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Duality Theory of Non-convex Technologies. Timo Kuosmanen. 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), 326-342.
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Duality Theory ofNon-convex Technologies Timo Kuosmanen
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), 326-342. • Cherchye, Kuosmanen and Post (2000a): What is the Economic Meaning of FDH? A Reply to Thrall, Journal of Productivity Analysis2000, 13(3), 259-263. • 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.
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?
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
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.
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)
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.
Special cases • The knowledge of the cost indirect revenue function Suffices for deriving the outer bound with convex input sets and convex output sets
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)
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.
