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: Timo.Kuosmanen@wur.nl

Thank you for your attention!