Visiting Researcher, Hirosaki University
Michiya NOZAKI
1. Introduction
CIRAC (Chubu Industrial and Regional Advancement Centor) compiled an Interregional Input-Output Table for Chubu region on 31th March, 2011. Outline of the Study Region is Prefectures as Toyama, Ishikawa, Fukui, Nagano, Gifu, Shizuoka, Aichi, Mie, Shiga, and the rest of Japan. The aim of compiling an Interregional Input-Output Table for Chubu region in Japan (for the year of 2005) is mainly to be utilized as the database of statistically exploring economic experiences in  Chubu region. CRISER (2014) compiled the updated interregional input-output table for Chubu region (the year table for 2010). CIRAC (2011; 2012) and CRISER (2013) are to consider such two features of Chubu region as the Openness (or the external conditions) and Diversity (or the internal situations), with an aid of newly compiled Interregional Input Output Table for Chubu region (the year table for 2005). This paper describes the production processes in the economy of Chubu region (i.e. the central part of Japan) by using an interregional input-output framework. Interdependencies in the production structure of different prefectures in Chubu region are relevant in a simultaneously progress of the globalization and localization where perturbations quickly propagate from one prefecture to another through industries and markets. We use the updated interregional input-output table for Chubu region (the year of 2010), with the compiling by CRISER (2014). Many studies deal with the question of how such interdependencies or linkages can be accurately measured (e.g. Chenery and Watanabe, 1958; Rasmussen, 1956; Miller and Lahr, 2001; Sonis and Hewings, 1992). These studies have proposed various alternative measures for such inter-industry linkages. In this paper, we will present the following contents. In section 2, we present the interregional input-output framework with two-region and two-industry for simplicity. In section 3, we present Average Propagation Length model according to Dietzenbacher, Romero, and Bosma (2005), and redefine it along with Oosterhaven and Bouwmeester (2013). In section 4, we use the concept of average propagation length (APL), which was presented by Dietzenbacher, Romero, and Bosma (2005), to study Chubu region empirically. At last section, we present the concluding remarks.
2i
The Interregional Input-Output Framework
For the purpose of explanation, we will consider a two region economy. Using r and s, for the regions, let there be two producing sectors(1,2) in region r, and two producing sectors(1,2) in region s. Suppose that one has information for region r on both intraregional flows,  z ij , and interregional flows,  z ij . Suppose, further that the same kind of information is available the use of inputs by firms located in region s,  z ij   and  z ij . This complete table of interregional data can be represented as
In the regional model as above, we used intraregional information only. We would like to incorporate much more explicitly the interregional linkages, as represented by information in  z rs  and  z sr . These off-diagonal matrices need not be square. The on-diagonal matrices are always square. While the elements in Z rs  represent exports from region r and simultaneously imports to region s, it is usual in regional input-output work to refer to these as interregional trade flows and to use the terms export and import when dealing with foreign trade that crosses national, not just regional, boundaries. Let us consider the basic equation for the distribution of sector i's product. In the two-region interregional input-output model, that part of  fi  that represents sales of sector i’s product to the productive sectors in the other region is removed from the final-demand category and specified explicitly (Miller and Blair (2009), pp.77-79).
3. A Reconsideration on Average Propagation Length (APL)
When we define average propagation, we analyse how a cost-push or a demand-pull propagates throughout the industries in the economy. According to Dietzenbacher, Romero, and Bosma, (2005, 411-412), an initial demand-pull in industry i increases the output value in industry j by  l ij  — δ ij  (neglecting the initial effects). δ ij  is the Kronecker delta; i.e. δ ij = 1  if i=j and 0 otherwise. The share  α ij /(lij − δ ij )  of this output increase requires only one round, but the share  [A 2 ] ij /(l ij − δ ij )  requires two rounds to get from i to j.  [A²] ij  denotes the element ( ij ) of matrix  Ak , which differs from (bij)k. That is, these effects go through one other industry k (= 1,..., n),   [A2] ij  = ∑ k aikakh. The share  [A³] ij /( l ij  − δ ij )  requires three rounds, going through two other industries, since  [A³] ij  = ∑ κ α κ h α h j .  The share of total effect that requires k rounds amounts to [Ak]į/(lįj − δįj), where the shares are non-negative and add to 1. The average number of rounds required to pass over a demand-pull in industry i to industry j yields
Let the numerator of the right-hand side of (3-1) be denoted by  h ij , with H =  ∑ k k A k . Then the terms  h ij  can easily be calculated by using H = ∑ k k A k = L(L − I). Equality can be shown as follows. Note that  L = (I − A)− 1 (where A represents  output coefficients); so
Dietzenbacher, Romero, and Bosma (2005, 411-412) define that the matrix V of APL is as follows
Oosterhaven and Bouwmeester (2013, 484) consider a one-sector economy in an extreme case. In a one-sector economy, aij  reduces to a = ∑j ∑i zij/ ∑j xj , while  b =Ei ∑j zij/ ∑i ¥i , where xj indicates the total output of industry j. In a one-sector economy, equation (3-1) simplifies to: v ij = {1 a + 2a² + 3a³ + … + Kak}/(a + a² + a³ + … + ak) (3-1-OB) For the limit of  a  ⇒ 0, Oosterhaven and Bouwmeester(2013, 484) get the outcome that, irrespective of the cutoff value of K, the APL  v ij ⇒ 1. Therefore, we redefine the matrix V of APL along with Oosterhaven and Bouwmeester (2013, 484) as
In the same way, alternatively, we can define the APL for a cost-push. Analysing how a one-yen cost-push increase in industry j affects the total output of industry i, we find bij+[B²]ij+[B³]ij + …=gij − δij. The APL for a cost-push yields {1bij + 2[B²]ij + 3[B³]ij + … }/(gij − δij)
Note that input matrix A and output matrix B are related to each other.
4.
The Empirical Studies for Chubu region
CIRAC (Chubu Industrial and Regional Advancement Centor) compiled an Interregional Input-Output Table for Chubu region on 31th March, 2011. Outline of the Study Region is Prefectures as Toyama, Ishikawa, Fukui, Nagano, Gifu, Shizuoka, Aichi, Mie, Shiga, and the rest of Japan. The aim of compiling an Interregional Input- Output Table for Chubu region in Japan (for the year of 2005) is mainly to be utilized as the database of statistically exploring economic experiences in Chubu region. CRISER (2014) compiled the updated interregional input-output table for Chubu region (the year table for 2010). In this paper, we are to test the methodology of APLs for an  Updated Interregional Input-Output Table for  Chubu region (the year table of 2010 for 34 sectors). To get a general idea of the role of a certain sector, consider the average. The column with averages gives the row-average forward APL of a sector. It is the largest for Transportation equipment and the smallest for commerce. This shows that Transportation equipment can be viewed as being situated at the beginning of production chains in the regional economy in Aichi Prefecture. Commerce is principally oriented towards final demand and there are not many important forward linkages from commerce. The row with averages gives the column average of the backward APLs of a sector. The largest average values are found for Petroleum and coal products and the smallest for construction. Summarizing, a large average forward APL and/or a small average backward APL for a sector, suggest that this sector is at the beginning of a set of production chains. Small average forward APLs and/or large average backward APLs point at a place near the end of some production chain. "In line with the development of the propagation length, our choice for the type of linkage is based on the total size of the effect of a cost-push and the effect of a demand- pull. Neglecting the initial effects, these effects are given by G一I and L一I, respectively. Instead of using the Leontief inverse for the backward linkages and the Ghosh inverse for the forward effects, we have taken the average. Recall that the propagation length is the same, no matter whether a cost-push or a demand-pull is considered. Therefore we also measure the linkages by taking both directions into account. So, the linkages are given by the elements of the matrix F " (Dietzenbacher, Romero, and Bosma, 2005, 415), which is defined as follows.
"The element f ij gives the size of the linkage and equals the average of the forward effect of a cost-push in sector i on the output in sector j and the backward effect of a demand-pull in sector j on the output in sector i (Dietzenbacher, Romero, and Bosma, 2005, 416). There seems to be an inverse relationship between APLs and elements f ji. The Pearson correlation coefficient equals -0.0613. Hence, lower APL values are, to some extent, associated with larger linkages. This implies that the largest impacts between two sectors are often those that are essentially direct. The APL indicates the “distance” between two sectors by expressing the average number of steps it takes to transmit a cost-push (or demand-pull) from one sector to the other. The procedure is to take APLs into account only if the linkage is sufficiently large, using a threshold value a. Further, the APLs are rounded to the nearest integar. From the matrix V with APLs and the matrix F with linkages we construct a new matrix S in following way (Dietzenbacher, Romero, and Bosma, 2005, 416).
where int( v ij ) is used to indicate the nearest integar to which  v ij  has been rounded. For the calculations with the 34 sector classification, we have used a threshold value a =0.00573. The Pearson correlation coefficient between APL and the matrix F of the Chubu region equals -0.0613. Hence, lower APL values are, to some extent, associated with larger linkages. This implies that the largest impacts between two sectors are often those that are essentially direct. Then, we can similarly construct a new matrix S of the Chubu region as (4-2) where int( v ij ) is used to indicate the nearest integar to which  v ij  has been rounded. For the calculations with the 34 sector classification, we have used a threshold value  a =0.00573 (See, Dietzenbacher, Romero, and Bosma, 2005, 416).
5. Concluding Remarks
We present our concluding remarks as follows: Firstly, Oosterhaven and Bouwmeester(2013, 484) consider a one-sector economy in an extreme case. In a one-sector economy,  a ij  reduces to a = ∑ j ∑ i Z i j / ∑ j x j , while  b =Σι ∑ j Zιj/ ∑ i × i  , where  x j  indicates the total output of industry j. In a one sector economy, equation (3-1) simplifies to  v ij  = {1 a + 2a 2 + 3a 3 + … + Kακ }/(α + α² + α³ + … + ακ ). For the limit of  α  ⇒ 0, Oosterhaven and Bouwmeester (2013, 484) get the outcome that, irrespective of the cutoff value of K, the APL  v ij ⇒ 1. Therefore, we redefine the matrix V of APL along with Oosterhaven and Bouwmeester (2013, 484) as:
Secondly, by analysing interregional trade with regard to the structure of 9 prefectures in Chubu region, we found strong economic relations between the four prefectures of this region (Aichi, Mie, Gifu, and Shizuoka prefectures).
Reference
Dietzenbacher, E., Romero, I. and Bosma, N.(2005),"Using Average Propagation Lengths to Identify Production Chains in the Andalusian Economy," Estudios de Economia Aplicada,Vol.23, No.2,pp.405-422. Dietzenbacher, E.and Romero, I.(2007), "Production Chains in an Interregional Framework: Identification by means of Average Propagation Lengths," International Regional Science Review , Vol.30, No. 4, pp.362-383. CIRAC (2011) An Interregional Input-Output Table for Chubu Region – Regional Economic Structures of Chubu Region - (in Japanese),Chubu Industrial and Regional Advancement Center (CIRAC). CRISER(2014) An Updated Interregional Input-Output Table for Chubu Region for the year of 2010, Chubu Region Institute for    Social and Economic Research (CRISER). Oosterhaven, J. and Maaike C. Bouwmeester (2013), 'The Average Propagation Length: Conflicting Macro, Intra-industry, and Interindustry Conclusions,' International Regional Science Review, Vol. 36 No.4, pp. 481-491.