EN :

The best known and probably the most frequently used of the models of this class is undoubtedly the Input-Output model of the Québec economy built, continuously updated and operated by the Bureau de la Statistique du Québec. However, the methodology, adapted and sometimes extended has found a number of other, in particular micro-economic applications.

Evidently, the basic inspiration of the methodology in question is to be found in the ideas put forward by Professor Leontief. Certain researches done in France, especially in the late 1950's, but also since then, have exerted considerable influence.

Although these models trace their origins to activity analysis in the sense that they start from the principle that to understand a complex system it is preferable to study in detail its inner structures and workings rather than the evolution over time of the great aggregates characterising the overall behaviour.

By abandoning the postulates of proportionality and of one-to-one correspondence between "products" and "industries", the models discussed here openly give up any pretence to mathematical elegance including the existence of "general solutions" of the kind of those associated with the Leontief inverses. They just become in effect simulation models and at the same time much more convenient and flexible frameworks for the collection, organization and the handling of data, data which are much closer to basic data than the highly processed data incorporated in the traditional Input-Output models. They are also much more easier to update and to incorporate "non-statistical" data.

Although more powerful, in many respects, than the traditional models, they share with them at least two basic weaknesses which, significantly, are not unrelated to each other. They are incapable of handling in a really comprehensive and systematic manner the confrontation of supply and of demand influences and they give no more than a most cursory treatment to the whole range of financial phenomena and a fortiori to the influence of these phenomena on the "real" ones. It is clear that the future work on this class of models will have to put heavy emphasis on trying to reduce these two weaknesses.

EN :

This article presents a methodology which draws heavily on the philosophy of the Input-Output models and having been made completely operational has already been used on three occasions in two countries for the purposes of regional development of construction materials industries.

This methodology, or more precisely the strictly formalized part of it is an extension of that of rectangular Input-Output models with modifiable coefficients. Thus, not surprisingly, it resembles fairly closely the approach by simulation, although the proposed model contains some simple optimizing sub-models. While obviously normative, these sub-models play a descriptive rôle in the model as a whole.

It is to be noted that the approach presented here can be applied to only one sector of the economy at a time. What is more, although capable of various extensions it will never be more than an auxiliary instrument destined to be used jointly with other analysis and planning instruments.

It is vital for any valid regional analysis not to restrict its investigations exclusively to what goes on in the region directly concerned. Even if the objective of the analysis is limited to a single region, one must take into account the interrelations between regions within the national economy and with foreign economies: important feedbacks affecting the region concerned may on occasion travel far beyond its limits before returning. The type of a model presented here, thanks to a great number of interrelations of which it can systematically keep track may turn out to be particularly useful here.

EN :

The author shows how, under certain conditions, results from the Quebec Input-Output model can be aggregated to obtain familiar components of the National Accounts and to derive income multipliers. These multipliers are also discussed and compared to other types of multipliers.

EN :

This paper describes a simple cost-push price model which has been developed at the Structural Analysis Division of Statistics Canada.

This price model is a traditional input/output cost-push model which has been adapted to utilize the rectangular industry by commodity input/output tables for Canada. It can be considered as the "dual" of the output model. Instead of analysing the propagation of demand through the economic system, the price model serves to analyse the propagation of factor prices throughout the system.

The purpose of such a price formation model is to determine the impact on industry selling prices and domestic commodity prices arising from a change in impart commodity prices and primary input prices.

This price model is of a static type; it accepts no substitutions and its structure is quite rigid. It is considered as being an annual model although it can be used for a different time period.

This model is fully operational and is widely used by many government and private agencies.

EN :

The most simple rectangular input-output models use two rectangular matrices: R a market coefficient matrix, A* a production coefficient matrix. A given exogenous demand X_o determines the sectorial activity levels X* = [I — RA*]^-1X_o. We assume that A* is random with expectation A. We study the distribution of the "error" X* — X with X = [I — RA]^-1X_o.

(1) For the statistically independent elements of A*, we analytically prove that X < EX*.

(2) In the more realistic case of statistically dependent elements of A*.

(a) One submatrix of A* with T non zero elements is chosen. The probabilistic model which generates the T coefficients is as follows: a* = (1 — μ)a + μ(S/n) b* où a* is the vector of the T random elements, a is the expectation of a* whose components are observed values of a real input-output model, S is the sum of components of a, μ is a parameter between zero and one, b* is a multinomial random vector with T components and parameters n, number of drawings during an experiment, and a/S, the corresponding probabilities.

We control the variability of a* through μ and n. For a given experiment, we get a realisation of A* and we compute X*. K independent experiments allow us to estimate the expectation and the variance-covariance matrix of X*, simultaneous confidence intervals for the expectation of the components of X*, and also a few global measures of errors on X*.

The Canadian model for 1961 (16 productive sectors, 40 commodities), is tested with that model.

The main result is: the relative errors, measured according to the variation coefficients, are greatly reduced when we pass from the "errors" on a* to the corresponding "errors" on X*.

(b) The same random model is also simultaneously applied to 2 or 3 sub-matrices of A*.

EN :

This article describes a model, developed by the Structural Analysis Division of Statistics Canada, that helps analyse the economic implications of policy decisions in the environment of a supply-constrained economy. The Canadian input-output model is modified to introduce constraints on the uses of some commodity or industry products. These constraints take the form of limits on the availability of commodities for some uses, constraints that ensure that some minimum levels of final demand for each commodity are satisfied, and capacity constraints on the outputs of industries. Given these constraints, a linear function of the activity levels is maximized. The resulting solution gives a vector of activity levels, and also corresponding final demands that are optimal in terms of the objective function.

The use of the model is illustrated by analyzing the 'optimal' allocation of industrial outputs in the face of a reduction in the availability of the commodity, 'crude mineral oils', for industrial uses. Two objective functions are used: total employment, and total wages, salaries and supplementary labour income. For each objective function, a ranking of the industries is defined by the solutions of the model.

Experience with this model leads us to conclude that it is useful in indicating which industries are of primary interest in a specific shortage situation, rather than in setting exact values of cutbacks to impose on industries. In the conclusion, relaxation of the major assumptions underlying the model and some possible extensions are discussed.