Productive Relations in the Northeast and the Rest-of-Brazil Regions in 1995: Decomposition and Synergy in Input-Output Systems

Abstract

Using a set of interregional input-output tables built by Guilhoto (1998) for 1995 for two Brazilian regions (Northeast and rest of the economy), the methodology developed by Sonis, Hewings, and Miyazawa (1997) is applied in the construction of a series of linkages such that it is possible to examine, through the nature of the internal and external interdependencies, the structure of trading relationships between the two regions. The methodology uses a partitioned input-output system and exploits techniques that produce left and right matrix multipliers of the Leontief Inverse. This procedure facilitates the classification of the types of synergetic interactions within a preset pairwise hierarchy of economic linkages subsystems. In general, the results show that the Northeast region has a greater dependence on the rest of the economy region than the rest of the economy has on the Northeast region, and at the same time the rest of the economy region seems to be more developed as it presents a more complex productive structure than the Northeast region.

In this paper, the methodology developed by Sonis, Hewings, and Miyazawa (1997) that classifies types of synergetic interactions is used to explore the structure of trading relations among regions. This methodology is applied to a set of interregional input-output tables built by Guilhoto (1998) for two Brazilian regions (Northeast and the rest of the economy). The objective is to explore the degree to which the structure of interactions is dominated by intraregional or interregional components and the extent to which the interregional interactions are symmetric in magnitude. The two-region system that has been chosen highlights important, strategic development issues in an economy that is struggling to address both equity and efficiency issues in a spatial context (see Baer, Haddad, and Hewings 1998). The Northeast of Brazil has received significant, continuing development initiatives over the past four decades; however, the results suggest only some modest improvements in welfare. By 1995, the Northeast's share in GDP had risen marginally to 13.4 percent from 13.2 percent in 1960 while per capita GDP grew from 42 percent to 55 percent of the national average. When attention is directed just to shares in industrial production, the Northeast declined from 8.3 percent (1959) to 7.9 percent (1994). The present paper attempts to explore some structural reasons that might shed light on this problem; while the focus will be on the economic structure of the Northeast and the Rest of Brazil (hereafter, NE and RB respectively) at one point in time, 1995, the findings will reflect long-term structural issues that have remained unresolved.

In the next section the theoretical background will be presented. In the second section the theory will be applied to the Brazilian interregional tables, while in the third section policy interpretations will be reviewed prior to the presentation of some concluding comments in the final section.

1. Theoretical Background¹

Consider a two-region, mutually exclusive division of a national economy. Following the adaptation of the Dixit-Stiglitz model by Fujita, Krugman, and Venables (1999), assume that there are two goods, a tradable and a nontradable, and that there are no interactions external to the national economy. Further assume that labor employed in the tradable commodity is mobile between regions and that labor moves to regions paying higher-than-average real wages. Given a transportation cost structure in which costs are assumed to be a linear increasing function of distance, then it can be shown that the equilibrium distribution of production will depend in large part on the magnitude of the transportation costs and their interaction with increasing returns at the firm level and labor mobility. Fujita, Krugman, and Venables (1999) show that with high transportation costs there will be a tendency for production to be divided between the two regions; if labor mobility is limited (by higher transportation or search costs), and the transportation costs are reduced, there is a tendency to develop a core-periphery outcome in which the tradable good becomes concentrated in one of the two regions.

Obviously, with a more complex system in which goods are all tradable to some extent, the search for greater variety by consumers may tend to exacerbate concentration tendencies, tendencies that will be reinforced by the existence of increasing returns. The competition between the NE and RB presents a very strikingly familiar scenario. Transportation costs between NE and RB are high but not high enough to create a protective, spatially monopolistic market in the NE; producers in the RB have been able to exploit scale economies and penetrate the NE market to the exclusion of NE producers. In this paper, the resultant interregional structure will be explored and interpreted using a set of input-output tables that have been manipulated (decomposed) to explore various facets of these structures.

Consider an input-output system represented by the following block matrix, A, of direct inputs:

where A₁₁ and A₂₂ are the quadrat matrices of direct inputs within the first and second regions, and A₁₂ and A₂₁ are the rectangular matrices showing the direct inputs purchased by the second region and vice versa.

The building blocks of the pairwise hierarchies of subsystems of intra/interregional linkages of the block-matrix input-output system are the four matrices A₁₁,A₁₂,A₂₁, and A₂₂, corresponding to four basic block matrices:

This paper will usually consider the decomposition of the block matrix (1) into the sum of two block matrices, such that each of them is the sum of the block matrices (2) A₁₁,A₁₂,A₂₁, and A₂₂. From (1), fourteen types of pairwise hierarchies of economic subsystems can be identified by the decompositions of the matrix of the block matrix A (see Figure 1 and Table 2).

A set of inner regional multipliers, the set of inverse matrices that are the "building blocks" of the synergetic interactions between the economic subsystems are presented in Table 1. There is no unique or preferred decomposition; various alternatives are explored to examine the degree to which the structure of the NE-RB interactions may be best presented and interpreted. Here, some comments are provided on the entries in this table (the bold numbering refers to the corresponding entries in this table).

1. The matrices B₁ = (I - A₁₁)^-1 and B₂ = (I - A₂₂)^-1 represent the Miyazawa internal matrix multipliers of the first and second regions showing the interindustrial propagation effects within each region, while the matrices, A₂₁B₁,B₁A₁₂,A₁₂B₂,B₂A₂₁ show the induced effects on output or input activities in the two regions.

Fig. 1.

Schematic Representation of the Possible Forms of the A₁ Matrix

Table 1.

Inner Regional Multipliers and Their Properties

Table 2.

Taxonomy of Synergetic Interactions between Economic Subsystems

Each entry consists of two levels: in the first level, a description of the structure and the corresponding form of the A₁ matrix is shown. In the second level the additive decompositions of the Leontief block matrix are shown.

2. The expressions

are usually referred to as the Schur complements. The inverses, D₁ and D₂ of the Schur complements (3), are referred to as the Schur inverses for the first and second regions. They represent the enlarged Leontief inverse for one region revealing the induced economic influence of the other region; that is, the Schur inverses represent total propagation effects in the first and second regions.

3. Miyazawa (1966) introduced the concept of left and right external matrix multipliers of the first and second regions, D^L₁₁,D^R₁₁,D^L₂₂,D^R₂₂. These multipliers are incorporated in the multiplicative decompositions of the Schur inverses and they represent the total propagation effects in the first and second regions as the products of internal and external regional matrix multipliers.

4, 5. By introducing the abbreviated Schur inverses, D₁₁,D₂₂, and the left- and right-induced internal multipliers for the first and second regions, B^L₁,B^R₁,B^L₂,B^R₂, one can obtain the multiplicative decompositions of the Schur inverses:

and their corresponding additive representations.

6-10. The formulae for this group of multipliers can be obtained by considering the block matrices:

Those represent the backward and forward linkages of the first region, the second region and the interregional relations of both regions.

The following Schur inverse,

may be referred to as the enlarged Leontief inverse, and the inverses

are called the left and right subjoined inverse matrix multipliers.

Consider the hierarchy of input-output subsystems represented by the decomposition A = A₁ + A₂. Introducing the Leontief block inverse L(A) = L = (I - A)^-1 and the Leontief block inverse L(A₁) = L₁ = (I - A₁)^-1 corresponding to the first subsystem, the outer left and right block-matrix multipliers M_L and M_R are defined by equalities:

The definition (8) implies that

In this paper, the following form of the Leontief block inverse will be used:

This expression can be verified by direct matrix multiplication, using definitions of the Schur inverses and their properties (see Table 1, entries 1 and 2). Further, the application of (9), (10), and (11) will be directed toward the derivation of a taxonomy of synergetic interactions between the two regions. The results are presented in the first and second levels of Table 2, while Figure 1 shows the schematic representation of the possible forms of the A₁ matrices.

Consider the hierarchy of input-output subsystems represented by the decomposition A = A₁ + A₂ and their Leontief block inverse L(A) = L = (I - A)^-1 and the Leontief block inverse L(A₁) = L₁ = (I - A₁)^-1 corresponding to the first subsystem. The multiplicative decomposition of the Leontief inverse L = L₁M_R = M_LL₁ can be converted to the sum:

If ƒ is the vector of final demand and x is the vector of gross output, then the decomposition (12) generates the decomposition of gross output into two parts: x₁ = L₁ ƒ and the increment δx = x - x₁. Such decomposition is important for the empirical analysis of the structure of actual gross output. In the second level of Table 2, the classification is revealed of possible additive decompositions of the Leontief block inverse for all decompositions of the input-output system into pairwise hierarchies.

While fourteen types of pairwise hierarchies of economic linkages have been developed (Figure 1 and Table 2), it is possible to suggest a typology of categories into which these types may be placed. The following characterization is suggested:

1. 1. Backward linkage type (VI, IX): power of dispersion;

2. 2. Forward linkage type (V, X): sensitivity of dispersion;

3. 3. Intra- and interlinkages type (VII, VIII): internal and external dispersion;

4. 4. Isolated region versus the rest of the economy interactions style (I, XIV, IV, XI);

5. 5. Triangular subsystem versus the interregional interactions style (II, XIII, III, XII).

By viewing the system of hierarchies of linkages in this fashion, it will be possible to provide new insights into the properties of the structures that are revealed. For example, the types allocated to category 5 reflect structures that are based on order and circulation. Furthermore, these partitioned input-output systems can distinguish among the various types of dispersion (such as 1, 2, and 3) and among the various patterns of interregional interactions (such as 4 and 5). Essentially, the five categories and fourteen types of pairwise hierarchies of economic linkages provide the opportunity to select according to the special qualities of each region's activities and for the type of problem at hand; in essence, the option exists for the basis of a typology of economy types based on hierarchical structure. Obviously, as this paper is restricted to one application, it is not possible to generalize about the degree to which specific economy types may appear in reality, the degree to which these types remain stable and the process of transformation from one type to another.

2. Application to Brazil

Using a set of interregional input-output tables built by Guilhoto (1998) at the level of forty sectors for the year of 1995 for two Brazilian regions (NE: Region 1; and the rest of the economy: Region 2), the methodology presented in section 1 is applied, and the results are presented in Figures 2 to 4 (based on Tables 3 to 5).

Gross output is decomposed into two parts: x₁ = L₁ƒ and the increment Δx = x - x₁. The values for x and x₁ are added for all sectors in regions 1 and 2 in such a way that it is possible to estimate the contribution of each interaction to the total production in each region. As the shares of x₁ in x take also into consideration the value of the final demand, it is interesting to isolate the shares of the final demand in each region to reveal how the pairwise interaction takes place in the regions.

For example, the value of the final demand in region 1 (NE) is responsible for 63.46 percent of the production in this region (the remaining 36.54 percent is generated in the process of production) while for region 2 (RB), this value is 60.25 percent (39.75 percent in the process of production). In a certain sense this is an indication that the rest of the economy is more developed than the NE region as the internal transactions in region 2 are responsible for a greater share of the total production than is the case in region 1.

In Figure 2, it is possible to see how intermediation in each region contributes to total production. For the NE region, of the 36.54 percent share of total production accounted for by intermediate demand, 66.03 percent of it is the result of intraregional demand, while 11.01 percent is the result of the NE region selling to the RB. Starting from the isolated regions (block matrices) and then adding the interactions among them it is possible to measure how each interaction adds to the total production. These results are presented in detail in Table 4 and Figure 3 for the NE and in Table 5 and Figure 4 for RB.

Fig. 2.

Results of the Synergetic Interactions between the NE and the RB Regions. Source: Tables 3, 4, and 5

Fig. 3.

Contribution (percent) of Each Pairwise and Block Matrix to the Total Share of (x₁-ƒ) in x - NE. Source: Table 4

Fig. 4.

Contribution ( percent) of Each Pairwise and Block Matrix to the Total Share of(x₁-ƒ) in x - RB. Source: Table 5

Table 3.

Results of the Synergetic Interactions between the NE and the RB Regions

Table 4.

Contribution (percent) of Each Pairwise and Block Matrix to the Total Share of (x₁-ƒ) in x - NE

Table 5.

Contribution (percent) of Each Pairwise and Block Matrix to the Total Share of (x₁-ƒ) in x - RB

Excluding final demand, the following summaries may be provided. To assist in the interpretation, the term production in the productive process (ppp) will be used. Consider the total sales made by a sector to both intermediate and final demand; ppp refers to the percentage of the total sales accounted for by intermediate demand.

Case I (A₁₁): When the NE region isolated, this value shows how much of the internal production is due to relations only inside the region; in this case, the value is 24.13%, which represents 66.03% of the ppp;

Case II (A₁₂): The purchases made by the industries in the RB region from the NE region generate 4.02% of the production in the NE region, 11.01% of the ppp, and by itself without having any interaction with the other block matrices generates no further production in the RB;

Case III (A₂₁): The purchases made by the industries in the NE region from the RB region generate 0.39% of the production in the RB region, 0.99% of the ppp, and by itself without having any interaction with the other block matrices generates no further production in the NE region;

Case IV (A₂₂): When the RB region is isolated, this value shows how much of the internal production is due to the relations only inside the region and in this case it is 38.65%, 97.24% of the ppp;

Case V (A₁₁ and A₁₂): The sales of production that the industries in the NE region sell to the production process of both regions generated a gross value of 30.30% of the production generated in the NE region, adding 5.88% to the ppp, and 0.00% in the RB region, as there is no feedback among the regions;

Case VI (A₁₁ and A₂₁): From the interactions of the inputs that the industries in the NE region buy from both regions, a gross value of 24.13% of the production in this region is generated, which means no addition to the ppp, and 0.51% of the production in the RB region, adding 0.30% to the ppp of this region;

Case VII (A₁₁ and A₂₂): When both regions are isolated, with no transactions between them, this value shows how much of the internal production is due to the relations only inside each region and in this case they contribute a gross value of 24.13% for the NE region and a gross value of 38.65% for the RB region, with no addition to the ppp of both regions;

Case VIII (A₁₂ and A₂₁): Considering only the interregional flows among regions, one finds a gross value of 4.07% of the production in the NE region is due to these flows, adding 0.13% to the ppp, while for the RB region this gross value is 0.42%, adding 0.08% to the ppp, showing again a greater dependence upon the production in the NE region in the interrelations among the regions;

Case IX (A₁₂ and A₂₂): From the interactions of the inputs that the industries in the RB region buy from both regions, a gross value of 38.65% of the production in this region can be revealed, with no addition to the ppp, and a gross value of 7.89% of the production in the NE region, adding 10.59% to the ppp. When these results are compared with the ones presented in Case VI, a greater dependence upon the NE region on the production process of the RB region is shown;

Case X (A₂₁ and A₂₂): From the sales of production that the industries in the RB region sell to the production process of both regions, a gross value of 39.37% of the production in the RB region is obtained, adding 0.82% to the ppp, and 0.00% in the NE region as there is virtually no feedback among the regions, in this case showing a greater value of internal multipliers in the RB region than in the NE region;

Case XI (A₁₁, A₁₂ and A₂₁): The relations inside the NE region and the sales and purchases that it makes from the RB region account for a gross value of 30.40% of the production in this region, adding 0.15% to the ppp, and a gross value of 0.56% in the RB region, adding 0.03% to the ppp, values greater than the ones presented in case VI since more transactions are now being taken into consideration;

Case XII (A₁₁, A₁₂ and A₂₂): The relations inside both regions and the purchases that the NE region makes from the RB region generate a gross value of 36.33% of the production in the NE region, adding 5.92% to the ppp, and a gross value of 38.65% in the RB region, with no addition to the ppp;

Case XIII (A₁₁, A₂₁ and A₂₂): The relations inside both regions and the purchases that the NE region makes from the RB region ascertain a gross value of 24.13% of the production in the NE region, with no addition to the ppp, and a gross value of 39.59% in the RB region, adding 0.26% to the ppp;

Case XIV (A₁₂, A₂₁ and A₂₂): The relations inside the RB region and the sales and purchases that it makes from the NE region generate a gross value of 39.49% of the production in this region, adding 0.21% to the ppp, and a gross value of 7.99% in the NE region, adding 0.13% to the ppp, values greater than the ones presented in case IX since more transactions are now being taken into consideration.

Case XV (A₁₁, A₁₂, A₂₁ and A₂₂): This case is not displayed in Table 4 that considers all the interactions in the economy; it is listed here only to call attention for the contribution that this last case has to the ppp, that is, adding 0.14% to the ppp in the NE region and 0.08% to the ppp in the RB region.

Tables 4 and 5 and Figures 3 and 4 show for both regions the contribution that each block matrix in each pair wise decomposition has to the ppp; they also present the total contribution of each block matrix. From these data, it is possible to see a greater dependence of the NE region on the RB region for, while 71.03 percent of the ppp in the NE region is due to interactions inside the region, the corresponding value for the RB region is 97.82 percent. Hence, it is possible to observe and to measure how the relations between the two Brazilian regions take place. The NE region has a greater dependence on the rest-of-the-economy region than the rest of the economy has on the NE region, and at the same time the rest-of-the-economy region seems to be more developed as it presents a more complex productive structure than the NE region.

3. Policy Implications

One of the major changes that has occurred within the economic structure of many economies is the apparent increase in specialization and diversification at the same time. Overall, regional economies are becoming more diversified, in terms of their macro structure. However, establishments (plants) within sectors are becoming more specialized, responding in large part to consumer demands for greater product variety. As a result, trade between regions tends to be concentrated in intraindustry rather than interindustry trade (see Krugman 1990). However, these developments are associated with trade between regions with similar levels of per capita income and with excellent transportation connections. Neither is the case for the NE-RB interaction; transportation costs are low enough to allow penetration from the other region but not sufficiently low enough to allow for the full realization of the benefits of increasing returns.

Having discerned significant imbalances in the trading relationships and the complexity of internal to the region intermediation, the next issue centers on the policy implications. Comparative analysis recently conducted for the NE economy with that of the Midwest of the United States (Magalhães, Sonis, and Hewings 2001) revealed dramatically significant differences in the level and volume of interactions for the two regions. While both regions account for about the same percentage of their nation's GDP, the Midwest U.S. economy's GDP per capita is above the national average in contrast to the NE Brazil economy (about 55 percent of the Brazil GDP per capita). While the Midwest region is highly connected to the rest of the U.S. economy (with an overall positive balance of trade), a huge volume of interactions flows between the member states; in the NE, the level of internal intermediation is lower and there is a negative balance of trade (imports > exports) with the RB. Clearly, appeals to development of clusters of activities to enhance the level of intermediation may not reflect the realities of an economy (NE) whose capacity to sustain further levels of activity may be circumscribed by poor internal transportation connectivities that reduce the effective demand for goods and services.

In addition, as noted by Baer, Haddad, and Hewings (1998), the promotion of more open markets within the context of WTO guidelines may make traditional forms of market intervention less feasible; in any case, the record from prior interventions suggest that the earlier policies had little success in significantly changing the structure of the NE region's economy to ensure that it would be in a position to compete successfully in the national and international marketplace in the next several decades. Asymmetric dependencies create major challenges for development strategy since it is difficult to break the hegemony of the dominant partner.

4. Conclusions

The main contribution of this paper is to show, using different synergetic interactions, that it is possible to analyze and to measure how the trading relationship between two regions takes place. This is accomplished using a two-region interregional input-output table constructed for the Brazilian economy for the year of 1995. From the results, it is possible to see that NE region has a greater dependence on the rest-of-the-economy region than the rest of the economy has on the NE region, and at the same time the rest-of-the-economy region seems to be more developed as it presents a more complex productive structure than the NE region. Given the relative sizes of the two regions, the results are not surprising; what the decompositions reveal is the degree to which the differences in structure reflect the patterns of trade, differences in the degree of intermediation, or combinations of both factors.

This study was conducted using one point in time and two regions; Guilhoto, Moretto, and Rodrigues (2001) have taken the next step in exploring interactions within a five-region division of the Brazilian economy, a process that potentially involves consideration of many million decompositions. Further research will have to await the publication of additional data to explore the degree of stability in structures of production and the structure of exchange.