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
Bad outputs (w)
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
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
v
w
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
max p v ' v  p w ' w  p x ' x
n
x   (i  i ) Xi
i 1
n
v   i Vi
i 1
n
w   i Wi
i 1
n
 (   )=1
i 1
i
i , i  0
i
Dual interpretation
• Equivalent dual formulation
max p v ' v  p w ' w  p x ' x
n
x   (i  i ) Xi
i 1
n
v   i Vi
i 1
n
w   i Wi
i 1
n
 (   )=1
i 1
i
i , i  0
i
min 
  πv ' Vi  π w ' Wi  π x ' Xi i
   π x ' Xi i
πv  pv
πw  pw
0  πx  px
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: [email protected]