Weak Disposability in Nonparametric

Production Analysis with Undesirable

Outputs

Timo 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 Disposability

Definition (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

bads

inputs

VRS

intensity 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 K

k K

outputs

bads

inputs

VRS

intensity 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

outputs

bads

inputs

VRS

intensity 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 C

v 8 3 5

w 6 4 1

x 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)NDP

ˆ (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)UAP

ˆ (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!