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# Weak Disposability in Nonparametric Production Analysis: Undesirable Outputs, Abatement Costs, and Duality PowerPoint PPT Presentation

Weak Disposability in Nonparametric Production Analysis: Undesirable Outputs, Abatement Costs, and Duality. Timo Kuosmanen MTT Agrifood Research Finland → Helsinki School of Economics. Background.

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

### Background

• 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

### Weak Disposability

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).

### Nonparametric production analysis

(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

### Nonparametric production analysis

• 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

### Illustration

• 3 observations, the same amounts of inputs

v

w

### Illustration

• Feasible set spanned by convexity

v

w

### Illustration

• Feasible set spanned by convexity and free disposability of v

v

w

### Illustration

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

### AJAE debate

• 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.

### Kuosmanen (2005)

• 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

### Färe and Grosskopf (2009)

• 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.”

### Kuosmanen and Podinovski (2009)

• 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.

### Dual interpretation

• 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

### Dual interpretation

• Profit function of the Kuosmanen technology

### Dual interpretation

• Equivalent dual formulation

### Dual interpretation

• Weak disposability has two important implications on the dual

• Limited liability: it is always possible to close down activity, accepting the sunk cost of inputs x

### Conclusions (KP 2009)

• 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.

### Conclusions (KP 2009)

“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.”

### Thank you for your attention!

• Time for questions and comments

• Further comments/feedback welcome. E-mail: Timo.Kuosmanen@mtt.fi