1、3 New Developments in Input-Output AnalysisFields of Influence of Changes, the Temporal Leontief Inverse and the Reconsideration of Classical Key Sector AnalysisMichael SonisRegional Economics Applications Laboratory, University of Illinois, USA and Bar Ilan University, Israele-mail: sonismmail.biu.
2、ac.ilGeoffrey J. D. HewingsRegional Economics Applications Laboratory, University of Illinois, USAe-mail: hewingsillinois.edu3.1 Introduction: Coefficient Change in Input-Output ModelsIn this chapter, a new temporal approach to the classical Leontief input-output analysis is elaborated that avoids t
3、he assumption of the constant direct input coefficients. The presentation builds on earlier work (Sonis and Hewings, 1989, 1991, 1992) that examined a variety of issues surrounding error and sensitivity analysis, decomposition and inverse important parameter estimation. These ideas are now brought i
4、nto a general form as a basis for a more complete, general fields of influence approach that is the main vehicle for describing the overall changes in economic relationships between industries created by combinations of changes in technological coefficients caused by diffusion of technological, orga
5、nizational and administrative innovations. The most important new concept based of the notion of the direct fields of influence is the multiplier product matrix and the corresponding artificial economic landscape which represents the classical key sector analysis and hierarchies of sectoral backward
6、 and forward linkages. The multiplier product matrix is the main part of the fundamental minimal information decomposition of Leontief inverse in the form of the difference between the global presentation of direct effects of the changes in inputs coefficients and the synergetic effects of the overa
7、ll interaction of the changes. The new Temporal 70 Michael Sonis and Geoffrey J.D. Hewings Leontief inverse is constructed such that it can serve as the basis for the temporal analysis of an evolving input-output system; the inverse depends on an evolutionary tail of changes in a highly non-linear m
8、anner. The detailed analytical structure of the Temporal Leontief inverse addresses the possibilities of tracing the impact of each change in the individual direct inputs in each time period through to the final state of the economy. This dynamic approach provides the basis for a new perturbation th
9、eory for matrix inversion. The celebrated Leontief input-output model (Leontief, 1951) has the following mathematical form:(3.1)xAfwhere x is the vector of gross output, f is the vector of final demand, is ijAathe matrix of direct inputs into the set of n production sectors, with properties:ija(3.2)
10、10,1,2. ijnijajnThe equivalent form of the Leontief model is:(3.3)1xIAfBwhere I is the identity (unit) matrix and the matrix is the Leontief 1BIAinverse matrix (see, Leontief, 1951, Miller and Blair, 1986, Hewings, 1985).The important assumption proposed by Leontief was the assumption of the constan
11、cy of the direct inputs during the time interval in question. Our intension in his paper is to analyze the case of the temporal coefficient change in the Leontief input-output model. Interest in the problem of coefficient change in input-output models is not a recent phenomenon; however, what is mos
12、t curious about input-output modeling is that analysts, for the most part, have not made the discussion of errors a prominent feature of the presentation of the model, in applications and, especially, in connection with results of impact analyses. However, while the sources of error (in data or in e
13、stimation) are often acknowledged, it is rare to find a presentation involving the use of an input-output model in which a statistical confidence interval is assigned to the level of output associated with any given change in final demand.3 New Developments in Input-Output Analysis 713.1.1 Three app
14、roaches to input coefficient changeInitial attention to change in coefficients in input-output models was directed to the issue of the effect of error or changes in individual coefficients on the elements of the associated Leontief inverse matrix (Evans, 1954; Simonovits, 1975; Lahiri and Satchell,
15、1986). Complementing this approach, there is the issue of coefficient stability and the effect of coefficient change induced by technology, changing markets, structural change and the general effects of economic growth and development.Contributions to this literature include the seminal papers by Se
16、valdson (1970) and Carter (1970) and the intriguing notions of Tilanus (1966) who, using the annual Dutch input-output tables, suggested a distinction between average and marginal input-output coefficients to parallel the distinctions made in individual consumer consumption theory. Lahiri (1976) app
17、roached this problem is a slightly different way, assuming that the choice of input coefficient was a function of the level of demand existing in any given industry. Clearly, Lahiris ideas provide the entre to the development of a micro-to-macro link in input-output systems in which production choic
18、e within the context of an establishment, firm or industry might be modeled in a behavioral setting with the general macroeconomic economy serving to condition choice. In some cases, the choices made at the micro level may, in turn, influence macro-level variables and thus the decision environment f
19、aced by other sectors of the economy. In the input-output literature, the models developed by Eliasson (1978) come as close as any to providing this link; the early developments of the transactions value social accounting models (TVSAMs) by Drud, Grais and Pyatt (1985) provided the precursors for ex
20、tensions towards a more general equilibrium modeling framework. The gradual adoption of computable general equilibrium models, in which the input-output framework is often embedded, has created an even more pressing need for identification of important parameters in the system and an assessment of t
21、he role of errors.For the most part, this work has not been generalizable to all input-output systems; at the regional level, the issue of coefficient change has been more problematical because so many regional and interregional models have been assembled from nonsurvey or partial survey data source
22、s. In this regard, the regional dimension provides the possibility for a new source of error not usually associated with the national level input-output models. The error usually arises in the transfer of the familiar input coefficients into trade coefficients; Stevens and Trainer (1976) suggested t
23、hat the problem was complicated by the possibilities of differences between the nation and the region in industrial technical structures, a finding confirmed by Israilevich and Hewings (1991).A more recent point of view, that of comparative macroeconomic dynamics (Sonis and Hewings, 1989, 1998a,b),
24、considers coefficient change to be the result of economic development connected with the emergence, spread and adaptation of 72 Michael Sonis and Geoffrey J.D. Hewings technological and organizational innovations as a part of the process of complication and self-organization of economy.At the region
25、al level, the debate has been important for focusing attention once again on the structure of input-output models and, in particular, on the methods that could be used to ascertain whether two structures were similar. Furthermore, derivative work emanating from this debate has also focused attention
26、 on the degree to which notions of importance within the input-output system could be identified. From this work, two complementary approaches to input coefficient change can be identified, namely (1) error analysis and (2) sensitivity analysis. While the two issues will be addressed separately, the
27、 distinction is, in many ways, somewhat artificial.Error Analysis. Theils (1957, 1972) pioneering work in entropy decomposition analysis provided a useful way of examining error or change in input structures. He suggested that change could be decomposed into a set of additive components. More recent
28、ly, Hewings (1984), Hewings and Syversen (1982), Jackson and Hewings (1984) and Jackson, et al. (1990) have explored this technique with reference to data for Washington State. On the other hand, West (1982) has approached error analysis from a relative change perspective, focusing, in particular, o
29、n the effects of coefficient error on the multipliers of the associated inverse matrix. Closely allied with this approach is that adopted by Jackson (1986) who developed the notion of a probability density distribution for each coefficient and showed how this “uncertainty” could lead to serious prob
30、lems in the utilization of the input-output model (Jackson and West, 1989; Wibe, 1982). The relative change approach has also been explored by Xu and Madden (1991).Lawson (1980) has approached the problem conceptually by considering various forms of error - additive and multiplicative - and the ways
31、 in which these might be used in a “rational” approach” to modeling. Closely allied with this line of reasoning would be the work of Stevens and Trainer (1976). Giarratani and Garhart (1991) developed further some propositions about the major sources of error. The notion of some “rationality” in the
32、 error or in the structure of coefficient change of course underlies the widespread application of the RAS or bi-proportional technique in the context of updating (especially at the national level) and estimation (at the regional level, where a national table is often used as a base). Bacharachs (19
33、70) work revealed a strong link between the RAS technique and the assumptions explicit in linear and nonlinear programming. Matuszewski, Pitts and Sawyer (1964) did in fact propose an LP-RAS technique; in their applications, several coefficients were “blocked out” in the updating algorithm because t
34、heir true values were either known or could be estimated with what Jensen and West (1980) have referred to as “superior data.” To this point, (early 1970s), however, no attempt had been made to assess the degree to which errors in individual coefficients could be ranked or rated in terms of their im
35、portance. West (1981) provided some important directions in this regard, suggesting a relationship between coefficient size and the associated multiplier. 3 New Developments in Input-Output Analysis 73Several of the techniques and approaches developed for error analysis were subsequently modified to
36、 perform sensitivity analysis; these are described in the next section.Sensitivity Analysis. Using a little-known theorem developed by Sherman and Morrison (1949, 1950), Bullard and Sebald (1977, 1988) were able to show that, in energy terms, only a very small number of the input coefficients in the
37、 US input-output model were analytically important. In applications at the regional level, Hewings (1984) referred to these as inverse important coefficients. In a similar fashion, Jensen and West (1980) found that the removal of a large percentage of the entries in an input-output table could be ac
38、complished with little appreciable effect on the results from the use of the model for impact analysis. Subsequently, West (1982) noted that the size and location of the coefficient within the input-output table provided the major determinant of an individual coefficients importance. Further work by
39、 Morrison and Thumann (1980) and Hewings and Romanos (1981) has extended the sensitivity notions to suggest that the censal mentality characterizing the developments of many input-output models (namely, that all entries need to be estimated with the same degree of accuracy) is probably misplaced. Th
40、is is especially true in the cases in which regional tables are derived from national tables or in the process of updating tables. The results of the sensitivity analysis in combination with statistical estimation techniques suggest that a more “rational” approach to coefficient change could be deve
41、loped (Jackson and West, 1989).Field of Influence of Changes Approach. A field of influence of changes approach to a large extent is independent of the type of coefficient change; the major objective is the provision of a methodology that is general enough to handle all types of changes - in single
42、elements, in all elements in a row or column or in all elements of the matrix. The procedure involves the calculation of the ratio of two polynomial functions of changes in contrast to the usual approach that is based on the infinite Taylor series expansion of the Leontief inverse. The linear approx
43、imation of this expansion without any synergetic effects in the form of the gradient field (see Xu and Madden, 1991) is identical to the first order field of influence. Moreover, the methodology provides a finite form, one that is eminently capable of realization in the form of a computer algorithm.
44、 This meso-level economic approach also provides the possibility for uncovering the hierarchical structure of changes through the identification of the intensity of influences, an alternative and complementary approach to the micro-level structural path analytical methods illustrated by Defourny and
45、 Thorbecke (1984). Thus, th present method is more general in that it can handle a complete range of changes. In particular, the ability to be able to examine the influence of changes in an arbitrary subset of elements is presented as a major feature of the methodology; it turns out that the familia
46、r RAS or bi-proportional adjustment technique is a special case of coefficient change (see Sonis and Hewings, 1989). In addition, as demonstrated by Sonis and Hewings (1991), the methodology may 74 Michael Sonis and Geoffrey J.D. Hewings be extended to issues of decomposition (see Kymm, 1990; Gillen
47、 and Guccione, 1990) or the updating of input-output matrices (Snower, 1990 reviews some of the recent work at the national level while Giarratani and Garhart, 1991 provide a similar review at the regional level; see also Dietzenbacher, 1990).The presentation below builds on earlier work (Sonis and
48、Hewings, 1989, 1991, 1992) that examined a variety of issues surrounding error and sensitivity analysis, decomposition and inverse important parameter estimation. These ideas are now brought into a general form as the basis for a more complete, general approach. The essential difference between the
49、field of influence approach and error and sensitivity analysis is that the former are considered as the main vehicle for describing the overall changes in economic relationships between industries created by combinations of changes in technological coefficients. Interpreted in a comparative static framework, it will then be possible to proceed to consideration of evolutionary economic dynamics. The field of influence of change represents conceptually the process of diffusion of innovations within input-output models. The changes in direct input coefficients are
