378.752
D34
,
W-96-2
Profit,
Directional
Distance
Functions,
And
Nerlovian
Efficiency
R.G.
Chambers,
Y.
Chung,
and
R.
Fare
Waite
Library
Dept.
of
Applied
Economics
University
of
Minnesota
1994
Buford
Ave
-
232
OaOff
St.
Paul
MN
55108-6040
USA
WP
#96-02
Scientific
Article
No.
,
Contribution
No.
,
from
the
Maryland
Agricultural
Experiment
Station
U
tf\div
s
'
Q.s
tr,t,r-cia_
3
7/
752
D
get
(
4)
—
PROFIT,
DIRECTIONAL
DISTANCE
FUNCTIONS,
AND
NERLOVIAN
EFFICIENCY
by
R.G.
Chambers,
Y.
Chung,
and
R.
Fare
.
1.
Introduction
Quite
some
time
ago,
Nerlove
(1965)
suggested
a
relative
efficiency
measure
based
on
profit.
The
essential
idea
behind
Nerlove's
(1965)
efficiency
measure
is
to
decompose
profit
maximization
into
two
stages:
In
the
first,
profit
is
maximized
for
a
given
production
function,
while
in
the
second
stage,
the
maximum
maximorum
of
profit
is
found
by
maximizing
over
all
possible
production
functions.
Overall
efficiency
is
then
judged
by
comparing
observed
profit
for
a
decisionmaking
entity
to
the
maximum
maximorum
profit.
And,
following
Farrell
(1957),
this
overall
efficiency
measure
is
decomposed
into
two
subsidiary
measures:
a
measure
of
price
or
allocative
efficiency
which
consists
of
comparing
observed
profit
to
the
profit
function
for
the
siven
production
function
and
technical
efficiency
which
measures
the
difference
between
the
profit
function
for
the
given
production
function
and
maximum
maximorum
profit.
A
particularly
striking
aspect
of
Nerlove's
(1965)
efficiency
measure
is
that,
unlike
virtually
all
other
existing
efficiency
measures,
it
is
expressed
in
difference
(as
opposed
to
ratio)
form.
Perhaps
this
explains
why
Nerlove's
(1965)
contribution,
despite
its
obvious
intuitive
appeal,
has
remained
dormant
for
so
long.
Apart
from
a
passing
reference
by
Lau
and
Yotopoulous
(1971),
the
profession
has
apparently
ignored
this
contribution.
The
purpose
of
this
paper
is
to
revisit
and
revitalize
Nerlove's
efficiency
measure
in
a
more
modern
framework
that
allows
for
multiple
inputs
and
multiple
outputs
in
a
natural
manner
while
using
a
representation
of
Financial
support
from
USEPA
is
acknowledged.
1
swi
the
technology
for
which
difference
(as
opposed
to
ratio)
measures
are
the
most
natural
measures
of
relative
efficiency.
That
representation
is
the
directional
technology
distance
function
which
generalizes
Luenberger's
(1992b,
1995)
shortage
function
and
Blackorby
and
Donaldson's
(1980)
translation
function.
The
directional
technology
distance
function
is
so
general
that
it
can
be
shown
to
encompass
all
known
distance
function
representations
of
the
technology
as
special
cases.
The
paper
proceeds
as
follows:
In
the
next
section,
we
introduce
the
directional
technology
distance
function
and
discuss
its
relationship
to
other
functional
representations
of
the
technology
(input
and
output
distance
functions,
McFadden's
(1978)
gauge
function,
and
the
directional
input
distance
function
(Chambers,
Chung,
and
Fare
(1995)).
Among
other
things,
we
show
that
the
directional
technology
distance
function
is
a
complete
function
representation
of
a
technology
exhibiting
free
disposal
of
inputs
and
outputs.
After
that,
we
discuss
the
dual
relationship
between
the
directional
technology,
distance
function
and
the
profit
function
while
providing
a
streamlined
proof
of
the
dual
correspondence
between
the
two.
We
also
show
that
this
dual
correspondence
has
all
previous
dual
correspondences
as
special
cases.
Then
we
take
up
efficiency
measures
and
show
how
the
directional
technology
distance
function
can
be
used
to
represent
the
Nerlovian
efficiency
measure.
The
final
section
concludes.
2.
Directional
and
Radial
Distance
Functions
In
this
section
we
define
and
contrast
Shephard's
radial
distance
functions
and
McFadden's
gauge
function
to
three
directional
distance
functions.
Shephard's
input
and
output
distance
functions'
respectively,
measure
the
largest
radial
contraction
of
an
input
vector
and
the
largest
radial
I
See
Shephard
(1953,
1970).
2
7
...
expansion
of
an
output
vector
consistent
with
each
remaining
technically
feasible.
McFadden's
gauge
function
measure
the
largest
radial
expansion
of
a
netput
vector
consistent
with
feasibility.
The
directional
distance
function
measures
the
size
of
an
input
and
or
output
v-!ctor
radially
from
itself
to
the
technology
frontier
in
a
preassigned
direction.
This
direction
can
differ
from
the
radial
direction
out
of
the
origin,
thus
making
the
directional
distance
function,
more
general
than
Shephards'
distance
functions
or
McFadden's
gauge
function.
The
directional
distance
functions
that
we
analyze
are
related
to
or
derived
from
the
shortage
and
the
benefit
functions
introduced
by
Luenberger
2
and
Blackorby
and
Donaldson's
(1980)
translation
function.
MI
functions
are
here
defined
in
terms
of
a
production
technology.
Let
x
e
M
N
+
denote
a
vector
of
inputs
and
y
E
91,
m
a
vector
of
outputs,
the
technology
T
is
given
by
(2.1)
T
=
{(x,
y)
such
that
x
can
produce
y).
In
this
paper
we
make
the
following
assumptions
on
T.
Ti
T
is
closed.
T.2
Input
and
outputs
are
freely
disposable,
i.e.,
if
(x,
y)
€
T
and
(x',—y')
__
(x,
—
y)
then
(x',y
1
)
ET.
T.3
There
is
no
free
lunch,
i.e.,
if
(x,
y)
e
T
and
x
=
0
then
y
=
0.
T.4
T
is
convex.
The
first
three
properties
are
imposed
throughout
the
paper,
while
convexity
is
only
assumed
when
we
discuss
dualities.
All
assumptions
are
standard
and
need
no
further
comments.
3
.,
-
See
Luenberger
(1992a,
1992b,
1994a,
1994b,
1995).
3
For
these
and
other
axioms
on
the
technology.
See
Fare
(1988).
3
t
Shephard's
input
and
output
distance
functions
are
defined
in.
terms
of
T
as
x
(2.2)
D
1
(y,x).sup{X>0:(—
a,
,y)el},
x
e91
1
4
'
1
.,y
0
-
l
i
ly
!.
).
and
(2.3)
D
0
(x,y)=1{0>
0:(x,y
/0)
ET},
x
E91
1
4
4
.,y
€91!‘
ii
,_
respectively.
Each
of
the
distance
function
is
a
complete
characterization
of
the
technology
T,
4
i.e.,
(2.4)
D
i
(y,
x)
?.
1
<r>
(x,
y)
ET,
(2.5)
D
o
(x,
y)
..
1
c>
(x,
y)
€
T.
Under
constant
returns
to
scale
the
following
simple
relation
holds
for
the
distance
functions:
(2.6)
D
i
(y,
x)
=
1/D
0
(x,
y).
McFadden's
gauge
function
is
defined
in
terms
of
outputs
with
inputs
entering
with
a
negative
sign
and
outputs
with
a
positive
sign.
Define
the
'mirror'
technology
T
=
{(-x,
y):
(x,
y)
e
T},
then
McFadden's
gauge
function
can
be
written
as
(2.7)
H(
-x,
y)
=
inf
{9
>
0:
(-
)
S
1
E
1"
-
}
x
e91
14
+
,y
E91
+
m
.
0
'
0
The
most
general
directional
distance
function
scales
inputs
and
outputs
simultaneously.
This
differs
from
the
definitions
of
the
above
distance
function.
The
input
distance
function
is
defined
by
scaling
inputs
and
the
output
distance
function
is
defined
by
scaling
outputs.
4
Fare
and
Primont
(1993,
pp.
15,
22)
show
that
weak
disposability
of
inputs
and
outputs
is
necessary
and
sufficient
for
the
input
and
output
distance
functions
to
completely
characterize
technology.
4