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Many nations, large and small, use an Input-Output Model to study the interdependence of the production plans and activities of the many industries which constitute an economy. But few, if any, have used this methodology for forecasting purposes.
This paper outlines a procedure for extending the static (open-loop and time independent) model to a dynamic and recursive (closed loop and time dependent) model suitable for forecasting. The procedure begins with the definition and analysis of the open-loop static model(i.e. the existing input-output model) which is then modified and converted to a closed-loop (recursive) model which is suitable for forecasting.
This conversion is done gradually, introducing only one possible change at a time, thus generating eight (8) total possible models. The cost associated with each of these eight models is also derived and it is found that the cost of the full recursive model is at least twice the cost of the simple static model.
Can such a cost be justified? The answer to this question depends on the ability of the revised input-output methodology to provide useful and credible forecasts.
Introduction
Input-Output analysis seeks to take into account the interdependence of the production plans and activities of the many industries which constitute an economy (4). This interdependence arises out of the fact that each industry employs the outputs of other industries as its raw materials. Similarly, its output is often used by other producers as a productive factor, sometimes by those very industries from which it obtained its ingredients.
Suppose:
[X.sub.i] = quantity of good i produced by industry i in a given period
[Y.sub.ij] = number of units of good i required by industry] in the given period
[b.sub.i] = exogenous demand for good i for a given period.
Therefore, in order for industry i to meet the demands on it, it must produce at least [X.sub.i] units, where:
[X.sub.i] = [Y.sub.i1] + [Y.sub.i2] ... + [Y.sub.in] + [b.sub.i] (1)
For industry j to produce [X.sub.j] units it would require inputs from other industries (i, j, k, ...). The number of units of goods that it requires from these industries depends, to a large extent, on the technology of these industries (2).
The basic assumption of Input-Output Analysis, for which we are indebted to Professor Leontief, states that: "... in any productive process all inputs are employed in rigidly fixed proportions and the use of these inputs expands in proportion with the level of output." Another way of stating this is to say that "the amount of good i required to produce good j is directly proportional to the amount of good j produced." These statements imply that
[Y.sub.ij] = [a.sub.ij][X.sub.j] (2)
where:
[a.sub.ij] = constant of proportionality, depending on the technology of industry j.
Basically, the Input-Output Analysis consists of nothing more complicated than the solution of a set of n simultaneous linear equations in n variables. These equations come about when the basic assumption, as expressed by equation (2), is substituted into equation (1). The result is (1), (8), (13):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
or: (I - A)X = B (4)
X = [(I - A).sup-1]B (5)
The model described above is static since everything is rigidly fixed, if the [a.sub.ij] coefficients are constant. It is a useful tool in understanding the interactions between the many industries of an economy. However, except for small time periods, it is quite likely that the coefficients of the model will change, even though the extend and rate of this change may not be known because not enough pertinent data may exist for measuring this change. As is, the model allows only a single look at the interactions among the segments of an economy. To obtain more looks we need more data, at different times.
If new sets of [a.sub.ij] coefficients and exogenous demands ([b.sub.i]) can be obtained, the model can be solved several times to obtain updated solutions. But the model remains open since no feedback is possible under the assumptions of the static model. However, the need for more data, implies increased cost.
To convert the open-loop model into a closed-loop model, suitable for forecasting, the output at time (t + 1) must be related to the output at time t. This can be accomplished by introducing the concept of "Stocks of Capital Goods" which, essentially, allows the buildup of Inventory, something that represents a departure from the underlying assumptions of the static model (12).
The static model can also be extended in other directions, thus converting the "simple static" model into a "complete recursive" model, but with a correspondingly higher cost.
Before we proceed with the development of the extended models it is necessary to pause briefly and derive the cost function associated with the static model. This cost function will serve as the reference against which the cost of the extended models will be compared to.
1. Cost Associated with the Static Model
The solution of equation (5) is procedural (9), (10), (11) once the [a.sub.ij] and [b.sub.i] coefficients have been evaluated, even if the size of the matrix which needs to be inverted is very large. The major cost of the static model is due to the difficulty associated with the determination of the technological coefficients [a.sub.ij] and the exogenous demands [b.sub.i] (3).
To obtain an estimate of the cost of the static model, let us define the costs [C.sub.ij] and [C.sub.i] where:
[C.sub.ij] = Cost of obtaining the technological coefficient [a.sub.ij]
and
[C.sub.i] = Cost of obtaining the exogenous demand [b.sub.i] for industry i
Then, the cost of obtaining all the technological coefficients can be defined by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Similarly the cost of obtaining all the exogenous demands can be defined by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The total cost of the static model is equal to the sum of the costs of the technological and exogenous demand coefficients and it, therefore, is given by:
[C.sub.total cost] = [n.summation over (i = 1)][n summation over (j = 1)][C.sub.ij] + [n.summation over (i = 1)][C.sub.i] (8)
If we now make the assumption that [C.sub.ij] = [C.sub.a] for all i and j and [C.sub.i] = [C.sub.b] for all i, then equation (8) can be rewritten as:
[C.sub.total cost] = [n.sup.2][c.sub.a] + n[C.sub.b] (9)
= n(n + 1)C, if [C.sub.a] = [C.sub.b] = C (10)
The last equation, which represents the cost of the static model under the stated assumptions, will be used as the reference against which the cost effectiveness of the extended models will be compared to.
2. Extensions to the Static Input-Output Models
To render the static model more useful for prediction, it can be modified by relaxing the "fixity" conditions of the technological coefficients. Also the model can be formulated in a "closed-loop" form so that future predictions depend on past predictions, thus resulting in a recursive model (15). In this paper the extensions to the static model proceed in three directions:
* The technological coefficients [a.sub.ij] are made function of time (T)
* The technological coefficients [a.sub.ij] are made random variables (R)
* The concept of "stock of capital goods" is introduced, which converts the "open-loop" static model into a "closed-loop" and recursive one suitable for forecasting (S).
The extensions can be made in any of these three directions, or combinations of them. Since there are three possible areas of extension with two levels each (i.e., present in the model; not present in the model), there are [2.sup.3] = 8 possible models. Using the letter symbols for each of the three model extension directions, the eight possible models are (5), (6), (7):
1)--present static model
2)T [a.sub.ij] = f(time)
3)R [a.sub.ij] is a random variable
4)R T [a.sub.ij] = f(time) & [a.sub.ij] is a random variable
5) S "simple" recursive model: "stocks of Capital Goods"
6) S T "simple" recursive model with [a.sub.ij] = f(time)
7) S R "simple" recursive model with [a.sub.ij] a random variable
8) S R T "complete" recursive model
3. Other Open-Loop Models and their Costs
Model 1 is the present static model which has been discussed in some detail above. Models 2, 3 and 4 represent various assumptions on the [a.sub.ij] coefficients which are shown on Exhibits 1 and 2.
[GRAPHIC OMITTED]
[GRAPHIC OMITTED]
Since we live in an era of great technological change in which manufacturing and industrial processes change continuously, it is apparent that the assumption of fixed [a.sub.ij] is not very realistic, especially when the period of prediction is relatively long. The [a.sub.ij] can be made functions of time and/or converted to random variables. This implies that functional relationships have to be obtained for each [a.sub.ij] Changes in the coefficients over time are mainly due to technological change and changes in consumer tastes which exert an effect on product mixes. The time variable should capture both these and other changes that may be introduced. Suppose that for each a. we take p measurements over a period of time. Then Least Square estimation can be used to find [a.sub.ij] (t) = a + [b.sub.t], as a function of time. To find the cost associated with this model we note that we need to evaluate matrix A, p times. Therefore, the total cost of this model is:
TOTAL COST = ([n.sup.2][C.sub.a])p + n[C.sub.b] = n(np[C.sub.a] + [C.sub.b]) (11)
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