Spatial aggregation in multiregional input-output models

Environment and Planning A, 1983, volume 15, pages 187-206

Spatial aggregation in multiregional input-output models

P Blair, R E Miller Regional Science Department, University of Pennsylvania, Philadelphia 19104, USA Received 19 October 1981; in revised form 19 February 1982

Abstract. This study continues the investigations into effects of spatial aggregation in connected, several-region input-output models that were initially reported upon in Miller and Blair (1980; 1981) for an interregional input-output model. In the present paper our interest is in the error that is introduced into a multiregional input-output model through aggregation of regions. As in the earlier paper, we investigate the question by means of both hypothetical and real data. In the present study, these are (1) randomly generated multiregional data incorporating varying degrees of regional linkage and (2) actual 1963 US multiregional data. To allow some comparison with the interregional experiments, the data for the random experiments in the present paper are derived from our original randomly generated interregional data. The same levels of spatial interconnectedness as in the interregional experiments are also used here, and the same measures of aggregation error are employed. As in the interregional case, we conclude that spatial aggregation in multiregional input-output models does not necessarily lead to unacceptable error.

1 Introduction This paper reports the results of a set of experiments designed to examine the effects of spatial, as opposed to sectoral, aggregation in multiregional input-output models. It is entirely similar in objectives to our earlier reports on the effects of such aggregation in interregional input-output models (Miller and Blair, 1980; 1981) (1) . One principal conclusion of this earlier work was that, by our measures, the amount of error introduced by spatial aggregation into the results of analyses using the interregional input-output framework was remarkably small in a rather wide variety of cases. The interregional model may be viewed as the most ideal and detailed form of completely connected regional input-output model. However, because of its immense data requirements it is seldom implemented for anything approaching a large real-world situation. On the other hand, the reduced amount of data necessary for the multiregional model makes it likely that this form of regionally connected input-output model may be widely used in actual studies. Since there is a well-known multiregional input-output model for the United States, it may be of real practical concern to know how important spatial detail is in the multiregional framework. The experiments reported upon here are very similar in structure to those described in our earlier papers, and readers are referred to either of those papers for details. In section 2 of this paper, we describe the initial experiments, by use of data from the earlier interregional cases. Sections 3 and 4 contain results in the form of mean absolute percentage errors from the experiments by use of the random data and the actual US multiregional data, respectively. In section 5 we offer some concluding thoughts. In appendix 1 we sketch and compare the structures of the interregional

^ Readers are referred to either of these papers for comments on earlier work on sectoral aggregation in regional input-output models, as in Doeksen and Little (1960), Williamson (1970), and Hewings (1972). In addition, our earlier papers contain comments on some of the standard results on sectoral aggregation in input-output models, as in Theil (1957) and Morimoto (1970), and in particular on the relevance of some of the standard theorems on sectoral aggregation bias to cases of spatial aggregation. See also appendix 2.

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and multiregional input-output models. The former is most closely associated with Isard, who described the model in detail some years ago (Isard, 1960, chapter 8). The latter has been presented by Polenske and her associates in a series of publications, perhaps the most comprehensive of which is Polenske (1980) (2) . This appendix also contains a numerical example to illustrate how the random data originally generated for the interregional model are converted for use in the multiregional case. In appendix 2 we note two measures of aggregation bias and present values of these measures for our experiments. Finally, appendix 3 contains the regional and sectoral breakdowns for the nine-region, ten-sector multiregional input-output model used in section 4. 2 Initial experiments: random matrices 2.1 Generating multiregional from interregional data In the interregional input-output model, it is necessary to have data on sales by sector i in region S to sector / in region T, z%T. Let ZST = [z%T ]. Given sectoral gross outputs by region, XjT, with XT = [XfT], one derives interregional input coefficient matrices containing elements a§T, a$T = zfjrlXjr\ thus AST = [a%T] = ZST(XT)~l, where X is the diagonal matrix formed from the vector X. (When S = T, this is an interregional coefficient matrix.) For the multiregional model, sales by sector /, regardless of location, to sector / in region T, z}f, are needed (with a 'dot' to indicate summation over that index). Then ZT=

[Zf],

(1) T

and a regional input coefficient matrix containing elements a'J — z\flXj AT = z ar ( X r r i .

is derived as (2)

Connections between regions in the multiregional model are reflected in commodity shipment data, qfT, which indicate sales of good i from region S to all purchasers (all sectors plus final demand) in region T. For commodity /, let

Q(0 =

qLL

qLM

qML

qMM

(3)

Dividing each element in, say, the first column of Q(z) by the sum of the elements in that column gives the proportion of the total amount of good i in region L that comes from each particular region. For example,

cfL = yir,

(4)

[where V^ is the sum of the elements in the first column of Q(j)] shows this proportion from region M. If it is assumed that there are n sectors in each region, these proportions are arranged into ^-element vectors that are specific to each origin-destination pair. For example, CLM = \c\M, ..., c£ M ], and the multiregional input-output model uses the product C LM A M as an approximation of ALM in the interregional model, where CLM is the diagonal matrix formed from vector CLM. The fundamental relationships for a ^ As in the empirically implemented model for the USA, we utilize the column-coefficient form of the multiregional model. This embodies assumptions regarding regional trade patterns that were explored by Chenery and by Moses some years ago; hence it is also known as the Chenery-Moses form of the multiregional input-output model. See Polenske (1970) for details.

189

Spatial aggregation in multiregional input-output models

two-region model are therefore ( I - CLLKL)XL

- QLM^MXM

=

—QML KL VL I Q _ QMMSM\YM

QLLyL

+

= QMLYL

QLMyM +

?

(5)

QMMyM

If one lets A =

~AL\

pLL i QLM

0

C =

QML\QMM

X =

the multiregional input-output model is ( I - C A ) * = CY ,

(6)

and, therefore, X = (I-CA^Cy.

(7)

(Appendix 1 discusses and contrasts the interregional and multiregional models in more detail.) To preserve as much comparability as possible with our previous interregional input-output experiments, we have attempted to utilize the uniformly distributed, randomly generated flow data that were originally produced for those experiments. Thus, for example, instead of generating random flow data zf that are needed, as in equation (1), for the regional coefficient matrices, A r , in equation (2), we have derived the ZT data for our current experiments from the earlier (interregional), more detailed ZST kinds of flows. The 50 x 50 Z matrices from our interregional system with five regions of ten sectors each were of the form -ZLL

JLM

ZLN

jLP

jLQT

ZNL

(8)

PL

Z

ZQL

ZQQ

where each submatrix, ZST, is 10 x 10. For the multiregional experiments, we form 10 x 10 matrices of the sort ZL—as in equation (1)—by adding together the matrices in the first (block) column of Z in equation (8). That is, ZL

=

ZLL

_|_ ZML

+

ZNL +

tfL

+

ZQL

^

(9)

These matrices record the zf information—the region of origin of the shipment is ignored. Along with the information, from the previous experiments, on XL, this allows calculation of the regional technical coefficient matrices, such as AL, as in equation (2). Derivation of the commodity flows, qf%T, as in equation (3), from the interregional data is more complicated, since both interindustry and final demand shipments must be accounted for in these qlT elements, whereas the interregional flow data, as in Z in equation (8), record only interindustry shipments. We define Q(z) = [qfT]—here of size 5 x 5—to be the counterpart to Q(0 in equation (3) which contains only interindustry sales; the elements of these matrices are easily found as the zth row sums in each of the twenty-five submatrices in Z in equation (8). What is needed next is some distribution of the final demand figures, from the interregional experiments, into 5 x 5 matrices F(z), F(i) = [ffT] ,

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P Blair, R E Miller

which record final demand shipments of good i between and within regions. In the interregional model, the z'th element of Ys contains final demand uses of good i within region S plus shipments of good i to all other regions for final demand purposes. We have to allocate this yf to each region in the same proportion as purchases in that region of good / for interindustry purposes are to total interindustry sales of good i by region S. That is, aST

ftST

Hl

—

(10)

,,S

r Then, our construction of the matrix Q(z), in equation (3), is Q(0 = Q(0 + F ( i ) .

(11)

The regional supply proportions are then found from these Q(/), as in equation (4), and the model is used, as in equation (7). Appendix 1 contains a numerical example that illustrates this derivation of a multiregional input-output data set from the original interregional input-output data. 2.2 General structure of the random matrix experiments Figure 1 indicates the overall structure of these experiments. There are twenty different initial 50 x 50 Z matrices, from the five-region, ten-sector interregional case, the elements of which were randomly drawn from a uniform distribution of the integers between 0 and 100. In conjunction with a random fifty-element final demand vector, Y (produced by drawing randomly from a uniform distribution of integers from 1 to 9, then multiplying by 100), a fifty-element gross output vector is found, as usual, as X = Zi + Y (where i is an appropriately sized column vector containing ones). In conjunction with the ZL matrices, as in equation (2), the regional technical coefficient matrices, AL are found.

* i , . . . , Ys

(I-CAr^C

Y6,...,Y10

' d-CA^cr,

(I-CA^Cld f

'region-r results

'all-region' results

Figure 1. General structure of multiregional spatial aggregation experiments.

Spatial aggregation in multiregional input-output models

191

In the first set of experiments (Yl9..., Ys), only the first ten elements of any particular Y were nonzero. These were the so-called 'region 1' experiments in the interregional case. That is, these were assumed to be (new) final demands for the outputs of region-1 sectors only; for example, new consumer demand for home computers made in California (region 1). In the multiregional input-output model, one can assess the impact of new consumer demand, in California, for output of the home computer manufacturing sector, but with the realization that the C 11 and C 21 , C 31 , ... matrices determine the proportions of the final demand amounts that are supplied by California manufacturers and imported into California from other regions in order to satisfy the new final demand. This is a consequence of premultiplication of Y by C on the right-hand side of the multiregional input-output model, in equations (5), (6), and (7). In these experiments, then, the random Yt {i = 1, ..., 5) are not distributed across 'receiving' regions. We interpret the nonzero elements in Y1 to be strictly multiregional final demands, satisfied from production in 1 or through imports to 1. Thus the label 'region-1' experiments has a somewhat different interpretation here than in the interregional input-output experiments. In the second set of experiments (Y6, ..., Y10), all fifty elements in each Y vector were nonzero. These were the 'all-region' cases in the interregional input-output model, and that terminology is still appropriate here. Each Yt (i = 6, ..., 10), from the interregional experiments, is 'reconstructed' (as illustrated in appendix 1) to a compatible multiregional final demand vector. In both region-1 and all-region experiments, spatial aggregation then proceeds, as before, from a five-region to a two-region model, with region 1 always spatially distinct. 2.3 Strength of spatial linkages In the interregional experiments, we introduced a measure of spatial interconnectedness, a (0 < a < 1), by which we multiplied the elements in the off-diagonal matrices in equation (8)—ZST, for S =£ T— so as to alter the level of interregional dependence. As a -> 0, the regions become less and less connected, economically; at a = 0, only the on-diagonal matrices—ZST, for S = T— contain nonzero elements and the regions are not connected (economically) at all. In the current multiregional experiments, the off-diagonal elements in each Q(/)— the qfT, for S =£ T—were multiplied, in turn, by each new a. This is equivalent to multiplying the elements in off-diagonal ZST matrices in equation (8) by a. This produced a new set of interindustry flows, Q(i)a. By means of the proportions from this modified Q(z) matrix, as in equation (10), the original final demands were redistributed to reflect the changed intensity of spatial linkages and a modified F(i)a matrix was formed. Then Q(0« = Q ( 0 * + F ( l ) t t ,

(12)

for each different a level. Notice that in the limit, when a = 0, there will be no zwterregional shipments, and the elements in YL will be interpreted as final demands in region L only; both Q(0 and

(A11>

a¥L

c?L = ^ r •

(A12)

Multiplied by 100, these are regional supply percentages. Arranging these proportions for all goods for a given pair of regions in a vector, one has CLM

=

[ ^

cLMf

CLM]

( A 1 3 )

And, similarly, one can construct CLL, CML, and C M M , all three-element vectors in our two-region, three-sector example. The crucial assumption in the column coefficient (Chenery-Moses) version of the multiregional input-output model is that these (observable) proportions represent average behavior for all purchasers of each good i imported from region L to sectors in region M. If c^M = 0-4, then all consumers of good 1 in region M are assumed to purchase 40% of their needs of that good from region L. If vectors such as CLM are

Spatial aggregation in multiregional input-output models

199

diagonaliz sd as in CLM

~c\M 0 0 = 0 c\M 0

0

ci

0

(A14) M

then the p>roduct (A15) is the estimate of the multiregional input-output model for the interregional relationships embodied in A LM . Similarly, C LL A L , C ML A L , and CMMAM are used as proxies for their interregional counterparts ALL, AML, and AMM, respectively. The trade proportions represented by vectors such as CLM in equation (Al3) also relate to final demand purchases in region M, since the underlying flow data, in equation (A9), measure both interindustry and final demand sales. Therefore the multiregional input-output equivalent to the two-region relationships in equation (A6) is (I - C LL A L )Z 1 ' - c L M A M Z M = CLLYL + CLMYM , _fML

AL vL

_j_ / T _ QMMAM\

VM

_

QMLYL

4- C\MMYM

] \

(A16)

\

In the multiregional model, YL and YM represent final demands (for example, of consumers) located in region L or region M for goods, irrespective of their regional origins. For YL, for example, the CLL and CML matrices determine how much of the final demand is satisfied from within L and how much is imported from M. If one lets AL .0

A =

\oM

(Al 7)

\A _

and -QLL \CLM

r = ^

(A18) '

QML\QMM

the multiregional counterpart to A in equation (A2) is QLLAL

I

QLMAM

(Al 9)

CA = and the counterparts to equations (A4) and (A5) are (I-CA)Z=

CY,

(A20)

and

x = (i-cA^cy.

(A21)

Generating multiregional from interregional data: an illustration For a two-region, two-sector interregional input-output example, let the randomlygenerated 4 x 4 matrix Z, as in equation (Al), be given by LL

Z ir =

~Z

LM

i Z

gML j 2,MM

~16 6 39 89 66 31 90 92

15 43 7 91

47~ 42 82 78 J

(A22)

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P Blair, R E Miller

(where the subscripts ir and mr will be used, where necessary, for the interregional and multiregional models). Let the randomly generated Y vector be given by 171 573 "32 804

-YL-

Yir =

Y'M

(A23)

Thus, regional gross outputs are 255" XL

Xir =

rM

= Zi+Y

786 "218 1155

=

(A24)

For the multiregional model, We find ZL and ZM, as in equation (A8), by adding together the matrices in a given (block) column of equation (A22), as in equation (9): 82 129

ZL

37 181

and

ZM =

22 .134

129 120.

(A25)

In conjunction with the gross outputs in equation (A24), we find AL and A"*, as in equation (A8): A

i

0-0471 0-2303

0-3216 0-5059

=

and

AM =

0 1009 0-6147

0-1117 0-1039

(A26)

Finally, then, A, as in equation (A 17), is given by

A =

0 0

0-3216 0-0471 0-5059 0-2303

0 0

o

b

o"i"6o9"6""iiT7

0

0

0-6147 0-1039

(A27)

In addition, from the row sums of the submatrices in equation (A22), Q(i) =

22 97

62 89

and

Q(2)

128 85 182 169

(A28)

Denote by P(i) = [pf.T] the matrices of the proportions of each of the elements in Q(0 to the row sums of those matrices. These are the proportions on the right-hand side of equation (10). Then

P(D =

0-2619 0-7318 0-5215 0-4785

and

P(2) =

0-6009 0-3991 0-5185 0-4815

(A29)

The individual elements in Yb [from equation (A23)] are: y{ = 171, y% = 573, 32> a n ( i yM = 394 T n e fjrst of these, y\, is to be distributed according to the proportions in the top row of P(l), that is, y\L = (0-2619)(171) = 44-79 and yLM = (0-7381)(171) = 126-21. Using the proportions in the second row of P(l) for y? = 32 gives y?L = (0-5215)(32) = 16-69 and yfM = (0-4785)(32) = 15-31. The distributions of final demands for good 2 are formed similarly, from the proportions in P(2). Thus yM =

F(l) =

44-79 16-69

126-21 15-31

and

F(2) =

344-34 416-89

228-66 387-11

(A30)

Spatial aggregation in multiregional input-output models

201

Therefore, as in equation (11), Q(D =

~ 66-19 113-69

188-21 104-31

and

Q(2) =

472-34 598-89

313-66 556-11

(A31)

[Notice, as pointed out earlier in this appendix, that the row sums in Q(7) equal the appropriate elements in the total gross output vector, X.] From the proportions of the elements in Q(l) and Q(2) to the column sums in those matrices, we find, as in equation (A 13) that CLL = [0-3701

0-4409],

CML = [0-6299 0-5591] ,

CLM = [0-6434 0 - 3 6 0 6 ] , CMM = [0-3566 0-6394]

'

and, therefore, C in equation (Al8) is given by "0-3701

0

i 0-6434

0 0-4409 i 0 6"