Geoscience Reference
In-Depth Information
what we will refer to as either its
impact analysis
or
planning
forms, which we
define as the following:
*
éù
=
ê
ê
ëû
*
éù
=
ê
ê
ëû
D
x
*
Impact Analysis Form
:
x
¼
Hf
where
H
and
x
and
D
¼
DL
L
x
*
éù
=
ê
ê
ëû
x
é
D
ù
Planning Form
:
x
¼
Gx
%
where
IA
and
x
G
= ê
ú
(
-
)
f
ë
û
The impact analysis form is most traditionally considered in input-output
applications where the question is what industry outputs and factors associated
with interindustry activity, such as energy use, environmental pollution levels and
employment, result from a given schedule of final demands presented to the
economy, as in Johnson and Bennett (
1981
), Hannon et al. (
1983
) and many others.
However, the planning form has advantages in applications where one seeks to
optimize an objective other than the objective implicit in a traditional input-output
model.
10
In the following we explore an example of extending this framework to
planning applications.
For the planning form, the basic input-output framework can be posed the
solution of a system of
n
linear equations in
n
unknowns, which is certainly one
of the most attractive features of the framework—a straightforward and unique
solution. Equivalently, but useful as a starting point, is posing the generalized
input-output model as an optimization problem. In general, this means relaxing
fundamental assumptions in the framework in order to adapt it to specific situations,
such as allowing technical coefficients to vary as a function of relative prices in the
case of econometric extensions to the basic model, adding capital coefficients in
dynamic input-output models, or adding trade coefficients in multi- or interregional
models, as examples. Also, in using input-output in planning applications, where
one seeks to optimize (maximize or minimize) some objective function related to
interindustry activity, it is useful to begin by formulating the input-output analysis
framework as a simple linear programming problem.
We start with the generalized input-output formulation presented in its planning
é
D
ù
form,
, where
D
is the matrix of direct impact coefficients relating
%
xGx
=
ê
x
ú
(
IA
-
)
ë
û
factors such as energy use, pollution emissions, and employment to industry output,
*
éù
==
ê
ê
ëû
x
xGx
f
*
%
i.e.,
, where
x
represents the levels of total impact of energy use,
pollution, and employment associated with output
x
and, of course, final demand
f
.
We use
to represent the more generalized structural relationships governing not
only the Leontief production possibilities but also the levels of energy consumed,
pollution discharged and employment generated that are associated with those
production possibilities. Hence, if we define
G
v
as the vector of value added
10
We will see later that the implicit objective function in an input-output model is to maximize the
sum of all final demands or, equivalently, to minimize the sum of all value added inputs.
