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Linear Programming Estimates of Changes in Input Coefficients*

Published online by Cambridge University Press:  07 November 2014

T. I. Matuszewski ,
P. R. Pitts  and
John A. Sawyer
T. I. Matuszewski
Affiliation:
Université de Montréal
P. R. Pitts
Affiliation:
Dominion Bureau of Statistics
John A. Sawyer
Affiliation:
University of Toronto
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Extract

The purpose of this study is to develop a procedure for estimating changes in the coefficients of an inter-industry input-output matrix. Every attempt to prolong the useful life of such a set of coefficients carries an obvious practical interest. We desire to find a procedure which does not call for large quantities of supplementary data, and which does not require more than a reasonable volume of calculations.

The method is applied to a sixteen-industry Canadian input-output matrix for the year 1949. This matrix is an aggregation of the forty-two-industry table published by the Dominion Bureau of Statistics. The adjustment of the coefficients (i.e., their up-dating) is done for 1956. Thus, the objective of the study is to estimate the 1956 matrix of input coefficients.

Projections of the output of each of the sixteen industries for the years 1950–59 using the unadjusted 1949 matrix of input coefficients and estimates of the actual final demand for these years produce discrepancies between the projected outputs and estimates of actual output which quickly become too large to be tolerated. The projection errors for 1957–59 are shown in columns 3, 7, and 11 of Table I, where the 1949 matrix is referred to as matrix A. It is the fact that the quality of these projections deteriorates as one moves away from the base year that led us to examine various methods of up-dating the base-period coefficients. Although at first glance it may seem inappropriate to use a matrix for projections eight to ten years away from the year to which it refers, it should be kept in mind that input-output tables are frequently at least five years out of date by the time they are published. For example, it is unlikely that the table for 1961 presently being constructed by the Dominion Bureau of Statistics will be published before 1967.

Type
Articles
Information
Canadian Journal of Economics and Political Science/Revue canadienne de economiques et science politique , Volume 30 , Issue 2 , May 1964 , pp. 203 - 210
Copyright
Copyright © Canadian Political Science Association 1964

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Footnotes

*

This is a revised version of a paper presented at the meetings of the Econometric Society at Ann Arbor, Michigan, on September 10, 1962. Part of the work for this paper was done at the Institute for Economic Research at Queen's University, Kingston, Canada, during the summers of 1960 and 1961. Assistance was also received from the Canada Council. Numerical calculations were made at the Centre de Calcul de l'Université de Montréal. We would like to thank the personnel of the Dominion Bureau of Statistics, and especially Miss B. J. Emery. Without their assistance the research presented here could not have been carried out.

References

1 Some simpler procedures than the one described in this paper are described by the present authors in “L'ajustement périodique des systèmes de relations inter-industrielles, Canada, 1949–1958,” Econometrica, XXXI, 01–April, 1963, 90–110 Google Scholar, and in “Alternative Treatments of Imports in Input-Output Models: A Canadian Study,” Journal of the Royal Statistical Society, Series A, CXXVI, no. 3(1963), 410–32.Google Scholar

2 Dominion Bureau of Statistics, Research Projects Section, “Sixteen Industry Aggregation” (Ottawa, 02, 1960)Google Scholar, obtainable on request from Mr. P. R. Pitts.

3 DBS, Publication 13–513, Supplement to the Inter-Industry Flow of Goods and Services, Canada, 1949 (Ottawa, 1960).Google Scholar

4 These projection errors are lower than those published previously by the authors because, in the meantime, better estimates of final demand and total industry output have become available. Moreover the earlier figures excluded industries 14 and 16 for which, until recently, no total output estimates were available. The estimates of final demand for 1957–59 include unallocated output as the same percentage of total final demand as in 1949. The estimates of final demand for 1959 were made prior to the revisions made to the DBS National Accounts in 1963.

5 In previous papers by the present authors, a circumflex (^) was placed over the symbol for a flow or a coefficient to indicate that it included intra-industry consumption of the output of an industry. In this paper all flows and coefficients include intra-industry consumption, but the circumflex has been omitted to simplify the notation.

6 K. J. Arrow and Marvin Hoffenberg also introduced constraints of this type in a similar situation. See their A Time Series Analysis of Interindustry Demands (Amsterdam, 1959), 55, 64–6.Google Scholar

7 See ibid., 56, and M. Simmonard, Programmation linéaire (Paris, 1962).

8 The method used for this paper is the method using concepts of “implicit costs” and “key values” described by Bowman, E. H. and Fetter, R. B., Analysis for Production Management (Homewood, Ill., rev. ed., 1961), 123–5.Google Scholar Thus, for a given basic solution z(X ij + = ui + vj = −z(X ij −) where ui and vj are shadow prices. See also Wagner, H. M., “On a Class of Capacitated Transportation Problems,” Management Science, V, 04, 1959, 304–18.CrossRefGoogle Scholar Summaries of the literature of the transportation problem may be fauna in Ford, L. R. Jr. and Fulkerson, D. R., Flows in Networks (Princeton, 1962), chap. 3Google Scholar; Hadley, G., Linear Programming (Reading, Mass., 1962), chap. 9–10Google Scholar; and Vajda, S., Mathematical Programming (Reading, Mass., 1961), chap. 6.Google Scholar

9 Dantzig, G. B., “Upper Bounds, Secondary Constraints and Block Triangularity in Linear Programming,” Econometrica, XXIII, 04, 1955, 174–83.CrossRefGoogle Scholar

10 There are 256 coefficients to be estimated and, hence, apparently 512 upper bound constraints since they are imposed on X ij + and X ij − separately. However, five of the coefficients were blocked” and thirty-eight of the base-year coefficients were zero and could not change without making the objective function infinite. This leaves 2(256-5-38) = 426 upper bound constraints which could become effective.

11 DBS, publication 61–505, Indexes of Real Domestic Product by Industry of Origin, 1935–61 (Ottawa, 1963).Google Scholar

12 These estimates were published in a paper by the present authors, “Inter-industry Estimates of Canadian Imports, 1949–1958” in Canadian Political Science Association, Conference on Statistics, 1961—Papers, Hood, Wm. C. and Sawyer, J. A., eds. (Toronto, 1963).Google Scholar Since then the estimates of exports by industry groups have been revised.

13 DBS, publication 13–201, National Accounts, Income and Expenditure, 1960, Tables 5 and 48. Some unpublished estimates were also available.

14 Our input-output models exclude the column and row for unallocated output and input from the system ot interdependence. In order to maintain the equality of the total value of final demand and of primary input, the “unallocated” output has been treated as if it were part of final demand.