Weak Disposability in Nonparametric Production Analysis: Undesirable Outputs, Abatement Costs, and Duality

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Weak Disposability in Nonparametric Production Analysis: Undesirable Outputs, Abatement Costs, and Duality

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Weak Disposability in Nonparametric Production Analysis:Undesirable Outputs, Abatement Costs, and Duality

Timo Kuosmanen

MTT Agrifood Research Finland

→Helsinki School of Economics

- Production activities often generate harmful side-products that are discharged to the environment, referred to as undesirable outputs
- pollution, waste, noise, etc.

Good outputs (v)

Inputs (x)

FIRM

Bad outputs (w)

Definition (Shephard, 1970):

- Technology T exhibits weak disposability iff, at any given inputs x, it is possible to scale any feasible output vector (v,w) downward by factor θ: 0 ≤θ≤1.
- If input x can produce output (v,w), then x can also produce output(θv,θw).

(a.k.a. Activity Analysis, Data envelopment analysis (DEA))

Minimum extrapolation principle:

Estimate production possibility set T by the smallest subset of (x,v,w)-space that

- Contains all observed data points (xi,vi,wi)
- Satisfies the maintained axioms

- Standard set of axioms:
- inputs x and (good) outputs v are freely disposable (monotonicity)
- outputs (v,w)are weakly disposable
- T is a convex set

v

w

- 3 observations, the same amounts of inputs

v

w

- Feasible set spanned by convexity

v

w

- Feasible set spanned by convexity and free disposability of v

v

w

- Feasible set spanned by convexity, free disposability of v, and weak disposability

- Kuosmanen (2005) Weak Disposability in Nonparametric Production Analysis with Undesirable Outputs, Amer. J. Agr. Econ. 87(4).
- Färe and Grosskopf (2009) A Comment on Weak Disposability in Nonparametric Production Analysis, Amer. J. Agr. Econ., to appear.
- Kuosmanen and Podinovski (2009) Weak Disposability in Nonparametric Production Analysis: Reply to Färe and Grosskopf, Amer. J. Agr. Econ., to appear.

- Points out that Shephard’s weakly disposable technology has a restrictive assumption that the abatement factor θ is same across all firms.
- It is usually cost efficient to abate emissions in those firms where the marginal abatement costs are lowest.

- Presents a more general formulation of weakly disposable technology that allows abatement factors to differ across firms

- Critique of Kuosmanen (2005)
- Main arguments:
- ”Shephard’s specification does satisfy weak disposability and is the “smallest” technology to do so.”
- ”the Kuosmanen technology is larger than required for it to be weakly disposable.”

- Response to critique by Färe and Grosskopf
- Show by examples that the Shephard technology violates convexity, one of the maintained axioms
- Formal proof that the Kuosmanen technology is the “true” minimal technology under the stated axioms.

- Shephard technology involves nonlinear constraints
- A nonconvex set does not have a natural dual interpretation
- The convex Kuosmanen technology can be presented as system of linear inequalities
- Provide new economic insights to weak disposability

- Profit function of the Kuosmanen technology

- Equivalent dual formulation

- Weak disposability has two important implications on the dual
- Shadow price of bad output can be negative
- Limited liability: it is always possible to close down activity, accepting the sunk cost of inputs x

- Shephard’s traditional weakly disposable technology, advocated by Färe and Grosskopf, is not convex and therefore violates one of the central assumptions underlying the method.
- Thus, it does not qualify as the minimal convex weakly disposable technology.
- Moreover, the Shephard technology is not the minimal weakly disposable technology even if we relax the convexity axiom entirely.

“A full axiomatic investigation undertaken by the authors has proved that:”

- Kuosmanen (2005) technology correctly represents convex technologies that exhibit joint weak disposability of bad and good outputs.
- It is therefore the smallest technology under the maintained set of axioms.

- “Kuosmanen introduces the property that the technology Yis convex. This is a condition that we do not invoke in our comment. … Yconvex does not enter our Proposition 4, and therefore lies outside the scope of our comment.”
- “the Kuosmanen model fails to satisfy the inactivity axiom, i.e., (0, 0, 0) єY.”

- Time for questions and comments
- Further comments/feedback welcome. E-mail: Timo.Kuosmanen@mtt.fi