<p>2222222222</p><h2 class="ft-title">Coefficient Change in Input-Output Models: Theory and Applications </h2><div class="html-ft-toc"><h3 class="small-bold" id="toc">Contents</h3><ol><li class="link-medium"><a id="hd_AN0009954671-2" title="1. Introduction " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#AN0009954671-2"><font color="#0033ff">1. Introduction </font></a></li><li class="link-medium"><a id="hd_AN0009954671-3" title="2. Theoretical Basis for Coefficient Change(n1) " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#AN0009954671-3"><font color="#0033ff">2. Theoretical Basis for Coefficient Change(n1) </font></a></li><li class="link-medium"><a id="hd_AN0009954671-4" title="3. Implications " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#AN0009954671-4"><font color="#0033ff">3. Implications </font></a></li><li class="link-medium"><a id="hd_AN0009954671-5" title="4. Empirical Applications " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#AN0009954671-5"><font color="#0033ff">4. Empirical Applications </font></a></li><li class="link-medium"><a id="hd_AN0009954671-6" title="5. Conclusions " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#AN0009954671-6"><font color="#0033ff">5. Conclusions </font></a></li><li class="link-medium html-toc-reft"><a id="reft_AN0009954671-7" title="Notes " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#AN0009954671-7"><font color="#0033ff">Notes </font></a></li><li class="link-medium"><a id="hd_AN0009954671-8" title="Table 1. Fields of influence, Washington State, 1963, 1967, 1972 " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#AN0009954671-8"><font color="#0033ff">Table 1. Fields of influence, Washington State, 1963, 1967, 1972 </font></a></li><li class="link-medium html-toc-reft"><a id="reft_AN0009954671-10" title="References " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#AN0009954671-10"><font color="#0033ff">References </font></a></li></ol></div><div class="full-text-content"><p class="body-paragraph">ABSTRACT A general theory of coefficient change in input-output and social accounting models is proposed. The major contribution is the introduction of the notion of a 'field of influence' as the basis for interpreting the effects of coefficient change. This basis is elaborated through a set of propositions. In Section 3, the implications are explored; first, the first-order changes in one row or column are examined. This approach is then generalized to changes in two or more rows or columns and the biproportional or RAS technique is shown to be a special case of coefficient change. Empirical applications are presented in Section 4, drawing on the work in regional and national economies. The paper concludes with some remarks about ways in which this work might be linked with parallel interests in decomposition of input-output systems.</p><a name="AN0009954671-2"></a><span class="medium-bold"><a id="hd_toc_7" title="1. Introduction " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#toc"><strong><font color="#0033ff" size="2">1. Introduction </font></strong></a></span><p class="body-paragraph">In this paper, we propose a general approach to the problems of coefficient change that is formed on the notion of a field of influence. This notion is to a large degree 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--single elements, all elements in a row or column or all elements in the matrix. The procedure involves the calculation of the ratio of two polynomial functions of changes, in contrast to the usual approach which is connected with the infinite Taylor series expansion of the Leontief inverse. The linear approximation of this expansion without any synergetic effects in the form of the gradient field (see Xu & Madden, 1991) is identical to the first-order field of influence presented in this paper.</p><p class="body-paragraph">Thus, the 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 familiar RAS or biproportional adjustment technique is a special case of coefficient change. In addition, as demonstrated in Sonis & Hewings (1991), the methodology can be extended to issues of decomposition (especially when the input-output model is embedded in a demographic-economic or social accounting system) and, by induction, to problems of aggregation and disaggregation (see Gillen & Guccione, 1990; Kymn 1990) or the updating of input-output matrices. (Snower (1990) reviews some of the recent work at the national level while Giarratani & Garhart (1991) provide a similar review at the regional level. See also Dietzenbacher (1990).)</p><p class="body-paragraph">The present paper builds on earlier work (Sonis & Hewings, 1988, 1989, 1990, 1991) which 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 a basis for a more complete approach. In the next section, the theoretical basis for a field of influence is articulated through a set of propositions. Section 3 examines the implications for inverse sensitivity, changes in rows and columns, first-order effects and synergetic changes moving to a consideration of the bi-proportional or RAS technique as a special case of coefficient change. The final section provides some empirical applications drawn from national and regional economies for which input-output and social accounting models are available. In particular, attention is focused on the way in which the field of influence changes as the structure of the same economy is described by an input-output and a social accounting model. Finally, some conclusions are offered that provide the basis for future explorations in this area.</p><a name="AN0009954671-3"></a><span class="medium-bold"><a id="hd_toc_15" title="2. Theoretical Basis for Coefficient Change(n1) " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#toc"><strong><font color="#0033ff" size="2">2. Theoretical Basis for Coefficient Change(n1) </font></strong></a></span><p class="body-paragraph">Let us introduce the following notation: define A = (a<sub>ij</sub>) as an n x n matrix of direct input coefficients and E = (e<sub>ij</sub>) as a matrix of incremental changes to these direct input coefficients. Further, define matrices C=(c<sub>(ij)</sub>), C(E)=(c<sub>(ij)</sub>(E)) such that C = I - A, C(E) = I - A - E, det C = Δ, det C(E)n = Δ(E); and matrices B=(b<sub>(ij)</sub>), B(E) = (b<sub>(ij)</sub>(E)) such that B = C<sup>-1</sup> = (I - A)<sup>-1</sup> and B(E) = C<sup>-1</sup> (E) = (I - A - E) <sup>-1</sup></p><p class="body-paragraph">Proposition 1. The ratio Q(E) = Δ(E)/Δ is the polynomial of the incremental changes e<sub>ij</sub> of the following form:</p><p class="body-paragraph">Q(E) = detB / detB(E)</p><p class="body-paragraph">= Δ(E) / Δ</p><p class="body-paragraph">(<a name="bib1up">
</a><a id="ref_linkbib1" title="1" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib1"><font color="#0033ff">1</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">where</p><p class="body-paragraph">[Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">is the k x k determinant that includes the components of the matrix C<sup>-1</sup> = B from the ordered(<a name="bib2up">
</a><a id="ref_linkbib2" title="n2" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib2"><font color="#0033ff">n2</font></a>) set of columns i<sub>1</sub>, i<sub>2</sub>,..., i<sub>k</sub> and rows j<sub>l</sub>, j<sub>2</sub>,..., j<sub>k</sub>. Furthermore, in the sum Σ' the products of changes (Multiple lines cannot be converted in ASCII text) which differ from each other only by the order of the multipliers are included only once.</p><p class="body-paragraph">Proposition 2. Let B = (I- A)<sup>-1</sup> and B(E) = (I - A - E)<sup>-1</sup> represent the associated Leontief inverse matrices; then the following decomposition holds:</p><p class="body-paragraph">(<a name="bib2up">
</a><a id="ref_linkbib2" title="2" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib2"><font color="#0033ff">2</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">where the fields of influence</p><p class="body-paragraph">[Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">of order k of changes e<sub>j[sub 1</sub>i<sub>1</sub>],..., e<sub>j[sub k</sub>i<sub>k</sub>] are the n x n matrices with elements</p><p class="body-paragraph">(<a name="bib3up">
</a><a id="ref_linkbib3" title="3" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib3"><font color="#0033ff">3</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">Proposition 3. The structure of the fields of influence can be ascertained in the following form.</p><p class="body-paragraph">The first-order field of influence</p><p class="body-paragraph">[Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">of the incremental change e<sub>j[sub 1</sub>i<sub>1</sub>], is the n x n matrix generated by the multiplication of the jth column of the matrix B with the ith row:</p><p class="body-paragraph">(<a name="bib4up">
</a><a id="ref_linkbib4" title="4" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib4"><font color="#0033ff">4</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">Moreover, the first-order field of influence includes the components of the gradient of the function b<sub>i[sub 1</sub>j<sub>1</sub>] considered as a scalar function of all components of the matrix A:</p><p class="body-paragraph">[Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">(here, the pqth component of the gradient is placed in the intersection of the qth row and pth column).</p><p class="body-paragraph">The second-order field of influence is created from the interaction within the product of two increments, e<sub>j[sub 1</sub>i<sub>1</sub>] e<sub>j[sub 2</sub>i<sub>2</sub>:</p><p class="body-paragraph">(<a name="bib5up">
</a><a id="ref_linkbib5" title="5" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib5"><font color="#0033ff">5</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">Obviously, if i<sub>1</sub> = i<sub>2</sub> or j<sub>l</sub> = j<sub>2</sub>, then</p><p class="body-paragraph">[Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">is a null matrix.</p><p class="body-paragraph">For each k = 2, 3,..., n - 1, the following recurrent formula is true:</p><p class="body-paragraph">(<a name="bib6up">
</a><a id="ref_linkbib6" title="6" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib6"><font color="#0033ff">6</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">This formula also provides the possibility of presenting the field of influence of order k through the use of fields of influence of lesser order 1,2,..., k - 1.</p><a name="AN0009954671-4"></a><span class="medium-bold"><a id="hd_toc_77" title="3. Implications " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#toc"><strong><font color="#0033ff" size="2">3. Implications </font></strong></a></span><p class="body-paragraph">3.1. The Sherman-Morrison Formula and Inverse-important Coefficients</p><p class="body-paragraph">The notion of inverse-important input coefficients (Bullard & Sebald, 1977, 1988; Hewings, 1984; Schintke & Staeglin, 1988) is based on a conception of change associated with only one input coefficient, a<sub>i[sub 1</sub>j<sub>1</sub>. If the change occurs in location (i<sub>1</sub>, j<sub>1</sub>) i.e.</p><p class="body-paragraph">(<a name="bib7up">
</a><a id="ref_linkbib7" title="7" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib7"><font color="#0033ff">7</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">then the Leontief inverse B(E) has the form (from equation (<a name="bib2up">
</a><a id="ref_linkbib2" title="2" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib2"><font color="#0033ff">2</font></a>))</p><p class="body-paragraph">(<a name="bib8up">
</a><a id="ref_linkbib8" title="8" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib8"><font color="#0033ff">8</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">where the field of influence is</p><p class="body-paragraph">(<a name="bib9up">
</a><a id="ref_linkbib9" title="9" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib9"><font color="#0033ff">9</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">The coordinate form of equation (<a name="bib9up">
</a><a id="ref_linkbib9" title="9" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib9"><font color="#0033ff">9</font></a>) provides us with the Sherman-Morrison (1950) formulation</p><p class="body-paragraph">(<a name="bib10up">
</a><a id="ref_linkbib10" title="10" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib10"><font color="#0033ff">10</font></a>) b<sub>ij</sub>(e) = b<sub>ij</sub> + b<sub>ii[sub 1</sub>] b<sub>j[sub 1</sub>j]e/ 1 - b<sub>j[sub 1</sub>i<sub>1</sub>e</p><p class="body-paragraph">Using the matrix form of the field of influence we can evaluate the cumulative effects of a change in the input coefficient a<sub>i[sub 1</sub>j<sub>1</sub>] on all components of the Leontief inverse and compare the cumulative effects of change in individual coefficients to identify those that may be said to be inverse important. These comprise the set of coefficients that will create the greatest impact on the rest of the economy given an initial change in one of them. It should be noted that the set of inverse-important coefficients will not necessarily be identical for different criteria (e.g. whether the focus is on income, employment or output) or different transformations of the transactions from which the coefficients are derived.(<a name="bib3up">
</a><a id="ref_linkbib3" title="n3" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib3"><font color="#0033ff">n3</font></a>)</p><p class="body-paragraph">The notion of inverse importance shares one major problem with the parallel idea of key sector (see Hewings, 1982), namely the formulation of the decision rule(s) for identifying the conditions under which a<sub>i[sub 1</sub>j<sub>1</sub> may be said to be inverse important. For example, one of several matrix norms may be used, namely</p><p class="body-paragraph">(<a name="bib11up">
</a><a id="ref_linkbib11" title="11" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib11"><font color="#0033ff">11</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">The choice of the norm |F| is the basis of the construction of the rank-size sequence of the elements au of the matrix A according to the numerical sizes of the norms</p><p class="body-paragraph">(Multiple lines cannot be converted in ASCII text)</p><p class="body-paragraph">The decision or cutting rule must be formulated in such a way that only a relatively small number of the elements of the rank-size sequence will comprise the set of inverse-important coefficients.</p><p class="body-paragraph">Furthermore, the formulations described above can serve as a basis for the calculation of column multipliers and their inverse importance (see also West, 1981). Define the column multipliers as the sums of the elements of rows of the Leontief inverse as follows:</p><p class="body-paragraph">[Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">For the case in which the change occurs in only one element (Multiple lines cannot be converted in ASCII text), the summation in equation (<a name="bib10up">
</a><a id="ref_linkbib10" title="10" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib10"><font color="#0033ff">10</font></a>) provides</p><p class="body-paragraph">(<a name="bib12up">
</a><a id="ref_linkbib12" title="12" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib12"><font color="#0033ff">12</font></a>) M<sub>k</sub>(e) = M<sub>j</sub> + b<sub>j[sub 1</sub>j]e/1 - b<sub>j[sub 1</sub>i<sub>1</sub>]e M<sub>i[sub 1</sub>]</p><p class="body-paragraph">Introducing the vector of column multipliers,</p><p class="body-paragraph">M = (M<sub>1</sub>, M<sub>2</sub>,..., M<sub>n</sub>)</p><p class="body-paragraph">M(e) = (M<sub>1</sub>(e), M<sub>2</sub>(e),..., M<sub>n</sub>(e))</p><p class="body-paragraph">and the vector rows of the Leontief inverse,</p><p class="body-paragraph">b<sup>r, sub j[sub 1</sup>] = (b<sub>j[sub 1</sub>], b<sub>j[sub 1</sub>2],..., b<sub>b[sub j[sub 1</sub>n])</p><p class="body-paragraph">then the vector form of equation (<a name="bib12up">
</a><a id="ref_linkbib12" title="12" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib12"><font color="#0033ff">12</font></a>) will be</p><p class="body-paragraph">(<a name="bib13up">
</a><a id="ref_linkbib13" title="13" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib13"><font color="#0033ff">13</font></a>) (M(E) = M + M<sub>i[sub 1</sub>] 1/1 - b<sub>j[sub 1</sub>i<sub>1</sub>]e b<sup>r, sub j[sub 1</sup>]</p><p class="body-paragraph">The vector norms ||M(E)|| of the column multipliers can serve as a measure of the cumulative effects of change in the input a<sub>i[sub 1</sub>j<sub>1</sub>].</p><p class="body-paragraph">3.2. Change in One Row (Column)</p><p class="body-paragraph">Consider the changes</p><p class="body-paragraph">e<sup>r, sub i[sub 1</sup>] = (e<sub>i[sub 1</sub>], e<sub>i[sub 1</sub>2],..., e<sub>i[sub 1</sub>n])</p><p class="body-paragraph">in the ith row of the matrix A. Then the matrix of increments E = (e<sub>ij</sub>) will have the components</p><p class="body-paragraph">(<a name="bib14up">
</a><a id="ref_linkbib14" title="14" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib14"><font color="#0033ff">14</font></a>) (Multiple lines cannot be converted in ASCII text)</p><p class="body-paragraph">Then, from Proposition 1, we have equation (<a name="bib1up">
</a><a id="ref_linkbib1" title="1" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib1"><font color="#0033ff">1</font></a>) in the form</p><p class="body-paragraph">(<a name="bib15up">
</a><a id="ref_linkbib15" title="15" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib15"><font color="#0033ff">15</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">while equations (<a name="bib2up">
</a><a id="ref_linkbib2" title="2" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib2"><font color="#0033ff">2</font></a>) and (<a name="bib3up">
</a><a id="ref_linkbib3" title="3" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib3"><font color="#0033ff">3</font></a>) may be shown in the form</p><p class="body-paragraph">(<a name="bib16up">
</a><a id="ref_linkbib16" title="16" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib16"><font color="#0033ff">16</font></a>) B(E) = B + 1 / Q(E) (F<sub>1</sub>e<sub>1</sub> + F<sub>2</sub>e<sub>2</sub> + ... + F<sub>n</sub>e<sub>n</sub>)</p><p class="body-paragraph">where</p><p class="body-paragraph">(<a name="bib17up">
</a><a id="ref_linkbib17" title="17" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib17"><font color="#0033ff">17</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">The coordinate form of equations (<a name="bib16up">
</a><a id="ref_linkbib16" title="16" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib16"><font color="#0033ff">16</font></a>) and (<a name="bib17up">
</a><a id="ref_linkbib17" title="17" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib17"><font color="#0033ff">17</font></a>) can be represented in a similar fashion to that presented by Sherman & Morrison (1949) for changes in one row:</p><p class="body-paragraph">(<a name="bib18up">
</a><a id="ref_linkbib18" title="18" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib18"><font color="#0033ff">18</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">Equation (<a name="bib18up">
</a><a id="ref_linkbib18" title="18" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib18"><font color="#0033ff">18</font></a>) provides a connection between the column multipliers M<sub>j</sub>(E) and M<sub>j</sub>:</p><p class="body-paragraph">(<a name="bib19up">
</a><a id="ref_linkbib19" title="19" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib19"><font color="#0033ff">19</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">or in vector form</p><p class="body-paragraph">(<a name="bib20up">
</a><a id="ref_linkbib20" title="20" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib20"><font color="#0033ff">20</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">Analogous formulations can be derived for changes in one column.</p><p class="body-paragraph">3.3. First-order Effects</p><p class="body-paragraph">In this section we describe the structure of the first-order effects of changes for the input coefficients of the A matrix. For the evaluation of first-order effects, we can choose the error matrix E = (e<sub>ij</sub>) with small increments e<sub>ij</sub> such that all pairwise products (Multiple lines cannot be converted in ASCII text) will be negligible and hence can be ignored. Under this condition, equations (<a name="bib1up">
</a><a id="ref_linkbib1" title="1" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib1"><font color="#0033ff">1</font></a>) and (<a name="bib2up">
</a><a id="ref_linkbib2" title="2" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib2"><font color="#0033ff">2</font></a>) provide</p><p class="body-paragraph">(<a name="bib21up">
</a><a id="ref_linkbib21" title="21" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib21"><font color="#0033ff">21</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">and</p><p class="body-paragraph">(<a name="bib22up">
</a><a id="ref_linkbib22" title="22" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib22"><font color="#0033ff">22</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">or, in coordinate form,</p><p class="body-paragraph">(<a name="bib23up">
</a><a id="ref_linkbib23" title="23" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib23"><font color="#0033ff">23</font></a>) b<sub>ij</sub>(E) ∼ b<sub>ij</sub> + S<sup>(1), sub ij</sup> (E) (<a name="bib23up">
</a><a id="ref_linkbib23" title="23" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib23"><font color="#0033ff">23</font></a>)</p><p class="body-paragraph">where the first-order effects are represented by a rational function S<sup>(1), sub ij</sup>(E) which is the ratio of two polynomials of the first degree such that</p><p class="body-paragraph">(<a name="bib24up">
</a><a id="ref_linkbib24" title="24" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib24"><font color="#0033ff">24</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">This formulation provides the basis for the description of the first-order effects of changes in a<sub>ij</sub> on the column multipliers such that</p><p class="body-paragraph">(<a name="bib25up">
</a><a id="ref_linkbib25" title="25" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib25"><font color="#0033ff">25</font></a>) M<sub>j</sub>(E) ∼ M<sub>j</sub> + S<sup>(1), sub j</sup> (E)</p><p class="body-paragraph">where</p><p class="body-paragraph">(<a name="bib26up">
</a><a id="ref_linkbib26" title="26" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib26"><font color="#0033ff">26</font></a>) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">It is possible to consider the second-order effects and, generally, the nth-order effects in a similar fashion.</p><p class="body-paragraph">3.4. The Linear Part of Change</p><p class="body-paragraph">Define</p><p class="body-paragraph">(27) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">Therefore the linear part of change in the component b<sub>ij</sub> of the Leontief inverse is</p><p class="body-paragraph">(28) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">which coincides with the numerator of the first-order effect S<sup>(1), sub ij</sup>(E).</p><p class="body-paragraph">3.5. Change in Two Rows</p><p class="body-paragraph">Consider the following matrix of increments:</p><p class="body-paragraph">(29) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">Then Proposition 1 provides</p><p class="body-paragraph">(30) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">Propositions 2 and 3 provide</p><p class="body-paragraph">(31) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">Similar analytical formulae are true for simultaneous change in two columns and a change in one column and one row.</p><p class="body-paragraph">3.6. Relative Coefficient Change and the RAS Adjustment Procedure</p><p class="body-paragraph">Relative coefficient changes can be introduced by choosing an increment e<sub>ij</sub> of the form</p><p class="body-paragraph">(32) e<sub>ij</sub> = a<sub>ij</sub>ε<sub>ij</sub></p><p class="body-paragraph">Here, the matrix ε = (ε<sub>ij</sub>) includes the shares of the relative changes for the direct coefficients a<sub>ij</sub>.</p><p class="body-paragraph">Substitution of equation (32) into equation (<a name="bib1up">
</a><a id="ref_linkbib1" title="1" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib1"><font color="#0033ff">1</font></a>) yields the following form of the determinant ratio:</p><p class="body-paragraph">(33) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">Analogously, substituting equation (33) into equation (<a name="bib2up">
</a><a id="ref_linkbib2" title="2" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib2"><font color="#0033ff">2</font></a>), the following form of the Leontief inverse results:</p><p class="body-paragraph">(34) [Multiple line equation(s) cannot be represented in ASCII text]</p><p class="body-paragraph">This form of relative coefficient change which can be used for purposes of sensitivity analysis can be shown to be a general form of the familiar bi-proportional or RAS procedure for adjustment of input matrices. Analytically, the RAS adjustment procedure provides the basis for the derivation of a new direct input coefficient from prior estimate, usually in the form</p><p class="body-paragraph">(35) ã<sub>ij</sub> = r<sub>I</sub>a<sub>ij</sub>s<sub>j</sub></p><p class="body-paragraph">Let</p><p class="body-paragraph">(36) ã<sub>ij</sub> = a<sub>ij</sub>(1 + ε<sub>ij</sub>) r<sub>i</sub> = 1 + ε<sub>i</sub> s<sub>j</sub> = 1 + δ<sub>j</sub></p><p class="body-paragraph">Then</p><p class="body-paragraph">a<sub>ij</sub>(1 + εij) = (1 + ε<sub>i</sub>a<sub>ij</sub>(1 + δ<sub>j</sub>) = a<sub>ij</sub>(1 + ε<sub>i</sub> + δ<sub>j</sub> + ε<sub>i</sub>δ<sub>j</sub>)</p><p class="body-paragraph">or</p><p class="body-paragraph">(37) ε<sub>ij</sub> = ε<sub>i</sub> + δ<sub>j</sub> + ε<sub>i</sub>δ<sub>j</sub></p><p class="body-paragraph">Thus the RAS adjustment procedure is equivalent to the problem of relative coefficient change with the share ε<sub>ij</sub> of the relative change given by equation (37). Moreover, for small relative changes in ε<sub>i</sub> and δ<sub>j</sub> the products ε<sub>i</sub>δ<sub>j</sub> can be ignored and the approximation</p><p class="body-paragraph">(38) ε<sub>ij</sub> ∼ ε<sub>i</sub> + δ<sub>j</sub></p><p class="body-paragraph">can be used.</p><p class="body-paragraph">This generalization can also be applied to extended input-output models, such as demographic input-output (see Sonis & Hewings (1991) for a formal statement).</p><a name="AN0009954671-5"></a><span class="medium-bold"><a id="hd_toc_265" title="4. Empirical Applications " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#toc"><strong><font color="#0033ff" size="2">4. Empirical Applications </font></strong></a></span><p class="body-paragraph">In this section some empirical applications of the concept of the field of influence will be illustrated with reference to Washington State, Sri Lanka, Brazil and Canada. The applications have been chosen to illustrate different insights into the structure of economies that may be revealed from the application of the field of influence.</p><p class="body-paragraph">4.1. Changes over Time: Single Elements</p><p class="body-paragraph">It has long been known that aggregate measures of structural change often hide very significant reorganizations that may have taken place in the nature and strengths of linkages between industries. This was particularly true for the regional economy of the State of Washington in the USA; only through an examination of the changes associated with individual coefficients could a real understanding be provided of the nature of change. The coefficients which generated the largest fields of influence for each of three years, 1963, 1967 and 1972 (years for which input-output tables were available), were identified using the same criteria. Figure 1 shows a representative portrayal for 1963 of the top 33 coefficients with the largest fields of influence while Table 1 summarizes the degree to which coefficients (purchases and sales) appeared in these top ranks for all three years. A more elaborate application to Canada is shown in Figure 2; here the top 20 elements are displayed (based on ranking in the initial year) for the period 1961-75. While the five top-ranked elements retained their rankings over time, there were considerable movements in rankings of elements 6-20.</p><p class="body-paragraph">In both cases, there appears to be the basis for useful distinction to be made between those elements that retained their ranking as important and those that either diminished or increased. The actual interpretation of causality in these cases is beyond the scope of this paper.</p><p class="body-paragraph">4.2. Changes in Fields of Influence Created by Model Extension</p><p class="body-paragraph">Not only do fields of influence change over time, they also change when the model is adjusted or expanded. Figure 3(a) portrays the elements with the largest fields of influence for the Sri Lankan economy interindustry account. When the same criteria are applied to the full social accounting system (Figure 3(b)), the non-interindustry elements dominate to the exclusion of the interindustry elements. This finding has been replicated in many studies.</p><p class="body-paragraph">4.3. Changes over Time: Synergetic Effects</p><p class="body-paragraph">The final illustration uses analysis of the Brazilian economy; Figure 4 shows the dominant two-element or synergetic fields of influence. In Hewings et al. (1989), these changes were related to some of the major economic development strategies promulgated by the government over a 20 year period.</p><a name="AN0009954671-6"></a><span class="medium-bold"><a id="hd_toc_283" title="5. Conclusions " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#toc"><strong><font color="#0033ff" size="2">5. Conclusions </font></strong></a></span><p class="body-paragraph">The theoretical procedures outlined in this paper provide significant opportunities for novel approaches to the interpretation of structure. Some earlier thinking has suggested that the field of influence might form one of the bases for a taxonomy of economies (see Jensen et al., 1988). The process of innovation diffusion and adoption is another logical candidate for consideration; Hewings et al. (1988) have provided some suggestions about the way coefficient change and the field of influence concept might be an important explanatory tool in the interpretation of the processes of innovation diffusion (see also Forsell, 1989).</p><p class="body-paragraph">The issue of structural change in general has received increasing attention, especially as economy-wide models, and not just the input-output part, become ever more complex and disaggregated. A more complete evaluation of the theory and its empirical applications, in comparison with other techniques, may be found in Hewings et al. (1991) and Sonis et al. (1991). In these cases, the field of influence approach is compared with structural path analysis, superposition decomposition principles and the familiar multiplicative decompositions of Pyatt & Round (1979).</p><p class="body-paragraph">Finally, it should be noted that the viewpoints presented here, especially as related to the methodology of coefficient change, are different from a Taylor series expansion. The methodology provides a finite form, one that is eminently capable of realization in the form of a computer algorithm. This meso-economic approach also provides the possibility of uncovering the hierarchical structure of change through identification of the intensity of influences, an alternative and complementary approach to the structural path analytical methods illustrated by Defourny & Thorbecke (1984).</p><p class="body-paragraph">The support of the US National Science Foundation Grants SES 84-10917 and SES 88-22459 to Hewings is gratefully acknowledged as is the provision of funds from the National Center for Supercomputing Applications to support computation on the Cray X/MP at the University of Illinois at Urbana-Champaign. Dr J. K. Lee provided computer programming assistance for some of the empirical applications.</p><a name="AN0009954671-7"></a><span class="medium-bold"><a id="ref_toc" title="Notes " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#toc"><strong><font color="#0033ff" size="2">Notes </font></strong></a></span><p class="body-paragraph"><em><a name="bib1"></a><a id="bib_upbib1" title="(n1.)" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib1up"><font color="#0033ff">(n1.)</font></a> In the interests of conserving journal space, the formal proofs are omitted; they may be obtained on request to the second author.</em></p><p class="body-paragraph"><em><a name="bib2"></a><a id="bib_upbib2" title="(n2.)" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib2up"><font color="#0033ff">(n2.)</font></a> It should be emphasized that the order of columns and rows in B<sub>or</sub> is essential.</em></p><p class="body-paragraph"><em><a name="bib3"></a><a id="bib_upbib3" title="(n3.)" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib3up"><font color="#0033ff">(n3.)</font></a> For example, if the transactions are represented in energy units rather than monetary flows, one would expect there to be differences in inverse importance.</em></p><a name="AN0009954671-8"></a><span class="medium-bold"><a id="hd_toc_295" title="Table 1. Fields of influence, Washington State, 1963, 1967, 1972 " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#toc"><strong><font color="#0033ff" size="2">Table 1. Fields of influence, Washington State, 1963, 1967, 1972 </font></strong></a></span><pre class="ct">Legend for Chart:
B - Purchases
C - Sales
A B C
Appeared in 1 year 40 42
Appeared in 2 years 13 15
Appeared in 3 years 9 7
Appeared only in 1963 and 1967 3 7
Appeared only in 1963 and 1972 2 2
Appeared only in 1967 and 1972 8 6
</pre><p class="body-paragraph">GRAPH: Figure 1. Cells with the largest fields of influence, Washington State, 1963.</p><p class="body-paragraph">GRAPHS: Figure 2. Fields of influence in the Canadian economy: changes in rank, 1961-75. Top 20: (a) ranks 1-5; (b) ranks 6-10; (c) ranks 11-15; (d) ranks 16-20.</p><p class="body-paragraph">GRAPHS: Figure 3. Fields of influence in Sri Lanka: (a) interindustry part of the social accounting matrix; (b) the complete social accounting matrix.</p><p class="body-paragraph">GRAPHS: Figure 4. Fields of influence for two-element changes in Brazil: (a) 1959; (b) 1970; (c) 1975.</p><a name="AN0009954671-10"></a><span class="medium-bold"><a id="ref_toc" title="References " href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#toc"><strong><font color="#0033ff" size="2">References </font></strong></a></span><p class="body-paragraph"><em>Bullard, C. W. & Sebald, A. V. (1977) Effects of parametric uncertainty and technological change in input-output models, Review of Economics and Statistics, 59, pp. 75-81.</em></p><p class="body-paragraph"><em>Bullard, C. W. & Sebald, A. V. (1988) Monte Carlo sensitivity analysis of input-output models, Review of Economics and Statistics, 70, pp. 705-712.</em></p><p class="body-paragraph"><em>Defourny, J. & Thorbecke, E. (1984) Structural path analysis and multiplier decomposition within a social accounting matrix framework, Economic Journal, 94, pp. 111-136.</em></p><p class="body-paragraph"><em><a name="bib4"></a><a id="bib_upbib4" title="" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib4up"></a>Dietzenbacher, E. (1990) Seton's eigenprices: further evidence, <strong><em>Economic Systems Research</em></strong>, 2, pp. 103-123.</em></p><p class="body-paragraph"><em><a name="bib5"></a><a id="bib_upbib5" title="" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib5up"></a>Forsell, O. (1989) The input-output framework for analysing transmission of technical progress between industries, <strong><em>Economic Systems Research</em></strong>, 1, pp. 429-445.</em></p><p class="body-paragraph"><em><a name="bib6"></a><a id="bib_upbib6" title="" href="http://web.ebscohost.com.ezproxy.lib.ucalgary.ca/ehost/detail?vid=3&hid=106&sid=98eefa23-f6bf-4e04-b4ba-8c0c5a46a42d%40sessionmgr107#bib6up"></a>Giarratani, F. & Garhart, R. E. (1991) Simulation techniques in the evaluation of regional input-output models: a survey, in: J. J. LI. Dewhurst, G. J. D. Hewings & R. C. 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