Weak Disposability in Nonparametric Production Analysis:

Undesirable Outputs, Abatement Costs, and Duality

Timo KuosmanenMTT 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.

Inputs (x)

Bad outputs (w)

Good outputs (v)

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

Illustration

• 3 observations, the same amounts of inputs

w

v

Illustration

• Feasible set spanned by convexity

w

v

Illustration• Feasible set spanned by convexity and

free disposability of v

w

v

Illustration• Feasible set spanned by convexity, free

disposability of v, and weak disposability

w

v

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.

Shephard technology

Kuosmanen technology

Example by Färe and Grosskopf

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

1

1

1

1

max ' ' '

( )

( )=1

, 0

p v p w p x

x X

v V

w W

v w x

n

i i ii

n

i ii

n

i ii

n

i ii

i i

Dual interpretation

• Equivalent dual formulation

1

1

1

1

max ' ' '

( )

( )=1

, 0

p v p w p x

x X

v V

w W

v w x

n

i i ii

n

i ii

n

i ii

n

i ii

i i

min

' ' '

'

π V π W π X

π X

π p

π p

0 π p

v w xi i i

xi

v v

w w

x x

i

i

Dual interpretation

• Weak disposability has two important implications on the dual

1) Shadow price of bad output can be negative

2) 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.

FG (2009) - addendum

• “Kuosmanen introduces the property that the technology Y is convex. This is a condition that we do not invoke in our comment. … Y convex 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) є Y.”

Thank you for your attention!

• Time for questions and comments

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