Geoscience Reference
In-Depth Information
*
, by defining a new vector of total impacts and concatenating the
vector of final demands with
total impacts,
x
*
x
to obtain this expanded vector of
total
impact,
*
éù
=
ê
ê
ëû
x
x
%
. We can similarly expand the matrix of direct-impact coefficients by
f
concatenating
D
and (
I A
) to yield a new expanded direct-impact coefficients
D
G
IA
. This formulation is particularly well suited to using input-
output with mathematical programming models.
8
As an alternative to the preceding formulation, we may wish to express total
impacts as a function of final demands in much the same way we traditionally use
the Leontief inverse. That is, we may find the total impacts in terms of energy,
pollution generation and employment associated with some given level of final
demand. In this case we can write our earlier expression for total impacts,
é
ù
matrix:
=
ê
ú
(
-
)
ë
û
*
x
¼
Dx
,
*
*
*
¼
DL
is the matrix of total-impact coefficients.
9
equivalently as
x
¼
D
f
, where
D
*
as a function of
f
while in
x
*
*
as a
In this case we compute
x
¼
Dx
we computed
x
function of
itself
in our vector of total impacts. This can be accomplished easily by concatenating
x
x
. Finally, once again for convenience, we may wish to include
x
*
with
with the vector of total impacts in the same manner we concatenated
x
f
in
*
éù
=
ê
ê
ëû
x
constructing
x
, i.e., we define a new expanded vector of total impacts to be:
x
.
x
As before with
, we can similarly expand the total-impacts coefficients matrix by
concatenating the Leontief inverse with the total-impact coefficients; we call the
G
*
éù
=
ê
ê
ëû
D
new expanded total-impacts coefficients matrix,
H
, so that
H
.
L
The expressions for
x
and
x
are equivalent descriptions of the same situation, of
course, since
x
and
f
uniquely define one another in a Leontief model—for every
given
, and vice versa. Note also that we can create a
matrix of impacts generated by each industry separately by
f
, there is one and only one
x
Hf
where
f
designates
the elements of
f
placed along the diagonal of a square matrix.
13.5
Summary: Generalized input-output Formulations
Recall that the generalized input-output model becomes possible with a set of
direct impact coefficients,
[
d
kj
], each element of which is the amount of an
impact variable
k
, for example, pollution or energy, generated per dollars' worth of
industry
j
's output. Using
D
¼
D
we can pose the generalized input-output model in
8
As examples, see Thoss (
1976
) and Blair (
1979
).
9
This formulation was originally applied by Just (
1974
) and Folk and Hannon (
1974
) to examine
the impacts of new energy technologies. Other more recent applications are summarized in
Forssell and Polenske (
1998
), including, in particular, Qayum (
1991
and
1994
), Sch¨fer and
Stahmer (
1989
) and Lang (
1998
).
