Duality Theory of Nonconvex Technologies

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Duality Theory ofNon-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),
  326-342.
• Cherchye, Kuosmanen and Post (2000a) What is the
  Economic Meaning of FDH? A Reply to Thrall,
  Journal of Productivity Analysis 2000, 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.

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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
  Tcom(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