Weak Disposability in Nonparametric Production Analysis with Undesirable OutputsTimo KuosmanenWageningen University, The Netherlands
14th EAERE Annual Conference, 23-26 June 2005, Bremen, Germany
Background Production activities typically generate
some environmentally detrimental undesirable outputs as side-products emissions, waste, noise, etc.
The treatment of undesirable outputs in nonparametric production analysis has recently attracted debate: Hailu and Veeman, AJAE 2001 Färe and Grosskopf, AJAE 2003 Hailu, AJAE 2003
This paper Shows that conventional formulations of
weak disposability implicitly and unintentionally assume that all firms apply uniform abatement factors. It is usually cost efficient to abate emissions in
those firms where the marginal abatement costs are lowest.
Presents an alternative formulation of weak disposability that allows for non-uniform abatement factors
Notation Firms transform inputs to (good) outputs, which
causes undesirable side-products (bads). Input quantities Output quantities Environmental bads
Production technology characterized by output set
1( ,..., ) NNx x x
1( ,..., ) MMv v v
1( ,..., ) JJw w w
( ) ( , ) can produce ( , )P x v w x v w
Weak DisposabilityDefinition (Shephard, 1970): Outputs are
weakly disposable if
and=>( , ) ( )v w P x 0 1
( , ) ( ) v w P x
Nonparametric production analysis(also known as Activity Analysis or Data
Envelopment Analysis (DEA)) Assume a sample of K observations Estimate output set P(x) by a set of output
vectors that consists of all observed output vectors output vectors that are feasible by the
maintained production assumptions and no other vectors
Nonparametric production analysis Maintained assumptions
inputs x and (good) outputs v are freely disposable
bad outputs w are weakly disposable outputs sets P(x) are convex for all x
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
Shephard’s formulation – uniform abatement outputs
badsinputsVRSintensity weights
abatement factor
1
ˆ ( ) ( , ) : , 1,...,
K
k km m
k
z v v m MP x v w
1
, 1,...,
K
k kj j
k
z w w j J
1
, 1,...,K
k kn n
k
z x x n N
1
1
0, 1,...,1, 1,...,
K
k
k
k
z
z k Kk K
outputsbadsinputsVRSintensity weights
K abatement factors
Generalized formulation
1
ˆ ( ) ( , ) : , 1,...,K
k k km m
k
z v v m M
P x v w
1
, 1,...,K
k k kj j
k
z w w j J
1
, 1,...,K
k kn n
k
z x x n N
1
1
0, 1,...,
0 1, 1,...,
Kk
k
k
k
z
z k K
k K
Partition the intensity weights as
where represents the part of firm k’s output
that is abated through scaling down of activity level, i.e.,
represents the part of firm k’s output that remains active, i.e.,
Linearization
k k kz
k
k(1 )k k kz
k k kz
outputsbadsinputsVRSintensity weights
Linearized formulation
1
ˆ ( ) ( , ) : , 1,...,K
k km m
k
v v m M
P x v w
1
, 1,...,K
k kj j
k
w w j J
1
( ) , 1,...,K
k k kn n
k
x x n N
1
( ) 1
, 0, 1,...,
Kk k
k
k k k K
Numerical example
Firm A Firm B Firm Cv 8 3 5w 6 4 1x 5 1 4
Output sets: w not disposable
ˆ (1)NDP
B
A
C
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7w
v
ˆ (2)NDPˆ (3)NDP
ˆ (4)NDP
ˆ (5)NDP
Output sets: weak disposability - Shephard
C
A
B
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7w
v
ˆ (1)UAP
ˆ (2)UAPˆ (3)UAP
ˆ (4)UAP
ˆ (5)UAP
Output sets: weak disposability – this paper
C
A
B
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7w
v
ˆ (1)P
ˆ (2)P
ˆ (3)P
ˆ (4)P
ˆ (5)P
Empirical significance Static environmental efficiency analysis Measurement of total productivity over time Estimation of abatement cost functions Etc...
Methodological significance Production assumptions interact The fundamental ”minimum extrapolation principle” by Banker et al. (Management
Science 1984) can fail The minimum set that satisfies the maintained assumptions and contains all observations may
exclude production vectors that are feasible by the same set of assumptions
=> Need to reconsider the main principle of data envelopment analysis
Further details... Paper accepted for publication in American
Journal of Agricultural Economics
Questions / comments are welcome to E-mail: [email protected]
Thank you for your attention!
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