input-output

Environment and Planning A, 1985, volume 17, pages 747-759

Shortcut 'input-output' multipliers: a requiem

R C Jensen Department of Economics, University of Queensland, St Lucia, Brisbane, QLD 4067, Australia

G J D Hewings Regional Science Program, University of Illinois at Urbana-Champaign, Urbana, IL 61801-3682, USA Received 21 April, 1983; in revised form 12 March 1984

Abstract. In this paper a critical review is presented of recent attempts to develop shortcut inputoutput-type multipliers from a variety of nonsurvey techniques. Particular attention is focused on the underlying theoretical assumptions and the dangers involved in adopting techniques which are not consistent with the underlying theoretical foundation of input-output models.

1 Introduction In recent years a number of shortcut approaches to multiplier calculation have been suggested. These can be classified into two categories, namely (1) nonmatrix multipliers, which can be developed in the absence of any input-output tables (although frequently categorized by their sponsors as input-output multipliers), and (2) matrix multipliers, which are derived from matrices with incomplete information. Category 2 includes the group of approaches generally known as nonsurvey inputoutput approaches. An attempt is made in this paper to provide a brief review of the more representative contributions to the literature on 'shortcut' multipliers and to provide some evaluation of these methods as contributions to the methodology of regional science.

2 A review of nonmatrix shortcut multiplier development In this review we will examine the published literature on nonmatrix shortcut multipliers, in an attempt to trace the development of the concept and to provide a taxonomy of approaches. Let: A represent the n x n input-output coefficient matrix, a typical element of which is aif; let B (by e B) represent the associated Leontief inverse matrix, (I - A)" 1 (where I is the identity matrix). Then, two column sums may be defined: W}, = E #,y, and jUj = X by. W;- is thus the sum of the direct or firsti

i

round input coefficients, whereas JU; is the more familiar column output multiplier since it represents the sum of the direct and indirect coefficients associated with a unit change in final demand in the industry which identifies column y.

Let

W = Z Wj/n, that is, the average column sum for the direct input coefficient matrix. In general terms, most approaches have in common the estimation of output multipliers jiij from estimates of total direct or first-round input-output coefficients Wj, or from variables related to Wj. The approaches differ in the method for converting the Wj into the jXj, and two stages, outlined in detail below, are recognizable in this respect. Stage 1 refers to efforts by earlier writers to develop methods to derive or approximate multipliers by simple tests of association between Wj (or related variables) and observed multipliers ptj. Stage 2 refers to efforts by more recent writers to propose more complex relationships between the Wj and the jUj in an attempt to develop more reliable formulae for the calculation of multipliers. One further feature in common with most writers in this area is the general belief that the input-output format provides a suitable format for accuracy testing of

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R C Jensen, G J D Hewings

shortcut multipliers. Commonly, the multipliers developed by the proposed formulation are calculated from the Wj of a particular input-output table and compared with the multipliers jUj derived by the normal process of matrix inversion of the same table. It will be argued below that, although the input-output format provides a suitable testing venue for shortcut multipliers, the use of this venue does not qualify the shortcut multiplier as an 'input - output' multiplier. This claim, which is frequently implicit in much of the writing on shortcut multipliers, implies also some 'authentication-by-association' with the input-output technique. 2.1 Stage 1: simple association tests between Wj and ju; The first two approaches were developed independently by Bromley (1972) and Salcedo (1972). Bromley's intent was simply to demonstrate "that a considerable amount of information about the economic structure of rural economies can be discerned without the implementation of a complete input-output study" (page 125). Bromley defined two indices: an index of internal purchases, defined as Wj above, and an index of internal sales, Sh defined as the proportion of total output of the ith sector sold for intermediate use. These indices were applied as required to illustrate three important properties of an economy, namely (1) a simple four-quadrant representation of what could be termed forward and backward linkages, using both indices, (2) triangularization, using Wj, and (3) relevant to this paper, the use of Wj as a "reasonable substitute for the derivation of the traditional business income multiplier" (page 129, our italics). Bromley's "traditional business income multiplier" is the output multiplier in conventional input-output terms. Having established the theoretical relationships between the Wj and the pij as the sums of the columns of the A and inverse matrices, respectively, Bromley used simple regression analysis on data from eight input-output studies ranging from smaller than a county to an entire state, to test the relationship between the Wj and the jUj. The intercept values of the regression equations were close to 1, and the values of the coefficient of determination (r2) were high. On this basis Bromley concludes that "the index of internal purchases could be used to rank sectors and it would seem safe to conclude that this ranking would not be too different from a ranking made on the more traditionally derived multiplier" (page 132). The usefulness of the Wj as a substitute for the ju; is established in the Bromley paper as indicating the probable ranking of sector multipliers in an input - output table and not the size of the sector multipliers. Bromley's regression coefficients range from 1.1 in the West Virginia table to 2.4 in the Southwestern Wyoming table, indicating a quite substantial range in the extent to which jUj varied with Wj from one table to another. Bromley is not concerned to estimate the multiplier size [ft-) in his approach. Salcedo (1972) was the first to speculate on the nature of the association between the multipliers and the characteristics of the input-output table. Salcedo began by observing the existence of an inverse relationship between the "proportion of leakage from an economic sector through final payment" and the (final demand) output multiplier for that sector, and that this relationship was a "well-defined linear one", with little scatter about a best-fit line (page 3). Salcedo then noted that if this linear relationship "truly holds", the multiplier for an economic activity could be estimated with information only on the proportion of payments leading from the economy. Salcedo's approach therefore used a linkage index for each sector, say /y (= 1 - Wj), or 1 minus Bromley's index. To analyze the impact of a proposed factory, "one merely has to determine the fraction of total payments for inputs going to taxes, saving, and depreciations, and imports to be able to get the corresponding multiplier from a plot" (page 3).


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Salcedo demonstrated the close fit of multipliers to the 'Salcedo line' (this name was coined by Nelson and Perrin, 1978) by plotting data relating to a thirteen-sector and a 183-sector model of the Texas economy, commenting on the "uncanny" (page 4) quality of the best-fit line. To describe the Salcedo line for any economy, Salcedo chose simply to define points at each end of the line. If the proportion of total inputs leaking from the economy was at the maximum of 1.0, the multiplier must similarly be 1.0; one endpoint is therefore fixed by these coordinates. At the other extreme, where no leakage occurred the multiplier was seen as "a reflection of the operation of the entire-economy" (page 4). Once / had been determined for an economy and the Salcedo line formed, one simply needed to 'read off' the multiplier corresponding to the /y for the sector in question. It was noted, however, that the method could underestimate or overestimate multipliers for some sectors, particularly where the lj value of sectors providing direct inputs differs significantly from lr Nelson and Perrin (1978) undertook some accuracy tests of the Salcedo method by comparing (from the Texas model for 1967 and 1972) multipliers developed by the Salcedo method with conventional input-output multipliers and with multipliers derived by a least squares line fitted to conventional multipliers. These findings showed that, although the magnitude of the differences between the Salcedo-based and regression-based multipliers with the conventional multipliers was quite similar, the Salcedo estimates yielded consistently smaller estimates. Yet Nelson and Perrin concluded that the short-run methods "can yield final demand multipliers which are in quite close agreement with those derived from traditional input-output analyses" (page 91). It is useful to consider further some aspects of the Salcedo approach. It was shown above that Salcedo used the inverse of the "proportion of leakage from the entire economy", or 1//, where / is the average leakage per sector and / = 1 — W. To find the multiplier for each sector juf Salcedo in effect solved the equation: 1 W,u,= 1 + W; = = 1+ *= . ^ ' 1- W 1-W This corresponds directly to the equation developed later by Burford and Katz (1977a) and described in more detail in the next section of this paper. 2.2 Stage 2: more complex relationships between W- and juf In this section we examine a number of attempts to develop multipliers from estimates of Wj9 based on establishing the relationship between W;- and fij. Two approaches to multiplier derivation in this category can be identified, namely (1) identification separately of the direct and indirect components of the multiplier /iy, and (2) examination of the immediate relationship between the direct or first-round component Wj and the multiplier juj. These are examined in turn.

2.2.1 Identification of multiplier components The main contribution in this category is the work of Drake (1976) which provided the basis for the Regional Industrial Multiplier System (RIMS) of the Bureau of Economic Analysis. Drake provided "input - output type multipliers" by "decomposing the input-output multiplier into three components" (page 1): 1 an initial change component which is 1; 2 a direct effect component, defined as W/, 3 an indirect effect component,

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In the case of a model closed with respect to households, the indirect effect also included the induced effect. Drake estimated the direct effect component (Wj) by techniques developed for the same purpose in the context of producing nonsurvey i n p u t - o u t p u t tables (Garnick, 1970; Morrison and Smith, 1974; Schaffer and Chu, 1969). These involved 'regionalizing' the 484-sector national US table (US Department of Commerce, 1975) with industries defined at the four-digit Standard Industrial Classification level by two steps, namely the deletion of inputs from national industries which did not exist in the region, and the application of the location quotient at the two-digit level. Drake defines and derives in mathematical terms an 'interdependency coefficient' (a*) which expresses the theoretical relationship between the total direct effect W;- and the total indirect effect Bp such that B} = a*Wj. This a* is common to all columns of a matrix and implies that the indirect component can be estimated as a linear homogeneous function of the direct component in a given model, provided that the covariance between column elements and sums can be assumed to be 0. D r a k e "was able to explain 8 0 - 9 0 per cent of the variance in the indirect components" (page 5) in testing of a number of regional models. The interdependency coefficient, subject to the covariance assumption, is given by:

«*=(i-iy>T'-i = - ^ - i . l A^^J i _ w n However, Wj is not known for regions without i n p u t - o u t p u t models, and suitable proxies need to be found for the calculation of regional multipliers. Drake undertook a regression analysis of seventeen regional i n p u t - o u t p u t models in an attempt to estimate a*, but the results were not successful. A s an alternative, the indirect component for every industry in each region was employed as the dependent variable in regression analysis, with several variables representing characteristics of the regional economies as explanatory variables. E a c h observation was obtained by decomposing an industry-specific multiplier from one of the seventeen i n p u t - o u t p u t tables in the sample analyzed. The regressions showed that the indirect effect' tended to decrease as the proportions of the economy devoted to agriculture and manufacturing both increased with the relative size of the economy. D r a k e then used these methods to estimate multipliers for fifty-three industrial matrices randomly selected from six i n p u t - o u t p u t models "in a structured scheme to provide for representation among the agricultural, manufacturing, mining and trade divisions" (page 12). These 'estimated' multipliers were then compared with 'survey' multipliers from the original i n p u t - o u t p u t table. His results are summarized in table 1. These results provided the basis for D r a k e to (1) claim that the multiplier estimates were of "the same order of magnitude as the survey tables" (page 12), and (2) imply (pages 12, 13) that the closeness of the means of the estimated and survey multipliers provides some support for the general methodology. Table 1. Distribution of ratio of estimated to survey multipliers (source: Drake, 1976, page 15). Cases Number 2 4 8 7 11

Ratio intervals % 3.8 7.5 15.1 13.2 20.8

Cases Number

0.63-0.69 0.70-0.79 0.80-0.89

6 6 9

Ratio intervals % 11.3 11.3 17.0 100.0

1.10-1.19 1.20-1.29 1.30-1.60


Although Drake's approach has apparently provided the methodology for a great deal of empirical and planning work, his approach and conclusions can be questioned on several grounds. The first question relates to the logic of the procedure and the implied relationship between direct and indirect relationships. The derivation of the direct effect, or the Wj9 by the location-quotient method has been the subject of a great deal of comment in the literature dealing with nonsurvey tables; these arguments have usually agreed that these approaches are generally unsatisfactory, and need not be repeated here. However, given that the location-quotient method has been applied at the regional level, it would seem that the analyst has created in effect a nonsurvey input-output table, of the same type tested by Schaffer and Chu (1969) and others. Given that this table is available, with all its defects, there would seem to be a strong argument for using the table in the normal way to calculate indirect effects, rather than to search for a statistical relationship between direct and indirect effects. This procedure has apparently been adopted in the RIMS II report (see Cartwright et al, 1980). Drake avoids this procedure on a "reliability argument" (page 16), namely that "whatever errors were introduced in the estimates of the A matrix will be carried over into the Leontief inverse". This argument appears somewhat unconvincing in the light of the fact that Drake's procedure also insures that "errors in the direct effect are also reflected in the estimates for the indirect effect component" (page 15). (On this issue, see also Evans, 1954, and Simonovuits, 1975.) The second question arises in a context of general multiplier accuracy. Table 1 shows that only 34.0% of estimated multipliers were within 10% of survey multipliers, that 26.4% showed 'errors' of 10-20%, and 18.8% showed 'errors' of 2 0 - 3 0 % , and that 20.8% showed errors in excess of 30%, with some quite substantial errors. One can seriously question the 'order of magnitude' claims, and the claim that such multiplier estimates can be considered to be 'reasonably reliable'. The third question relates to the closeness of the means of the estimated and survey multipliers. This closeness simply confirms that the procedure produces a set of multipliers with a distribution which displays a mean which is close to the mean of the survey multipliers. This could indicate that the estimated multipliers could be an acceptable sample of survey multipliers, but does not imply that the methodology will predict any single multiplier in a reliable way. The main value of Drake's paper would appear, therefore, to lie less in the reliability of the procedure to produce accurate multiplier estimates and more in the derivation of the concept of the interdependency coefficient a* and the covariance conditions under which this coefficient will indicate linearity between the components of tbe input-output multiplier. 2.2.2 Relationship between Wj and jUj In this section we review those methods which purport to p r o d u c e 'input - output' or

'input-output type' multipliers directly from the Wp or from variations of Wj. The methods appear to accept almost as an article of faith, an assumption that the Wj are the prime determinant of the jUj. This assumption appeared in the work of Stevens and Trainer (1980). It is discussed in more detail in section 2.3 below, where it is argued that this assumption (termed here the Wj -• jUj assumption) is inappropriate and misleading. Several papers published by Burford and Katz (BK) provide details of this approach (namely, BK, 1977a; 1977b; 1978a; 1978b; 1981a; 1981b; and Katz and Burford, 1982). We begin with the original paper (BK, 1977a), and proceed to consider some subsequent contributions from these authors.


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The BK (1977a) paper is important as a basic statement of the approach. Three purposes are stated, namely "(i) [to demonstrate] empirically that multipliers computed from randomly generated I - O coefficient matrices of fixed size and with fixed columns have small variances relative to their means and that these multipliers are relatively insensitive to the specific values of the coefficients in the matrix; (ii) show the specific form of the a* coefficient suggested by Drake (1976), demonstrate the relationship of this coefficient to the standard direct and indirect output multiplier, and demonstrate how this can be used to compute multipliers without having an I - O matrix so long as the proportion of industry purchases within the region can be estimated and (iii) use the multiplier equations developed to demonstrate that, not only the output multipliers but income and other multipliers as well can be estimated without a full I - O matrix" (page 22, our italics). In pursuit of goal (ii), BK show (page 27) that Drake's a* or interdependency coefficient is in effect: I

*-> *-> n )

1- W

and that provided Drake's condition (namely, that the covariance of the a,y and Wj is 0) holds, then (/>,) = ( I - A ) " 1 = I + * A =

I + ^ - A .

There appears to be no empirical evidence on the validity of assuming Drake's covariance condition. Presumably the fact that Drake's multipliers and the BK multipliers both contain considerable 'errors', suggests that the condition does not hold as closely as Drake and BK assume. However, this question aside, the multiplier estimates for column j will be

^

i- w

J

'

which was the BK "equation 30" as termed for ready reference. As explained in section 2.1 above, it is also equivalent to the earlier-developed Salcedo (1972) method for estimating multipliers. BK contend then that equation 30 "can be used as a quick computational formula for the approximate multiplier, even if there is an I - O matrix available and a very close approximation to the true multiplier if there is no I - O matrix for the area in question" (page 27). Although one would wonder at the point of calculating approximate multipliers by these methods if an input-output matrix existed, given the technology of input-output and the additional informative value of input-output multipliers, we proceed to examine in more detail the empirical section of BK (1977a) in pursuit of their goal (i) above. BK generate random samples of fifty input-output matrices, with fixed matrix dimensions and fixed column totals. For this set of fifty six-sector tables, sector multipliers, mean sector multipliers, and variances were calculated to determine the effects of "random coefficient errors on multipliers with column totals and dimensions fixed" (page 27). On the basis of this analysis (BK, 1977a), BK conclude that: "(i) It is clear that the mean of the multipliers from the sample of matrices can be estimated very closely by the use of equation 30, and that the amount of variation resulting from specific values of coefficients is also small. (ii) The key determinants of the values of multipliers are (1) the average value of the column totals in the interindustry coefficient matrix and (2) the specific values of the individual column totals.


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(iii) The distribution of coefficient matrices in the columns has a relatively small role in determination of specific multiplier values, though multipliers do tend to be affected by the general coefficient structure within columns" (pages 28-29). It is our contention in this paper that these three claims by BK are inappropriate in the context of the BK argument and data; the three claims are discussed in turn. Claim (i) implies that equation 30 has some special virtue since it is able to estimate very closely the mean of fifty observations of the multiplier generated by random sampling. Clearly, this is a property of sampling theory rather than of equation 30; from the central limit theorem we would expect this result with the 'closeness' reflecting the properties of the process of random number generation. Second, their results cannot show "that the amount of variation resulting from specific values of coefficients is also small", since the effect of specific values of any coefficient is not demonstrated in any way. Claim (ii) is even more worrying. There does not appear to be any reason for concluding that the key determinants of the value of the multipliers can be either (1) the average value of the column totals in the interindustry coefficient matrix (which is just one entry in the input-output table) or (2) the specific values of the individual column totals. The tables show some degree of correlation or tendency to a relationship between the Wj and the jup but certainly do not indicate the Wj as the 'key determinant' of the jUj, particularly when the first are held constant during the experiment. Claim (iii) is similarly debatable. The reader is provided with no evidence on the multipliers generated from columns with different distributions of coefficients, or any evidence on the effect of coefficient structure, other than the limited evidence of maximum and minimum multiplier values. We believe, therefore, that these claims by BK are not supported in any way by the published data. Other aspects of BK (1977a) deserve comment. First, BK conclude that "a small increase in the average of column totals results in comparatively large increases in average multipliers for all columns even for columns with unchanged column totals" (page 29). This conclusion would seem to be derived from the line of causation which begins with the average of column totals (W), rather than with the individual column totals (Wj), or preferably the individual atj. The second further aspect deserving comment lies in the empirical application of equation 30 to Schaffer's (1976) multipliers computed from the Georgia input-output model. BK report that equation 30 reproduced Schaffer's direct and indirect multipliers "very closely" (page 33), but that the error is largest when a sector "has an exceptionally large diagonal element in its coefficient matrix or because a high percentage of that industry's purchases are from one or two other industries with larger than average ... Wf (page 33). This is some prima facie evidence that claim (iii) above is questionable and that the role of individual coefficients is of some importance in the matrix. In a later paper, BK (1977b) establish the theoretical point that, under certain conditions, the expected value of the distribution of multipliers from random inputoutput tables is equal to the multiplier calculated by equation 30. The empirical section of the paper rests on similar work to BK (1977a), except that a random sample of five hundred matrices was used. An additional claim is made that the paper "justifies the empirical evidence ... pertaining to the lack of sensitivity of the Leontief multipliers to errors in the coefficient estimates" (page 457). This claim is not based on the observation of the effect of errors in individual coefficients on the multipliers (on this, see West, 1983). Presumably, it is possible only because the constraining of the Wj to particular levels ensures that any large 'errors' in one coefficient must be balanced by equally large 'errors' of opposite

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sign in another coefficients). It is difficult to understand how BK can observe, from their published evidence, that "this observation should lead one to re-evaluate the usefulness of determining all the coefficients of a regional input-output matrix" (page 457). We, similarly, take exception to the claim that the BK "formula ... allows input-output type analysis on all types and sizes of economies" (page 458). The distinction between multiplier analysis and input-output type analysis has been sufficiently well established to contradict this type of statement. A third paper explaining the BK approach (BK, 1981a) prompted a perceptive response by Harrigan (1982) and a reply by Katz and Burford (1982). In essence Harrigan, although appreciating the general BK approach, questioned the use by BK of random matrices, that is, the assumption that each (nonnegative) matrix is equally likely. Harrigan argues (page 376) that it is only if this assumption of diffuse prior probabilities for the population of A matrices used by BK is appropriate that the values of ju will be unbiased estimates of actual gross multipliers. Harrigan argues that "a more plausible procedure therefore would be to dispense with the assumption of diffuse or noninformative priors and to substitute for it some information about the probable structure of the unobserved A matrix" (page 377). Harrigan's point is that knowledge of real-world A matrices is not negligible, since a considerable amount is known about input-output structures, and that the inclusion of such information in the form of prior probabilities improved the accuracy of the shortcut multiplier formulation in the case of a Scottish example. The Katz and Burford (1982) reply revealed an emergence of various versions of shortcut multipliers according to the amount of information which might be available to the analyst. They refer (page 384) to a formulation for use when (1) row and column totals are both known, (2) when column totals and one column of coefficients are available, and (3) to a Phibbs and Holsman (1980) version which uses column totals and all diagonal elements of the A matrix. In a later paper (BK, 1981b) they provide a version (4) with the replacement of W with a weighted average of column totals, and (5) two multiplier versions for the inclusion of induced effects in the multipliers. Indeed the stage has been reached where Phibbs and Holsman (1981) have felt able to formulate guidelines or rules of thumb which will identify situations in which the BK approach is likely to produce 'errors' of an unacceptable level. 2.3 An evaluation of shortcut multiplier development Preceding sections have outlined the development of the so-called shortcut 'input output' or 'input-output-type' multipliers in the form of a rather selective literature review. Any evaluation of these methods must, however, range beyond the rather narrow confines of the limited scope of papers on the topic and establish first the general perspective of the approach. In this section we briefly address this task and then proceed to comment on more specific aspects of the literature. The search for reliable and operational multipliers in economic analysis has, of course, a long and honorable history, and research which contributes positively to this cause has a high value to the profession and to society in general. Multiplier research should, therefore, be judged less on semantic points of debate and more on the contribution to knowledge and pragmatic planning requirements. The following comments do, however, combine elements of both of these aspects. The first comment is a general claim that the literature on shortcut multipliers presents virtually no new or original concepts and is essentially a rediscovery of ideas which have been with us for some decades. Although some of the theoretical developments have been helpful, in an operational sense nothing new has been added to the armory of regional analysis. Macroeconomists have for some time accepted


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the concept of Keynesian multipliers based on national propensities to consume, and a rather impressive work has emerged in the development of regional Keynesian multipliers using regional propensities to consume. This work (for example, by Greig, 1971; Brownrigg, 1971; 1974) has involved extensive modification of multiplier formulae to accommodate various characteristics of economic impact. The shortcut multiplier approach by using W (the average propensity of productive sectors to consume locally) has simply followed this timeworn path without recognition of the fact, and without distinction between marginal and average propensities. In this sense there should be no implication that the shortcut multipliers are original in any conceptual context. It is indeed one of these strange twists of fate and academic research that the more primitive approach has been rediscovered as a 'shortcut' to, or a substitute for, the recently hailed development of regional input-output tables. The second comment turns on terminology, but is not simply semantic. The input-output table is a detailed account of the interactions between sectors in an economy; the multipliers resulting from this table are the result of this interaction, both as statements of effects on each sector in the economy and on the economy as a whole. Such is the nature of input-output multipliers, where every element of the A matrix and the Leontief inverse is an integral component of an input-output multiplier. The total inadequacy of the shortcut method in embodying the full richness of the interconnectedness of the input-output table in any operational sense should preclude the use of the term Input - output' or 'input-output-type' multipliers as is common in the literature reviewed in sections 2.1 and 2.2. Certainly the use of the input-output multipliers as a measure of the accuracy of shortcut formulae is legitimate, but this practice should not be used to justify an inappropriate title or for the implication of 'authentication-by-association'. Further, attempts to develop formulae to measure indirect or induced effects are common to most methods of multiplier calculation in one form or another, and do not apply uniquely to the input-output method. The third comment address the role of the W;- in the determination of multiplier size, or the extent to which the Wj -• ju;- assumption or the conclusions of BK can be valid. These assert that the W;- are a 'key determinant' of the value of the multiplier, implying that the chain of causation in multiplier formation begins with the Wj. The logic of the input-output model, would, however, establish that the Wj is simply the sum of the atj forming the relevant columns and has no role, in itself, in the process of multiplier formation through the usual matrix-inversion procedures. It is simply the sum of the propensities of each sector to consume locally, with no analytical significance in an input-output context. The atj are the initial point of multiplier formation, both within each column and in the matrix as a whole and are the complete determinants of the multiplier. The BK logic would be equivalent to the claim that, in input-output studies of interregional or international trade, only local propensities to consume are needed to describe satisfactorily mutual trading links. The role and importance of the individual ai;- has been demonstrated by Jensen and West (1980), Hewings and Romanos (1981), Hewings (1985), and West (1983). Although the input-output multiplier can clearly be approximated by the use of the Wj in one form or another, it bypasses the logic of the input-output model to assert the dominance of the Wj in multiplier determination. In particular, the BK assertion that they have demonstrated this dominance cannot be accepted. Fourth, although some of the theoretical development in the literature, particularly the BK work, is useful in developing a deeper perspective on input-output multipliers, their equation 30, as earlier established, is simply a restatement of Salcedo's (1972) earlier work, and further, BK's and Phibbs and Holsman's work is simply a 'refinement' of this. The main problem in the development of multiplier

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analysis seems to lie, not so much in the further refinement of multiplier formulae, but in the collection and collation of data of a quality consistent with the quality of our existing models. Whether this be in the form of the more demanding atJ for input-output analysis or the supposedly less demanding Wj for shortcut Keynesian methods is a matter of choice for the analyst. 3 Further considerations In the recent exchange between Katz and Burford (1982) and Harrigan (1982), the issues of (1) the role of prior information and (2) the use of additional data in developing more accurate multipliers were discussed. These issues highlight some fundamental differences in the ways in which regional analysts view the structure of the regional economy and use models of that economy for impact and forecasting purposes. On the first issue it should be clear, as Harrigan notes, that our knowledge about the structure of real-world A matrices is not negligible. Not only do we have information derived from survey-based models for a number of regions, we also have available national models. To ignore this collection of prior information in the development of nonsurvey multipliers would seem difficult to justify on theoretical grounds. Whether, as BK claim, it is 'technically unfair' in comparison of methods, would appear to be irrelevant. The costs of data collection are minimal—in fact BK test all their methods against known I - O matrices, so why ignore this rich collection of data in the development of their formulae? There is, however, a more fundamental reason for not ignoring these data. The input-output production function is characterized as one which is highly skewed (see Jackson, 1983): BK's random matrices cannot accommodate such a distribution. They mistakingly confuse the accurate estimate of a scalar (the output multiplier) as somehow implying that the elements of A have either been estimated correctly or their estimation is irrelevant. On this matter, the reader is referred to Hewings (1977). Using randomly generated matrices which were adjusted by the RAS technique to known row and column marginal values, estimates of sectoral output multipliers were produced. These compared very favorably with those obtained from the survey-based models. However, when the column elements of the multipliers (that is, the Leontief inverse) were examined, considerable differences were noted between observed and estimated values. This is important because BK claim that they want to use their method "in measuring the economic impact of a firm or industry in situations where no reliable input-output model for the region exists" (page 384). If concern is focused only on the total effect, then the BK formulae will do just as well as many others. On the other hand, if concern is focused on the direct and indirect impact of an individual industry on the individual sectors of the economy, then the BK formulae provide little comfort that the estimates will be acceptably accurate. Since the sectoral distribution of impacts provided by the input-output model are the raison d'etre for the use of input-output models, the BK systems are obviously deficient. The second issue concerns the use of additional information in developing more accurate multipliers: this issue is closely related to the first one since, in many cases, this additional information may include a prior A. BK (1981b) show that they can produce better estimates if row totals are known as well as column totals. Bacharach (1970) showed that even with information for only one margin (row or column), then we can develop a matrix B from A which is as close as possible (in an informationtheoretic sense). In this case, matrix B will have column sums equal to known values, the multipliers will be as close to the observed ones as possible, and the analyst will have greater confidence in the use of matrix B for impact analysis. When column


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and row constraints are both available, then the usual RAS or biproportional solution is obtained. The Phibbs-Holsman technique (in which the au are known) and the BK technique in which the individual column coefficients for the /th sector are known are special cases of the more general RAS solution. The procedure has been tested by Matuszewski et al (1964) and many authors subsequently. Whatever the deficiencies of biproportional techniques, they do focus on a fundamental property of input-output systems which BK have tended to overlook. Harrigan (1982), as noted earlier, drew attention to the problem of the assumption that E{ai}) = c in a given column j (where E is the expectation value). Furthermore, an input-output model, as the name implies, deals with both sides of the productionconsumption interface. In such a context, it is difficult to justify an assumption that the coefficients in different columns are independently distributed. Although an assumption of linear independence is necessary to avoid problems of matrix singularity, the issue remains troublesome. The methodology provided through the BK technique is only able to document the effects of sector-specific changes in final demand on the rest of the economy but not on specific sectors. In essence, it provides a vector representation of the impacts normally associated with the Keynesian-type or economic base multiplier in which information is only known for aggregate propensity to consume locally. Although the BK vector is useful, more often than not our interest is in the output vector which is sector impact specific. The earlier discussion suggests that if we strive to achieve this vector through nonsurvey techniques, we have also the BK output information available to use. However, the reverse is not true. 4 Concluding remarks Section 2.3 provides a more detailed evaluation of the so-called shortcut multiplier approaches. These concluding remarks are limited therefore to some general observations on the practice and philosophy of shortcut multipliers. Seldom in the literature extolling the virtues of shortcut multipliers is any serious reference made to the important problems of calculating the Wj. There seems to be an implied belief that it will be easier to calculate each Wj with some degree of accuracy than to calculate the component atJ- of that Wj, because each Wj is simply one number. This is troublesome since accuracy in each Wj can only be in practice assured by collecting data relating to each atj with the required degree of accuracy, and summing to calculate the Wj. If each at- is calculated, the need for a shortcut method will not occur. Finally, we need to ponder briefly on the question of the shortcut approaches from the point of view of professional practice, whether these be multiplier calculation or nonsurvey tables for multiplier calculations. Any technique which involves 'black box' type mechanical operations as a substitute or replacement for a professional concern or input must be professionally suspect. There seems to be a tendency for the shortcut methods to engender some hope that new horizons for multiplier calculations have been unearthed and that the 'new' formulae can be mechanistically applied in a manner which is reminiscent of coin-in-the slot weighing or fortune-telling machines, whereas what all of these new approaches point to is a return to earlier days of more primitive regional multipliers. Acknowledgements. The comments of Frank Giarratani, especially with reference to the assumptions and distributions implied in many shortcut techniques, are gratefully appreciated as are the comments of two anonymous referees. The support of NSF Grant SES 82-05961 is also acknowledged.

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References Bacharach M, 1970 Bi-proportional Matrices and Input-Output Change (Cambridge University Press, Cambridge) BK see Burford R L, Katz J L Bromley D W, 1972, "An alternative to input-output models: a methodological hypothesis" Land Economics 48 125-133 Brownrigg M, 1971, "The income multiplier: an attempt to complete the model" Scottish Journal of Political Economy 18 281-297 Brownrigg M, 1974 A Study of Economic Impact: The University of Stirling (Scottish Academic Press, Edinburgh) Burford R L, Katz J L, 1977a, "Regional input-output multipliers without a full I - O table" Annals of Regional Science 11 2 1 - 3 8 Burford R L, Katz J L, 1977b, "An estimator of regional interindustry multipliers without an I - O matrix" Business and Economics Section Proceedings of the American Statistical Association pp 4 5 3 - 4 5 8 Burford R L, Katz J L, 1978a, "Regional input-output multipliers without a full I - O table: reply" Annals of Regional Science 12 9 9 - 1 0 2 Burford R L, Katz J L, 1978b, "Regional input-output multipliers without a full I - O table: (communication)M«A2fl/5 of Regional Science 12 105-106 Burford R L, Katz J L, 1981a, "A method for estimation of input-output-type output multipliers when no I - O model exists" Journal of Regional Science 21 151-161 Burford R L, Katz J L, 1981b, "The Burford-Katz approach to input-output multiplier estimation" invited paper for presentation at the Seventh Pacific Regional Conference, Surfer's Paradise, Australia; copy available from R L Burford, Department of Quantitative Analysis, Louisiana State University, Baton Rouge, LA Cartwright J, Beemiller R, Gustely R, 1980 RIMS II, A Disaggregated Regional Input-Output Modelling System for the US paper presented to the First World Regional Science Congress, Cambridge, MA; copy available from J Cartwright, Bureau of Economic Analysis, Washington, DC Drake R L, 1976, "A short cut to estimates of regional input-output multipliers: methodology and evaluation" International Regional Science Review 1 1-17 Evans W D, 1954, "The effect of structural matrix errors on interindustry relations estimates" Econometrica 22 461-480 Garnick D H, 1970, "Differential regional multiplier models" Journal of Regional Science 10 35-47 Greig M A, 1971, "The regional income and employment multiplier effects of a pulp mill and paper mill" Scottish Journal of Political Economy 18 3 1 - 4 8 Harrigan F J, 1982, "The estimation of input-output-type output multipliers where no input-output model exists: a comment" Journal of Regional Science 22 375-381 Hewings G J D, 1977, "Evaluating the possibilities for exchanging regional input-output coefficients" Environment and Planning A 9 927-944 Hewings G J D, 1985, "The role of prior information in updating regional input-output models" Socio-Economic Planning Sciences 18 319-336 Hewings G J D, Romanos M C, 1981, "Simulating less-developed regional economics under conditions of limited information" Geographical Analysis 13 373-390 Jackson R, 1983 A Distributional Approach to the Modelling and Simulation of Industrial Systems unpublished PhD dissertation, Department of Geography, University of Illinois at Urbana-Champaign, Urbana, IL Jensen R C, West G R, 1980, "The effect of relative coefficient size on input-output multipliers" Environment and Planning A 12 659-670 Katz J L, Burford R L, 1982, "The estimation of input-output type multipliers when no input-output model exists: a reply" Journal of Regional Science 22 383-387 Matuszewski T I, Pitts P R, Sawyer J S, 1964, "Linear programming estimates of changes in input coefficients" Canadian Journal of Economics and Political Science 30 203-210 Morrison W I, Smith P, 197'4, "Non-survey input-output techniques at the small area level: an evaluation" Journal of Regional Science 14 1-14 Nelson P.E, Perrin J S, 1978, "A short cut for computing final demand multipliers: some empirical results" Land Economics 54 8 2 - 9 1 Phibbs P J, Holsman A J, 1980, "A short cut method for computing final demand multipliers for small regions" Environment and Planning A 12 1001-1008


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Phibbs P J, Holsman A J, 1981, "An evaluation of the Burford-Katz short cut technique for deriving input-output multipliers" Annals of Regional Science 15 1 1 - 1 9 Pleeter S (Ed.), 1980 Economic Impact Analysis, Methodology and Applications (Martinus Nijhoff, The Hague) Salcedo D, 1972, "A short cut method for determining the impact of an economic activity on an entire economy" unpublished report, Office of Information Services, State of Texas, Austin, TX Schaffer W A, 1976 Studies in Applied Regional Science, Volume 1. On the Use of Input Output Models for Regional Planning (Martinus Nijhoff, The Hague) Schaffer W A, Chu K, 1969, "Non-survey techniques for constructing regional interindustry models" Papers, Regional Science Association 23 83-101 Simonovuits A, 1975, "A note on the underestimation and overestimation of the Leontief inverse" Econometrica 43 4 9 3 - 4 9 8 Stevens B H, Trainer G A, 1980, "Error generation in regional input-output analysis and its implication for non-survey models" in Economic Impact Analysis, Methodology and Applications Ed. S Pleeter (Martinus Nijhoff, The Hague) US Department of Commerce, Bureau of Economic Analysis, 1975 Input-Output Structure of the US Economy (US Government Printing Office, Washington, DC) West R G, 1983 Sensitivity and Probability Analysis in Regional Input-Output Models unpublished PhD dissertation, Department of Economics, University of Queensland, St Lucia, Brisbane, Queensland

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