Directional output distance functions: endogenous directions
based on exogenous normalization constraints
Färe, R., Grosskopf, S., & Whittaker, G. (2013). Directional output distance
functions: endogenous directions based on exogenous normalization constraints.
Journal of Productivity Analysis, 40(3), 267-269. doi:10.1007/s11123-012-0333-8
10.1007/s11123-012-0333-8
Springer
Version of Record
http://hdl.handle.net/1957/46915
http://cdss.library.oregonstate.edu/sa-termsofuse
Directional output distance functions: endogenous directions
based on exogenous normalization constraints
R. Fa
¨
re
•
S. Grosskopf
•
G. Whittaker
Published online: 8 January 2013
Ó Springer Science+Business Media New York 2013
Abstract In response to a question raised by Knox
Lovell, we develop a method for estimating directional
output distance functions with endogenously determined
direction vectors based on exogenous normalization con-
straints. This is reminiscent of the Russell measure pro-
posed by Fa
¨
re and Lovell (J Econ Theory 19:150–162,
1978). Moreover it is related to the slacks-based directional
distance function introduced by Fa
¨
re and Grosskopf (Eur J
Oper Res 200:320–322, 2010a, Eur J Oper Res 206:702,
2010b). Here we show how to use the slacks-based func-
tion to estimate the optimal directions.
Keywords DEA Directional distance functions
Slack-based measures
JEL Classification C61 D24 D92 O33
1 Introduction
This paper is inspired by a question raised many years ago
by Knox Lovell when we were presenting work on direc-
tional distance functions: namely, how should the
researcher choose the direction vectors when estimating the
directional distance function other than in some ad hoc
way? We have struggled with this issue for a number of
years and here suggest that one might determine the
direction vectors endogenously. The approach is reminis-
cent of the structure of the Russell measure proposed in
1978 by Fa
¨
re and Lovell. More recently we show how this
model is related to the slacks-based directional distance
function introduced by Fa
¨
re and Grosskopf (2010a, b) and
show how to use the slacks-based function to estimate the
optimal directions.
In the standard case in which the researcher chooses the
directional vector, the resulting efficiency scores depend on
that vector.
1
By endogenizing the direction vector, i.e.,
optimizing over them, the efficiency scores are in some
sense ‘optimal’ rather than ad hoc choices of the
researcher. Directional distance functions, see Chambers
et al. (1998), are defined on a technology and were intro-
duced by Luenberger (1992, 1995) as shortage functions;
our model also applies to these functions.
In a recent paper (Zofio et al. 2012), the authors state:
‘When market prices are observed and firms have a
profit maximizing behavior, it seems natural to choose as
the directional vector that projecting inefficient firms
towards profit maximizing benchmarks.’ This leads the
authors to optimize over the directional vector and hence
endogenize it (see expression 9). Here we address the
case of the directional distance function without appeal
to profit maximization, and therefore without requiring
price data.
R. Fa
¨
re
Department of Agricultural and Resource Economics, Oregon
State University, Corvallis, OR 97331, USA
R. Fa
¨
re S. Grosskopf (&)
Department of Economics, Oregon State University, Corvallis,
OR 97331, USA
e-mail: [email protected]
G. Whittaker
Agricultural Research Service, Corvallis, OR 97331, USA
1
See Briec (1997, 1998), Chung et al (1997) and Fa
¨
re and Grosskopf
(2004) for alternative choices for the directional vectors.
123
J Prod Anal (2013) 40:267–269
DOI 10.1007/s11123-012-0333-8
2 Main results
In order to keep our exposition as simple as possible we
assume that we have one input x 3 0 which is used to produce
two outputs (y
1
, y
2
) 3 0. Moreover we assume that there are
two decision making units (DMUs) or firms. The output set
may then be formalized using Activity Analysis or Data
Envelopment Analysis as
PðxÞ ¼fðy
1
; y
2
Þ : z
1
y
11
þ z
2
y
21
=y
1
z
1
y
12
þ z
2
y
22
=y
2
z
1
x
1
þ z
2
x
2
5x
z
1
; z
2
=0g;
ð1Þ
where z = (z
1
, z
2
) are the intensity variables.
Let g = (g
1
, g
2
) 3 0 be a nonnegative directional output
vector and assume that the components belong to the unit
simplex,
2
i.e.,
g
1
þ g
2
¼ 1; ð2Þ
which ensures compactness, and guarantees that the
problem we specify in (4) below has a solution. Among all
possible normalizations of the directional vector we have
followed Luenberger (1995, p. 78), which is also a stan-
dard approach in economics. This normalization also
serves our purpose in our proof, see expression (8) which
follows.
We may now formulate a directional output distance
function with (g
1
, g
2
)asvariables. Their values will be
endogenously determined through the following optimiza-
tion problem,
max
z;g;b
fb : ðy
11
þ bg
1
; y
21
þ bg
2
Þ2PðxÞ; g
1
þ g
2
¼ 1;
g
1
; g
2
¼
[
0g;
ð3Þ
which we formulate for DMU 1 as
max
z;g;b
b s:t: z
1
y
11
þ z
2
y
21
=y
11
þ bg
1
z
1
y
12
þ z
2
y
22
=y
12
þ bg
2
z
1
x
1
þ z
2
x
2
5x
1
g
1
þ g
2
¼ 1
z
1
; z
2
=0;
b =0; g
1
; g
2
=0:
ð4Þ
This is a nonlinear optimization problem, and we would
like to transform it into a linear optimization problem,
namely into
max
z;b
1
;b
2
b
1
þ b
2
s:t: z
1
y
11
þ z
2
y
21
=y
11
þ b
1
1
z
1
y
12
þ z
2
y
22
=y
12
þ b
2
1
z
1
x
1
þ z
2
x
2
5x
1
z
1
; z
2
=0; b
1
; b
2
=0
ð5Þ
Expression (5) is the output-oriented version of the slacks-
based directional distance function introduced by Fa
¨
re and
Grosskopf (2010a, b), which bears some resemblance to the
Russell measure proposed by Fa
¨
re and Lovell (1978). In the
Fa
¨
re-Grosskopf formulation, g
1
= g
2
= 1whichimpliesthat
they are in the same units as the outputs. However, here the bs
are unit free, and therefore may be added. This distance
function takes value zero if and only if the output vector
belongs to the efficient output set, also referred to as the
Pareto-Koopmans efficient subset, see references.
In order to use (5) to estimate (4) we need to show that
they are equivalent. We begin by showing that the model in
(4) can be derived from (5). There are two possible cases to
consider:
(1) both b
1
and b
2
= 0
(2) at least one b
i
, i = 1, 2 is positive.
In the first case, which is when the DMU is efficient, the
direction vector is not uniquely determined. Thus we may
set the direction vector arbitrarily in a positive direction,
for example, let g
1
= g
2
= 1/2. If at least one
b
i
[ 0, i = 1, 2, then in (5) take b
i
¼ bg
i
=0; i ¼ 1; 2
with g
i
=0 and g
1
? g
2
= 1, then we can rewrite (5)as
max
z;g;b
bg
1
þ bg
2
s:t: z
1
y
11
þ z
2
y
21
=y
11
þ bg
1
z
1
y
12
þ z
2
y
22
=y
12
þ bg
2
z
1
x
1
þ z
2
x
2
5x
1
g
1
þ g
2
¼ 1
z
1
; z
2
=0; b =0
g
1
; g
2
=0:
ð6Þ
and transform (6) into
max
z;g;b
b s:t: z
1
y
11
þ z
2
y
21
=y
11
þ bg
1
z
1
y
12
þ z
2
y
22
=y
12
þ bg
2
z
1
x
1
þ z
2
x
2
5x
1
g
1
þ g
2
¼ 1
z
1
; z
2
=0
b =0; g
1
; g
2
=0:
ð7Þ
i.e., expression (4).
Next we show the converse, i.e., that (5) can be derived
from (4). Let b 3 0 and multiply b in the objective
function of (3) with (g
1
? g
2
) = 1 then we have
2
The unit of measurement problem that can occur is trivially
corrected by introducing appropriate weights.
268 J Prod Anal (2013) 40:267–269
123
max
z;g;b
b ðg
1
þ g
2
Þ s:t: z
1
y
11
þ z
2
y
21
=y
11
þ bg
1
z
1
y
12
þ z
2
y
22
=y
12
þ bg
2
z
1
x
1
þ z
2
x
2
5x
1
z
1
; z
2
=0; b =0:
g
1
; g
2
=0:
ð8Þ
If we take b
1
= bg
1
1, b
2
= bg
2
1, then (5) follows,
namely
max
z;b
1
;b
2
b ðb
1
þ b
2
Þ s:t: z
1
y
11
þ z
2
y
21
=y
11
þ b
1
1
z
1
y
12
þ z
2
y
22
=y
12
þ b
2
1
z
1
x
1
þ z
2
x
2
5x
1
z
1
; z
2
=0; b
1
; b
2
=0: ð9Þ
Thus the two models can be derived from each other
allowing us to use the ‘linear’ model in (5) to find the
optimal g
1
and g
2
. Solving (5) yields optimal b
1
and b
2
,
and with at least one b
i
[ 0 (if both are zero assign any
positive value to g
i
, i = 1, 2). To continue, let’s assume
that both b
i
’s are greater than zero, then we have
b ¼
b
1
g
1
¼
b
2
g
2
ð10Þ
This together with g
1
? g
2
= 1 can be used to solve for
optimal ðg
1
; g
2
Þ:
We have
g
1
g
2
þ 1 ¼
1
g
2
; and
b
1
b
2
¼
g
1
g
2
ð11Þ
thus
g
2
¼
1
b
1
=b
2
þ 1
¼
b
2
b
1
þ b
2
ð12Þ
and g
1
¼ 1 g
2
or g
1
¼ b
1
=ðb
1
þ b
2
Þ so g
1
þ g
2
¼ 1: If
one b
i
¼ 0; our conclusion still applies.
Thus by solving model (5) we can find the optimal
directional vector for each DMU or firm. It is straightfor-
ward to generalize the above results to the case of multiple
inputs and outputs as well as alternative input/output
orientations.
Finally, let us consider a simple numerical example with
two DMUs:
DMU 1 2
y
1
12
y
2
12
x 11
Based on these data, model (5) yields the following
results for DMU 1:
b
1
¼ b
2
¼ 1:
which we can use to solve for the optimal direction vector,
i.e.,
g
1
¼ g
2
¼ 1=2:
For DMU 2, which is efficient, model (5) yields b
1
¼
b
2
¼ 0; which implies that we may assign arbitrary
positive values to g
1
and g
2
.
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