of the technology Boils down to the established duality theory if T=com(T), but this generalization also applies to non-convex and congested technologies.

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Idea #2 Profit maximization under exogenous quantity / budget constraints. Instead of defining a different economic model for alternative constraints, consider a general model of constrained profit function 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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An interesting finding The constrained profit function implicitly contains an exact and complete representation of the technology, i.e. Applies to convex and non-convex technologies. Highlights the pivotal role of the constraints in determining the duality relationship: –the more information we have on profitability under alternative constraint structures, the more we know of the underlying technology. –In the extreme case of the full information, T can be recovered exactly.

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Special cases The knowledge of the cost indirect revenue function Suffices for deriving the outer bound with convex input sets and convex output sets

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Testing hypotheses Possible to test convexity and disposability hypotheses by using price/profit/constraint data, without data on input or output quantities produced. Depends on the diversity of constraints! Attention on the economic selection effects: –At the firm level (selection of profit maximizing netputs) –At the industry level (selection of the fittest firms)

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Model specification in DEA External constraints as a source of non-convexity: –Price-taking behavior in the competitive market environment does not suffice to justify labeling deviations from convexity as inefficiency. Extra care with modeling the constraints! Connection between the Petersen technology and the indirect, cost/revenue constrained approach to efficiency analysis.

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Illustration

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Unconstrained profit maximization

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Input constrained profit maximization

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