Regionalization of Input–Output Matrices with Limited Information: Application to the State of Rio Grande do Sul, Brazil

Abstract

The regionalization of input–output tables enables a granular understanding of economic systems, allowing for interregional and interindustry analysis for goods and services in a local economy. This paper details the construction of an intermunicipal IO matrix for the state of Rio Grande do Sul (Brazil), a region marked by both economic diversification and significant territorial disparities. Using the 16-sector state IO matrix (base year 2019) provided by the state-level treasury (SEFAZ-RS) as a starting point, we adapt the Interregional Input–Output Adjustment System (IIOAS), integrating gravity-based trade modelling and RAS balancing, to produce a disaggregated structure for 497 municipalities. The regionalization follows three main steps: (i) generation of an initial matrix assuming proportional municipal shares in sectoral supply and demand; (ii) iterative RAS-based adjustments to align with municipal and state-level constraints; and (iii) incorporation of complementary municipal data—including employment, GDP, household consumption, and exports—to refine final demand and value-added allocations. The results demonstrate the feasibility of deriving spatially intermunicipal IO structures from limited data. The results show that, while industrial and service activities are concentrated around the Porto Alegre metropolitan area, rural subregions remain specialized in low value-added primary sectors.

1. Introduction

Input–output (IO) analysis has been recognised as a tool for examining the interdependencies within an economy, allowing the determination of how sectors are interconnected through trade flows of goods and services [1,2]. While country-level IO tables allow one to analyse the structure of a specific economy, it might be insufficient to capture the economic linkages that unfold across subnational territories, especially in countries marked by regional disparities [3,4,5]. This is the case of the state of Rio Grande do Sul (RS), in southern Brazil, which exhibits a highly diversified economic structure combined with pronounced territorial disparities. As of 2021, Rio Grande do Sul’s state accounted for 6.5% of Brazil’s gross domestic product (GDP), ranking as the fourth-largest state economy in the country [6]. Its productive structure encompasses strong agro-industrial chains (such as soybeans, beef, poultry, and dairy), a strong manufacturing sector (including food processing, chemicals, and machinery), and a diversified service sector concentrated in metropolitan areas such as the capital city, Porto Alegre, and Caxias do Sul, and Pelotas. However, this sectoral heterogeneity is unevenly distributed across municipalities. For instance, the Metropolitan Region of Porto Alegre—comprising 34 municipalities and accounting for 38% of the state’s population and 47% of its total GDP—concentrates services and industrial clusters, as shown in Figure 1. Otherwise, much of the interior municipalities remains specialised in agriculture and agro-industry, often with lower value-added and reduced industrial complexity [6]. Furthermore, rural municipalities exhibit greater sensitivity to climatic variability and market shocks, revealing structural regional disparities.

This paper aims to present the theoretical foundations and methodological procedures used to estimate an intermunicipal input–output matrix for the Brazilian state of Rio Grande do Sul. The Interregional Input–Output Adjustment System (IIOAS) approach proposed by [7,8,9,10] is considered. Using the 2019 state-level IO table (16 sectors) provided by the State Treasury of Rio Grande do Sul (SEFAZ-RS) as a starting point, the procedure involves spatial allocation of production and demand flows across 497 municipalities through proportionality assumptions, iterative RAS adjustments, and integration of auxiliary regional data, such as GDP, employment, exports, and household consumption [4,7,9,11,12,13,14,15,16,17,18,19].

Thus, a spatially disaggregated IO model is needed to capture the regional interdependencies, and to identify sectoral spillovers and regional multipliers. The IO literature has been providing different approaches for regionalisation [3,9,20,21,22,23,24,25,26,27,28,29,30]. Most of the efforts have focused on single-region or intraregional IO models, particularly in contexts where comprehensive survey data are lacking. Within this tradition, the Location Quotient (LQ) approach has become the most widely adopted non-survey method [24,31,32,33,34]. Among the alternatives, the Cross-Hauling Adjusted Regionalisation Method (CHARM), proposed by [35,36], addresses two-way trade flows and has been particularly relevant in European applications. However, when the goal is to simulate economic impacts in relatively small regions—such as municipalities or sector-specific levels—there are considerable limitations of single-region tables, which do not allow for interregional feedback effects. As noted by [37], such feedback is important to understanding the linkages and the economic propagation of a specific final demand shock [38]. Consequently, the case for adopting interregional IO models (IRIO) becomes relevant [1,39,40].

The IIOAS approach has been widely applied in Brazilian regions as it might be considered a flexible methodology for regionalising IO systems in data-limited situations. The method has supported the construction of interregional IO matrices for all 26 Brazilian states and the Federal District [10], allowing comparative assessments of self-sufficiency and interdependence. For instance, São Paulo and Rio de Janeiro have been identified as the most self-sufficient economies, while states like Roraima and Tocantins—both in Northern Brazil—display high external dependence. IIOAS has also been used to analyse regional spillovers and final demand drivers. For example, the Midwest agribusiness sector studied by [41], the MATOPIBA frontier region economic analysis by [42,43,44], and the analysis of sectoral impacts in Brazilian regions by [13,45]. Furthermore, IIOAS has been used for evaluating the territorial effects of fiscal policies (e.g., distribution of oil royalties) and investment in education and healthcare [11], as well as in examining Brazil’s participation in global value chains and regional trade asymmetries [46,47,48].

Therefore, this paper has two main contributions. First, it details a comprehensive application of the IIOAS to estimate a disaggregated intermunicipal IO matrix for the State of Rio Grande do Sul, Brazil. Second, it generates new empirical evidence on the spatial structure of production, demand, and interdependencies within Rio Grande do Sul.

This paper is organised into five sections. Following this introduction, Section 2 reviews the literature on regionalisation methods. Section 3 outlines the methodological steps for applying the IIOAS model. Section 4 presents the main results, and Section 5 offers concluding remarks.

2. Background

Regionalization of input–output (IO) tables is a process in regional economic analysis allowing us to understand interdependencies within and between subnational areas. There are different approaches and methods used to estimate regional IO matrices. The most used procedures for input–output regionalisation are based on non-survey techniques and hybrid models. Among these, the Location Quotient (LQ) family of methods—especially Flegg’s Location Quotient (FLQ) and its extensions, such as the Sector-Specific FLQ (SFLQ) and the two-dimensional LQ (2D-LQ)—has gained empirical improvements [32,34,49]. These models adjust national technical coefficients to regional conditions by incorporating assumptions about interregional trade and intraregional production structures. In this regard, more recent advances, such as the 2D-LQ, enhance model fit by separately adjusting supply- and demand-side components [15,49]. Likewise, the SFLQ introduces industry-specific adjustments, better reflecting heterogeneous trade structures across sectors [33,35]. However, the effectiveness of these techniques depends on the calibration of the δ parameter—that is usually based on interregional road freight transport (IRFT) or imports from the rest of the world (IROW) [32].

Hybrid approaches improve accuracy by combining non-survey and survey-based information. Some techniques have been developed, including the modified cross-entropy (MCE) technique and the Cross-Hauling Adjusted Regionalisation Method (CHARM), developed by [36], which aim to adjust for cross-hauling biases [35]. These methods vary in computational complexity and data requirements. Nevertheless, challenges remain regarding parameter estimation and data availability for small regions, which continue to both LQ-based and hybrid models [35].

Despite the LQ approach being the most adopted non-survey technique, the limitation of single-region tables remains. As [1,37] highlight, single-region models fail to capture cross-regional feedback effects. In such cases, interregional input–output (IRIO) models provide a more theoretically grounded alternative. In this regard, refs [39,50] argue that the magnitude of local multipliers increases with the spatial scope of analysis—implying that neglecting interregional interactions may significantly underestimate the economic impacts.

In this regard, the IRIO methods, including gravity–RAS hybrids and Bayesian spatial models, aim to capture trade flows explicitly between regions. The Interregional Input–Output Adjustment System (IIOAS), developed by [15,51], is a hybrid methodology for regionalising national or aggregated IO tables under limited data. The framework integrates principles from gravity modelling, regional economics, and iterative matrix adjustment to construct consistent interregional systems. Therefore, IIOAS recognises that regional economies are interconnected through flows of goods, services, labour, and capital.

For Brazil, and some Latin and non-Latin countries, the approach is used for regionalisation of national-based tables. Theoretically, IIOAS is based on gravity-based models. In this regard, the allocation of trade flows is based on the interaction between regions, which increases with economic size and decreases with distance [52]. Moreover, the core IO framework [52] provides the foundation for tracing intersectoral linkages. Its interregional extension (IRIO) enables the analysis of spatial spillovers and multiplier effects. Applied for regions, convergence theories perspectives allow for incorporating how regions evolve in response to capital accumulation, technological diffusion, and structural transformation [53]. New Economic Geography (NEG) highlights agglomeration, transport costs, and increasing returns to scale as determinants of spatial economic structures [17]. The IIOAS indirectly captures these forces through spatial trade elasticities and cross-regional dependence. Finally, in network theory—considering regions as nodes and trade flows as links—the IIOAS structure reflects the topology of regional economies, permitting shock effects analysis.

3. Regionalisation Procedure

This section presents the methodological procedures applied to estimate an intermunicipal input–output system for the state of Rio Grande do Sul, Brazil, based on the Interregional Input–Output Adjustment System (IIOAS) developed by [10,52]. The objective is to redistribute sectoral flows from the state-level IO matrix across 497 municipalities, ensuring regional coherence between supply, demand, and interregional trade flows.

3.1. Data Sources

Table 1. Data sources.

Data Component	Source	Spatial Unit	Website of Reference
State IO Matrix (16 sectors)	SEFAZ-RS (2019)	State level	https://dee.rs.gov.br/matriz-insumo-produto (accessed on 10 March 2025)
GDP by municipality	IBGE-Regional Accounts	Municipality (497)	https://www.ibge.gov.br/estatisticas/economicas/contas-nacionais/9088-produto-interno-bruto-dos-municipios.html (accessed on 10 March 2025)
Employment by sector	RAIS/MTE	Municipality × sector	https://bi.mte.gov.br/bgcaged/ (accessed on 10 March 2025)
Household consumption	IBGE-Family Budget Survey (POF) + ICMS municipal receipts	Municipality (proxy)	https://bi.mte.gov.br/bgcaged/ (accessed on 10 March 2025)
Government expenditure	SIOPE + SIOPS (Education/Health)	Municipality	https://www.gov.br/saude/pt-br/acesso-a-informacao/siops (accessed on 10 March 2025)
Investment (FBCF) proxy	ISS tax base and sectoral wages	Municipality × sector	https://prefeitura.poa.br/carta-de-servicos/issqn-imposto-sobre-servicos-de-qualquer-natureza (accessed on 10 March 2025)
Exports	MDIC/SECEX trade database	Municipality × product	https://www.gov.br/mdic/pt-br/assuntos/comercio-exterior/estatisticas (accessed on 10 March 2025)
Distances between cities	IBGE + Google Maps (road distance, average driving time)	Inter-municipal pairs	https://www.ibge.gov.br/geociencias/cartas-e-mapas/mapas-municipais.html (accessed on 10 March 2025)
Population and geolocation	IBGE, mobile data (proxy for induced effects)	Municipality	https://www.ibge.gov.br/geociencias/cartas-e-mapas/mapas-municipais.html (accessed on 10 March 2025)

shows the data sources used thought the regionalisation process. The state input–output matrix for Rio Grande do Sul, with base year 2019, was developed by the State Treasury Secretariat (SEFAZ-RS) and constructed from the National Supply and Use Tables published by IBGE. The model is structured with $s= 16$ economic sectors (

Table 2. Industries within IO model.

Sector ID	Economic Sector	Economic Sector (Original Language)	Abbreviation
S1	Agriculture	Agropecuária	Agro
S2	Extractive industries	Indústrias extrativas	Ind.Extr
S3	Manufacturing industries	Indústrias de transformação	Ind.Tran
S4	Electricity, gas, water, sewage, and waste management	Eletricidade e gás, água, esgoto e gestão de resíduos	SIUP
S5	Construction	Construção	Cons
S6	Trade	Comércio	Com
S7	Transport, storage, and postal services	Transporte, armazenagem e correio	Transp
S8	Accommodation and food services	Alojamento e alimentação	Aloj.Alim
S9	Information and communication	Informação e comunicação	Info
S10	Financial, insurance, and related services	Atividades financeiras, de seguros e serviços relacionados	Finan
S11	Real estate activities	Atividades imobiliárias	At.Imob
S12	Professional, scientific, technical, administrative, and support services	Atividades profissionais, científicas e técnicas, administrativas e serviços complementares	At.Prof
S13	Public administration, defence, public education, health, and social security	Administração, defesa, educação e saúde públicas e seguridade social	Adm
S14	Private education and health	Educação e saúde privadas	Educ
S15	Arts, culture, sports, recreation, and other service activities	Artes, cultura, esporte e recreação e outras atividades de serviços	Art
S16	Domestic services	Serviços domésticos	Serv.Dom

) and includes four final demand agents: households, government, gross capital formation (GCF), and exports (both to other Brazilian states and abroad). In the regionalisation process, exports are treated as a net vector, reflecting production originating in Rio Grande do Sul but consumed externally (outside the state or internationally).

Municipal economic indicators were derived from the 2019 GDP estimates published by Brazilian Institute of Geography and Statistics (IBGE), disaggregated by sector. Sectoral correspondence with the original IO matrix is presented in the Appendix A. Georeferenced coordinates provided by IBGE allowed the calculation of distances between municipality centroids, which were used in the gravity modelling of trade flows.

3.2. Estimation Procedures and Assumptions

Figure 2 shows the regionalisation process and the estimation procedure. Overall, the methodology follows three steps: (i) generation of an initial intermunicipal IO structure; (ii) iterative adjustments to ensure consistency with observed totals; and (iii) integration of auxiliary data to improve spatial allocation accuracy. Therefore, the key features of IIOAS include the initial allocation based on proportional shares of regional supply and demand; gravity-based trade matrices that incorporate distance penalties and autarky coefficients; the RAS adjustments ensuring conformity to observed marginal totals; and a final calibration using municipal data on final demand and value-added components. This method is usually applied for federative countries like Brazil, where decentralised statistical capacity varies and reliable interregional trade data are scarce.

Economic Assumptions

Four assumptions were made regarding economic assumptions. First, proportionality, which assumes that sectoral structures (technical coefficients) are uniform across municipalities—a simplification known to bias results in more heterogeneous economies. Second, isotropy of economic agents’ preferences—households across municipalities are presumed to exhibit similar consumption patterns. Third, there are no cross-hauling adjustments—unlike the CHARM method [36], this approach does not explicitly address simultaneous two-way trade in the same commodity between regions. Fourth, IIOAS assumes independence between sectors and locations during the initial allocation phase. These assumptions are necessary due to data limitations but are partially mitigated through RAS adjustments and the use of auxiliary indicators. Nonetheless, structural biases may persist in sparsely populated or highly specialised municipalities [52].

The state-level IO matrix is disaggregated to the municipal level through a proportional allocation mechanism. The demand generation is supported by regionalised vectors of output ($x_{reg}$), exports ($e_{reg}$), and final demand ($f_{reg}$) were constructed. These were used to estimate the structure of domestic sales by each region. Thus, each municipality’s share of sectoral output is used to estimate its role in the production and consumption structure as follows:

$$z_{ij}^{r,s}=z_{ij}·\left({\displaystyle \frac{x_i^r}{x_i}}\right)·\left({\displaystyle \frac{x_j^r}{x_j}}\right)$$

(1)

where $x_i^r$ and $x$ denote the output of sector i in municipality r and sector j in municipality s, respectively. This allocation preserves the aggregate consistency of the original matrix while assigning flows to specific localities. Imports across all users (excluding exports) were estimated based on proportional shares derived from the structure of domestic sales presented in the national Supply and Use Tables (SUTs) from IBGE. Additionally, margin coefficients and product tax rates were assumed to be uniform across all user categories, varying only by product, as a simplifying assumption.

3.3. Gravity-Based Trade Matrix

To estimate intermunicipal trade flows, a gravity model is applied to the estimated supply and demand vectors. The model assumes that trade between two municipalities is proportional to their economic size and inversely related to the distance between them, as follows:

$$T_{ij}=G·{\displaystyle \frac{M_i·M_j}{D_{rs}^{\kappa }}}$$

(2)

where $T_{rs}^i$ is the trade flow of good $i$ from municipality $r$ to $s$; $M_r^i$ and $M_s^i$ represent sectoral economic masses—in this case using the GDP by municipality and sector; $D_{rs}$ is the road distance between the centroids of the two municipalities; and $\kappa$ is a distance decay parameter [35]. Flows are penalised by spatial frictions and modulated by self-sufficiency assumptions. A higher $\kappa$ implies stronger penalisation of distant trade. The elasticity parameter $\sigma$ captures the substitutability of trade partners; lower values imply greater preference for local inputs. Following [54], we adopt $\kappa =0.5$ and $\sigma =0.3$, consistent with empirical evidence for Brazilian interregional flows.

A matrix of domestic supply (SUPPLY) and estimated sectoral demand by region (DEMAND) is constructed to determine potential interregional transactions. The Hybrid System of National Interactions (SHIN) defines the share of each municipality in satisfying demand from others, based on trade frictions, local production capacity, and relative autarky levels, as show Equation (3). Specifically, SHIN tables were constructed to derive total demand by destination, distinguishing between locally supplied demand (domestic/local production) and that met through imports from other states or the rest of the world.

$$P_{rd}^s={\left({\displaystyle \frac{1}{D_{rd}^s}}\right)}^{\sigma }·\left({\displaystyle \frac{q_r^s}{{\sum }_r^s{q}_r^s}}\right)·\left(1-autar{k}_d^s\right)$$

(3)

where $P_{rd}^s$ is the share of sector $s$ in region $r$ meeting demand from region $d$, $D_{rd}$ is the road distance, $\sigma$ is the sector-specific spatial elasticity, and $q_r^s$ is the regional output share.

Once SHIN tables, containing estimated intermunicipal transactions by sector, are derived from the gravity-based allocation, they must be adjusted to ensure that their row and column totals match observed aggregates of output and input for each municipality and sector. Thus, a preliminary interregional matrix $Z_{rs}^0$ is estimated and then the biproportional RAS algorithm is applied, a widely used iterative procedure for matrix balancing [1]. The RAS algorithm operates by sequentially scaling the rows and columns of the matrix $Z$ using two adjustment vectors: $R^n$, a diagonal matrix of row multipliers that adjusts the supply side (i.e., the total output of each municipality-sector), and $S^n$, a diagonal matrix of column multipliers that adjusts the demand side (i.e., the total use by each municipality-sector). At each iteration $n$, the matrix is updated as follows:

$$Z^{n+1}=R^n\cdot Z^n\cdot S^n$$

(4)

The procedure begins with the gravity-estimated matrix $Z^0$, and iteratively recalibrates it to satisfy two marginal constraints: row—the sum of each row must equal the estimated total supply (gross output) of the corresponding sector and municipality, and column—the sum of each column must equal the estimated total demand (intermediate + final) of the corresponding sector and municipality. This iterative process is continued until convergence is achieved—typically defined as when the relative difference between successive iterations falls below a pre-specified tolerance level (equals to 1 × 10⁻⁵). Mathematically, this implies that both the adjusted row sums ${\sum }_j{z}_{ij}^n$ and column sums ${\sum }_i{z}_{ij}^n$ closely approximate their respective target vectors, ensuring the internal consistency of the IRIO model.

Thus, IIOAS procedure reconciles bottom-up estimates (from regional proxies) with top-down constraints (from the state IO matrix). Moreover, when applied to highly disaggregated systems such as 497 municipalities × 16 sectors, it provides a tractable and statistically consistent method for calibrating large interregional systems in the absence of direct survey data. Finally, the gravity-based allocation assumes isotropic spatial interaction, meaning that trade friction is homogeneous in all directions and not influenced by specific topographic or infrastructure constraints. While this simplification facilitates tractable estimation, it may overlook regional peculiarities in logistics and transport. Furthermore, cross-hauling—the simultaneous export and import of similar products between regions—is not explicitly corrected in the model. This choice follows the assumption that, at the aggregated sectoral level, such flows tend to cancel out.

3.4. Final Demand and Value-Added Components

The final stage in constructing the intermunicipal input–output system involves the spatial allocation of the components of final demand and value-added. This step allows for completing the accounting identities and the Leontief multipliers. According to [10,52], the proportional allocation principle is achieved and the IRIO matrix is consistent with the specific context of Rio Grande do Sul. Therefore, final demand compromises four main components: household consumption ($\mathrm{H}\mathrm{H}$), government expenditure ($\mathrm{G}$), gross capital formation ($\mathrm{I}\mathrm{N}\mathrm{V}$), and exports ($\mathrm{E}$). Each of these elements is distributed across the 497 municipalities using proxies derived from municipal-level data, ensuring that their aggregated totals remain consistent with the values recorded in the original state-level matrix.

(a)

Household Consumption: Household consumption by municipality and sector is estimated by applying the sectoral structure from the state-level final demand vector to each municipality’s share in total household income or ICMS (Brazilian tax revenue from consumption). In the absence of microdata, this proxy approximates the following spatial pattern of consumption:

$${HH}_{ij}^r={HH}_{ij}·\left({\displaystyle \frac{{hh}^r}{\Sigma {hh}^r}}\right)$$

(5)

where $H{H}_{ij}^r$ is the estimated household consumption in municipality $r$ for sector $j$, $H{H}_{ij}$ is the aggregate household consumption for sector $j$ at the state level, $h{h}^r$ accounts for the proxy indicator for household consumption in municipality $r$—such as income, payroll, or ICMS tax base, and $\Sigma h{h}^r$ is the total of the proxy indicator across all municipalities. This equation distributes state-level household consumption by sector across municipalities proportionally to a local consumption proxy.

(b)

Government Expenditure: Public sector expenditure is allocated using sectoral data from federal systems, focusing on health and education spending by municipality. When no detailed breakdown is available, public payroll and municipal population weights are used as follows:

$$G_{ij}^r=G_{ij}·\left({\displaystyle \frac{g^r}{\Sigma g^r}}\right)$$

(6)

where $G_{ij}^r$ accounts for the estimated government expenditure in municipality $r$ for sector $j$, $G_{ij}$ is the total government expenditure for sector $j$ at the state level, $g^r$ is the proxy for government spending in municipality $r$—such as public payroll or municipal population), and $\Sigma g^r$ is the total proxy across all municipalities. It allows allocates government spending across municipalities proportionally to their share of a selected proxy.

(c)

Gross Capital Formation (Investment): Investment demand is spatially distributed using indicators such as construction activity, municipal ISS tax base on services and civil works, or sectoral wage mass for capital-intensive sectors, as follows:

$${INV}_{ij}^r={INV}_{ij}·\left({\displaystyle \frac{{inv}^r}{\Sigma {inv}^r}}\right)$$

(7)

where $I_{ij}^r$ is the estimated gross capital formation in municipality $r$ for sector $j$, $I_{ij}$ represents state-level investment for sector $j$, $i^r$ is the proxy for investment in municipality $r$—such as Services Tax base and capital-intensive wages), and finally $\Sigma i^r$ is the total investment proxy across all municipalities.

(d)

Exports: Exports are treated as a net demand vector representing the value of goods and services produced in the state and consumed outside Rio Grande do Sul. SECEX data (Trade, Industry, and Development Ministry-MDIC) at the municipal-product level is used to allocate export values to origin municipalities. Equation (8) disaggregates trade data to allocate state-level exports to their municipal origins as follows:

$$E_{ij}^r=E_{ij}·\left({\displaystyle \frac{x_j^r}{\Sigma x_j^r}}\right)$$

(8)

where $E_{ij}^r$ represents the estimated exports from municipality $r$ for sector $j$, $E_{ij}$ is the total state-level exports for sector $j$, $x_j^r$ is the export value of sector $j$ in municipality $r$ based on trade data, and $\Sigma x_j^r$ is the total export value of sector $j$ across all municipalities.

(e)

Value-added: Value-added is decomposed into labour compensation (wages), gross operating surplus and mixed income, and taxes less subsidies on production and imports. In this regard, value-added is allocated proportionally to sectoral output and employment at the municipal level, using Ministry of Labour data (RAIS) estimates as primary proxies. Labour compensation is assigned using total wages by municipality-sector cells; gross operating surplus and taxes are estimated residually, ensuring coherence with total output and intermediate inputs, as follows:

$$V{A}_{ij}^r=X_{ij}^r-\Sigma Z_{kj}^r$$

(9)

where $V{A}_{ij}^r$ is the value-added in municipality $r$ for sector $j$, $X_{ij}^r$ is the total output of sector $j$ in municipality $r$, and $\Sigma Z_{kj}^r$ accounts for the total intermediate inputs used by sector $j$ in municipality $r$.

Therefore, this identity ensures that value added is calculated as the residual between gross output and intermediate consumption, satisfying the accounting identity as follows:

$$X = Z + VA=Z+ FD$$

(10)

All final demand components are subsequently rebalanced through a final RAS procedure to ensure that municipal-level sums align with the totals of the original state-level matrix. It allows preserving aggregate control while embedding regional heterogeneity into the system.

This step completes the construction of the intermunicipal IO matrix, enabling the calculation of technical coefficients, the Leontief inverse, and spatially disaggregated multipliers for policy simulation, impact assessment, and structural analysis.

3.5. Hypothetical Extraction Procedure

To assess the sensitivity of the intermunicipal production system in Rio Grande do Sul, we implemented a series of hypothetical extraction simulations by applying uniform 20% negative shocks to four aggregate sector groups: (i) agriculture, (ii) industry/manufacturing, (iii) trade, and (iv) services.

Table 3. Groups and uniform shocks.

Sector ID	Economic Sector	Aggregate Shock	Extraction Level
S1	Agriculture	Agriculture	−20%
S2	Extractive industries	Agriculture	−20%
S3	Manufacturing industries	Industry	−20%
S4	Electricity, gas, water, sewage, and waste management	Industry	−20%
S5	Construction	Industry	−20%
S6	Trade	Trade	−20%
S7	Transport, storage, and postal services	Trade	−20%
S8	Accommodation and food services	Services	−20%
S9	Information and communication	Services	−20%
S10	Financial, insurance, and related services	Services	−20%
S11	Real estate activities	Services	−20%
S12	Professional, scientific, technical, administrative, and support services	Services	−20%
S13	Public administration, defence, public education, health, and social security	Services	−20%
S14	Private education and health	Services	−20%
S15	Arts, culture, sports, recreation, and other service activities	Services	−20%
S16	Domestic services	Services	−20%

shows the applied shocks. The aim is to measure the direct, indirect, and total effects of sectoral supply and demand constraints on municipal and sectoral gross output ($\mathsf{\Delta }X$). The baseline follows the standard IO as follows:

$$x_s^r =(I-A_{rs})-f_s^r$$

(11)

where $x_r^s$ is the baseline production vector for municipality $r$, sector $s$; $A_{rs}$ represents the baseline matrix of technical coefficients (input requirements per unit of output); and $f_r^s$ is the baseline of final demand vector.

Applying all rows and columns of $A_{rs}$ corresponding shock of 20% (representing lower input demand and supply capacity), and a restriction in $f_s^r$ by 20% for those same sectors, it was constrained to

$${\overline{x}}_s^r=(I-{\overline{A}}_{rs})-{\overline{f}}_s^r$$

(12)

where the bars represent the IO elements under shock scenario. The total impact is given by the difference in output before and after shock, as follows:

$$T =x_s^r-{\overline{x}}_s^r$$

(13)

Each simulation targets one aggregate group at a time (e.g., all activities coded under agriculture or trade/services). The shock is applied uniformly across all municipalities, reflecting a stylised systemic disruption. Elements of the technical coefficients matrix A and the final demand vector f for the affected sectors are multiplied by (1 − δ), where δ = 0.20. Specifically, $\overline{A}=\left[{\overline{a}}_{ij}\right]=\left(1-{\delta }_i\right)a_{ij}$ and $\overline{f}=\left[{\overline{f}}_{ij}\right]=\left(1-{\theta }_i\right)f_{ij}$. The impacts were decomposed as follows: (a) direct impact is given by $\mathsf{\Delta }f=\overline{f}-f$, (b) indirect impact: $\mathsf{\Delta }A=(I-\overline{A})-\mathsf{\Delta }f$; and (c) total impact given by $\mathsf{\Delta }x=\mathsf{\Delta }f+\mathsf{\Delta }A$. This allows identifying both internal production losses in the shocked sectors and the spillover effects transmitted to other sectors and municipalities through intermediate linkages.

All estimations were performed using the R programming language. Custom scripts were developed for matrix manipulations, gravity model calibration, and RAS iterations, relying on the following packages: ioanalysis, tidyverse, Matrix, and sf for spatial visualisation. The scripts are available upon request and will be published as Appendix A.

4. Results

4.1. Structural Analysis

Figure 3 shows a sectoral comparison of total gross production between the original state-level matrix and the sum of the disaggregated intermunicipal estimates from IIOAS. Thus, the regionalisation process preserved the aggregate output. Discrepancies are negligible and fall within the tolerance range imposed during the RAS iterative adjustment.

Figure 4 and Figure 5 present the indices of backward linkages (BLs) and forward linkages (FLs) for the regionalised system. These structural indicators reflect, respectively, the dependence of each sector on inputs from the rest of the economy (BL), and the degree to which the sector provides inputs to other sectors (FL). The results show that manufacturing of transport equipment (Ind.Tran) and transport services (Transp) are among the most interconnected sectors, with strong multipliers in both dimensions. These sectors operate as core nodes within the regional production network, stimulating upstream and downstream activities across and within sectors and municipalities of Rio Grande do Sul’s state. In contrast, sectors such as domestic services (Serv.Dom) and real estate activities (At.Imob) exhibit weaker linkages, acting more as terminal sectors with limited spillover capacity.

The cross-classification in Figure 5 reinforces these findings: sectors positioned in the upper-right quadrant, such as Ind.Tran and Transp, are strategic due to their dual role in transmitting economic stimuli. Conversely, sectors in the lower-left quadrant demonstrate marginal systemic influence.

4.2. Hypothetical Extraction Results

To illustrate the applicability of the IRIO system for impact assessment, a hypothetical extraction exercise was carried out. A uniform shock of 20% reduction was applied to each of four sectoral group across all municipalities. Technical coefficients and final demand vectors for each sector were uniformly reduced, and the production system was re-estimated using the modified Leontief inverse. Therefore, the decomposition of impacts into direct, indirect, and total effects revealed spillover and propagation mechanisms.

(a)

Agriculture-related sectors (Sectoral group 1) across all: The first hypothetical extraction exercise focused on the agriculture sector, which plays an important role in the economic structure of Rio Grande do Sul. Results are shown in

Table 4. Agricultural shocks results.

Sector ID	Economic Sector	Direct	Indirect	Total
S1	Agriculture	−10.15	−0.93	−11.08
S2	Extractive industries	0	−0.54	−0.54
S3	Manufacturing industries	0	−0.52	−0.52
S4	Electricity, gas, water, sewage, and waste management	0	−0.58	−0.58
S5	Construction	0	−0.02	−0.02
S6	Trade	0	−0.34	−0.34
S7	Transport, storage, and postal services	0	−0.51	−0.51
S8	Accommodation and food services	0	−0.02	−0.02
S9	Information and communication	0	−0.14	−0.14
S10	Financial, insurance, and related services	0	−0.41	−0.41
S11	Real estate activities	0	−0.05	−0.05
S12	Professional, scientific, technical, administrative, and support services	0	−0.32	−0.32
S13	Public administration, defence, public education, health, and social security	0	−0.01	−0.01
S14	Private education and health	0	−0.01	−0.01
S15	Arts, culture, sports, recreation, and other service activities	0	−0.05	−0.05
S16	Domestic services	0	0	0

, which reveal both the direct contraction in agricultural output and the indirect spillover effects across the rest of the economy.

The agricultural sector itself experienced a direct reduction of −10.15%, corresponding to the calibrated shock, with an additional indirect contraction of −0.93%, resulting in a total output decline of −11.08%. This indicates that agriculture is not only a large final demand sector but also a critical supplier to upstream industries. According to IBGE, the agricultural sectoral is relevant for southern Brazil economy. The largest indirect losses were observed in electricity, gas, water, and waste services (−0.58%), extractive industries (−0.54%), transport and logistics (−0.51%), manufacturing (−0.52%), financial services (−0.41%), trade (−0.34%), and professional and administrative services (−0.32%). These results reflect the broad production and logistical interdependencies of agriculture, particularly in supplying raw materials to processing industries, relying on energy inputs, and fuelling distribution networks.

It is important to note that the sensitivity of manufacturing, especially in agribusiness-linked sub-sectors (e.g., food processing), is expected due to direct sourcing from agriculture. Likewise, transport and storage services are heavily exposed to fluctuations in agricultural output, as are trade and financial services, which are tightly coupled to primary sector turnover. Non-tradable and public services, such as education, health, and public administration reported negligible impacts (<−0.01%), confirming their low dependency on agricultural supply chains and their role as relatively insulated service providers. Similarly, domestic services were unaffected, highlighting the sector’s self-contained nature and detachment from intersectoral production flows.

Figure 6 illustrates the spatial distribution of total production impacts across the IRIO’s regions. The colour gradient represents the percentage reduction in total production per municipality, revealing heterogeneity in the spatial propagation of the shock. The most severely affected areas are in the central-western and northern interior of the state. These regions are characterised by high agricultural dependency and limited economic structure diversification, making them relatively most vulnerable to disturbances in primary production. Municipalities such as Pedras Altas, Jari, André da Rocha, Santa Cecília do Sul, and Jacuizinho registered total production declines exceeding 8%, significantly above the state average. These localities have high shares of agricultural GDP, and their local economies are strongly dependent on agribusiness supply chains.

The propagation of losses beyond the agricultural base is visible in the case of Rolador, Boa Vista do Cadeado, and Capão do Cipó, where secondary linkages with sectors such as transport, trade, and manufacturing intensified the downturn. This spatial pattern reinforces the structural asymmetries within the Rio Grande do Sul’s Brazilian state. While some urban and service-oriented municipalities (especially in the metropolitan area) experienced marginal or negligible effects, vast rural regions showed acute sensitivity to sectoral shocks.

(b)

Manufacturing-related industries (Sectoral group 2): As before, a hypothetical −20% uniform shock was applied to the group of industrial sectors—encompassing both extractive industries and manufacturing. The overall shock resulted in a total contraction of 16.05% in manufacturing, confirming the high sensitivity of this sector to internal disturbances, as shown

Table 5. Manufacturing-related shocks.

Sector ID	Economic Sector	Total Impact
S1	Agriculture	−6.72
S2	Extractive industries	−10.32
S3	Manufacturing industries	−16.05
S4	Electricity, gas, water, sewage, and waste management	−2.82
S5	Construction	−0.20
S6	Trade	−3.01
S7	Transport, storage, and postal services	−6.85
S8	Accommodation and food services	−0.36
S9	Information and communication	−1.92
S10	Financial, insurance, and related services	−2.98
S11	Real estate activities	−0.65
S12	Professional, scientific, technical, administrative, and support services	−4.94
S13	Public administration, defence, public education, health, and social security	−0.15
S14	Private education and health	−0.15
S15	Arts, culture, sports, recreation, and other service activities	−0.75
S16	Domestic services	-

. Due to its central role in intersectoral input supply and demand chains, the industrial sector transmitted significant negative spillovers to the rest of the sectors: transport and logistics (−6.85%), reflecting the strong link between industrial production and distribution infrastructure; agriculture (−6.72%), due to reduced demand for agro-industrial processing and supply chain services; professional and technical services (−4.94%)—which serve as inputs to industrial production; and trade (−3.01%) and financial services (−2.98%), due to the contraction in business activity and investment. The public sector and personal services sectors were marginally affected (<−0.20%), highlighting their relative independence from industrial dynamics.

Figure 7 presents the spatial pattern of total production losses across municipalities in response to the industrial shock. The distribution shows a markedly different geography compared to the agricultural scenario. The most affected areas are concentrated in northern and northeastern regions, with significant clusters of vulnerability in municipalities that host medium-sized industrial hubs.

The ten most impacted municipalities include Roque Gonzales, Aratiba, Pinhal Grande, Alpestre, Entre Rios do Sul, Lindolfo Collor, Tupandi, Salto do Jacuí, Pinhal da Serra, and Imigrante. These municipalities exhibit high industrial concentration relative to their economic base and are less diversified, making them more exposed to sector-specific downturns. Some are located in corridors of industrial production linked to metalworking, machinery, food processing, and furniture, which are sensitive to both demand and supply chain disruptions.

Interestingly, the metropolitan region and larger urban centres—while economically relevant—show a more moderate response due to greater economic diversification and service-based absorption capacity. Conversely, smaller municipalities with strong industrial specialisation but limited functional economic networks exhibit higher vulnerability.

(c)

Trade and Services:

Table 6. Trade-related sector’s impacts.

Sector ID	Economic Sector	Total Impact
S1	Agriculture	−6.72
S2	Extractive industries	−10.32
S3	Manufacturing industries	−16.05
S4	Electricity, gas, water, sewage, and waste management	−2.82
S5	Construction	−0.20
S6	Trade	−3.01
S7	Transport, storage, and postal services	−6.85
S8	Accommodation and food services	−0.36
S9	Information and communication	−1.92
S10	Financial, insurance, and related services	−2.98
S11	Real estate activities	−0.65
S12	Professional, scientific, technical, administrative, and support services	−4.94
S13	Public administration, defence, public education, health, and social security	−0.15
S14	Private education and health	−0.15
S15	Arts, culture, sports, recreation, and other service activities	−0.75
S16	Domestic services	-

shows the results of a shock imposed in retail and wholesale trade activity, and service-related sectors (

Table 7. Services-related sector’s impacts.

Sector ID	Economic Sector	Total Impact
S1	Agriculture	−0.31
S2	Extractive industries	−0.45
S3	Manufacturing industries	−0.56
S4	Electricity, gas, water, sewage, and waste management	−1.01
S5	Construction	−0.16
S6	Trade	−0.50
S7	Transport, storage, and postal services	−7.76
S8	Accommodation and food services	−16.83
S9	Information and communication	−11.20
S10	Financial, insurance, and related services	−10.88
S11	Real estate activities	−0.59
S12	Professional, scientific, technical, administrative, and support services	−2.60
S13	Public administration, defence, public education, health, and social security	−0.06
S14	Private education and health	−0.13
S15	Arts, culture, sports, recreation, and other service activities	−0.49
S16	Domestic services	-

). Regarding trade, the direct contraction of −14.44% in the trade sector dominates the scenario. Despite being considered a final demand-facing sector, trade also acts as a critical intermediary and demand generator for a wide range of services and infrastructure.

As such, it exerts substantial indirect effects across multiple other activities, especially the following: transport and storage (−1.88%), reflecting the tight integration of logistics and distribution services with wholesale and retail trade; professional and administrative services (−2.84%), indicating the reliance of trade firms on marketing, legal, accounting, and support services; ICT and financial services (−1.78% and −1.07%), driven by the demand for information systems, e-commerce infrastructure, and credit systems; and real estate and utilities (−1.21% and −1.38%) (physical infrastructure of retail spaces and energy consumption). Industrial and agricultural sectors were affected to a much lesser extent (≤−0.40%), as their production links to trade are mostly downstream and less dependent on short-term trade dynamics.

Figure 8a shows the spatial pattern of total production losses resulting from the shock to trade. The impact is widely dispersed but reveals a clear pattern of higher concentration in central and northern regions, including some urban and peri-urban zones. This contrasts with the industry or agriculture scenarios, where impacts were more clustered along production corridors or primary sector belts.

The geography of impact reflects that the location of the trade sector across municipalities and its position as a service sector are sensitive to both local demand and systemic economic contraction. While trade activity is present in nearly all municipalities, the intensity of the contraction varies, being highest in local economies, and is heavily reliant on retail commerce without diversified production bases. Municipalities with strong trade-employment shares and low public sector buffers showed the most pronounced effects.

As shown in Table 7, the uniform −20% shock imposed on the broader service sector encompasses activities such as accommodation and food services, information and communication, finance and insurance, professional and administrative services, and other personal services. Despite being traditionally considered non-tradable and less spatially mobile, the service sector plays a critical systemic role due to its backward linkages and urban concentration. The sectors most severely affected are those at the core of the urban service economy: accommodation and food services experienced the most intense contraction (−16.83%), as expected, given their reliance on both discretionary income and tourism-related activity. ICT (−11.20%) and financial services (−10.88%) follow closely, underscoring their embeddedness in commercial ecosystems and interdependencies with both final consumers and corporate clients.

The indirect effects also reach logistics, public utilities, and manufacturing sectors, illustrating that services are not isolated: they drive input demand and coordinate production systems, particularly in advanced urban economies. Spatially, Figure 8b presents the spatial distribution of total production losses due to the service sector shock. The geography of impact is markedly urban-centric, with the most affected municipalities being those with strong urban infrastructure, service employment concentration, or tourism reliance. The ten most affected municipalities include Eldorado do Sul, Gramado, Machadinho, Porto Alegre, São João do Polêsine, Pouso Novo, Butiá, Canela, Marcelino Ramos, and Torres. Several of these municipalities—Gramado, Canela, and Torres—are central nodes in the state’s tourism economy, while Porto Alegre and Eldorado do Sul are important metropolitan centres, housing information technology hubs and financial institutions. The results reflect the exposure of such cities to demand-side shocks in services, magnified by their strong reliance on flows of people, capital, and consumption.

In contrast, peripheral rural municipalities showed lower sensitivity, due to their smaller service sectors and more production-based economic structures. However, this also reveals potential developmental gaps, as service-based value chains are more concentrated in the urban core areas.

5. Summary and Conclusions

This study contributes to both methodological and applied regional analysis by adapting the Interregional Input–Output Adjustment System (IIOAS) to a highly granular, intermunicipal context comprising 497 municipalities in the state of Rio Grande do Sul, Brazil. By integrating proportional allocation, gravity-based trade modelling, and RAS-type iterative adjustment, the model achieves a spatially disaggregated system that remains consistent with state-level technical coefficients while introducing differentiation across space in supply and demand flows.

The framework allowed for a comprehensive simulation of sector-specific shocks—agriculture, industry, trade, and services—enabling the estimation of spatial multipliers and local spillover effects. Therefore, regarding regional economic analysis, the agricultural shocks shown that municipalities with a high concentration of agricultural GDP and employment suffered not only from direct output reductions but also from their role as transmission nodes for indirect losses across sectoral boundaries. Spillovers were observed in manufacturing, logistics, and services, particularly in rural and agro-industrial municipalities.

From manufacturing/industry shocks, the sector proved be central to the regional economy, with strong backward linkages to agriculture and forward linkages to services and logistics. Disruptions in manufacturing propagated across the state through supply networks, disproportionately affecting municipalities specialised in food processing, machinery, and energy sectors. Regarding trade and services, simulations revealed that even final-demand-oriented sectors like trade and services can trigger wide-reaching economic effects due to their dense intersectoral linkages, especially in urban areas. Municipalities reliant on tourism, ICT, or financial services—such as Gramado, Porto Alegre, and Torres—showed notable vulnerabilities.

Together, these sectoral findings demonstrate the strategic value of intermunicipal IO models for evidence-based policymaking. Rather than relying on aggregate metrics, the IIOAS framework enables localised diagnostics and simulations of policy scenarios, offering analytical support for territorial equity, resilience planning, and disaster preparedness. However, while the regionalisation of IO tables offers powerful tools for territorial analysis, several challenges remain. Non-survey methods, such as IIOAS and FLQ, provide feasible alternatives but may introduce estimation bias where proxies are weak or incomplete. Capturing trade flows, especially cross-hauling and spatial feedback effects, remains difficult in the absence of direct data. Advances such as the CHARM model and gravity–RAS offer methodological improvements but demand spatially rich datasets. For small-scale regions, municipalities often lack the statistical infrastructure to support accurate disaggregation. Hybrid models and sector-specific LQ refinements (e.g., SFLQ) are promising, but they require additional empirical validation. Future work should integrate environmental extensions (e.g., environmentally extended IO/SDA) to quantify embodied emissions and resource footprints at municipal scale, alongside sector-specific cross-hauling corrections (e.g., CHARM-style adjustments) and basic parameter sensitivity checks for κ and σ. Future work could include a sensitivity analysis testing changes in some parameters., such as κ and σ, or an alternative distance metric to show that spatial patterns and validation metrics remain robust.

Finally, the regionalisation of input–output matrices plays a fundamental role in understanding the structural interdependencies and spatial dynamics of economic systems. As demonstrated in this paper, such models are not only instrumental for academic inquiry but also essential for the design of targeted, equitable, and resilient public policies. In economic context marked by regional inequalities, analytical tools such as IIOAS—particularly when applied at high spatial resolution—enable governments to anticipate systemic risks and tailor responses at the appropriate scale. In this line, future research should continue improving the accuracy and transparency of hybrid regionalisation methods, while also exploring how IO-based models can be dynamically integrated with real-time data sources to better inform policy in evolving territorial contexts.