Policy Research Working Paper 10650 Spatial Misallocation of Complementary Infrastructure Investment Evidence from Brazil Fidel Pérez-Sebastián Rafael Serrano-Quintero Jevgenijs Steinbuks Infrastructure Chief Economist Office December 2023 Policy Research Working Paper 10650 Abstract How does the misallocation of complementary public costs. Simulation results for the Brazilian economy point capital affect the spatial organization of economic activ- to significant welfare gains from reallocating infrastructure ity? To answer this question, this paper endogenizes the investment. Spatial and fiscal complementarities in hetero- government’s decision to invest in the transport and elec- geneous infrastructure provision determine a sizeable part tricity networks. A novel multi-sector quantitative spatial of those gains. Misallocation of both infrastructure invest- equilibrium model incorporates the quality of the electric ments is positively associated with local political support power and the road transportation infrastructure net- for the incumbent authority. works, which determine sectoral productivities and trade This paper is a product of the Infrastructure Chief Economist Office. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted atjsteinbuks@worldbank.org. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced by the Research Support Team Spatial Misallocation of Complementary Infrastructure Investment: Evidence from Brazil∗ Fidel Pérez-Sebastián1 , Rafael Serrano-Quintero2 , and Jevgenijs Steinbuks3 1 Department of Economics (FAE), University of Alicante 2 Department of Economics, University of Barcelona 3 Office of the Chief Economist, Infrastructure Practice, The World Bank Keywords: Brazil; electrical grid; infrastructure investment; public roads; spatial misallo- cation JEL Classification: H54; O18; O41 ∗ We thank Luis Andrés, Rafael Araújo, Rodrigo Bomfim de Andrade, Yongyang Cai, Kerem Coşar, Manuel García-Santana, Jaqueline Oliveira, Esteban Rossi-Hansberg, Edouard Schaal, Stéphane Straub and the audiences at the 1st University of Barcelona Macro Workshop, the Monthly Seminars of the Armenian Economic Asso- ciation, the 1st International Workshop on Economic Growth and Macroeconomic Dynamics, the 16th North American Meeting of the Urban Economics Association, the 11th European Meeting of the Urban Economics As- sociation, the Internal Seminar of the University of Barcelona, the 4th Workshop of the Spanish Macroeconomics Network, the 2nd LAUrban Annual Meeting, and the World Bank InfraXchange Workshop for their helpful com- ments and suggestions. All remaining errors are of the authors alone. 1 Introduction Though the significance of public infrastructure in economic development is well established, how to efficiently allocate different types of infrastructure projects to meet development needs remains an important research and policy question. There is an urgent need to understand how different infrastructure networks interact and evolve over space. How much can fiscally con- strained governments improve economic welfare by investing across different infrastructure types? How large are the economic gains of coordinating infrastructure investments in differ- ent regions? Should national governments prioritize coordination between different sectoral infrastructure planning agencies? To answer these questions, we build a novel multi-sector spatial equilibrium model in which the evolution of public infrastructure is endogenously determined by a government that invests in the quality of public capital to maximize welfare. We study two types of public infrastructure—the electricity and road transportation networks—in a concurrent setting to assess the extent of misallocation and spatial complementarities in infrastructure provision. The endogeneity of infrastructure investment is important because economic activity and the different infrastructure networks affect one another, which can lead to biased estimates in reduced-form settings. Improvements in public infrastructure can also have spillover effects that amplify the aggregate impact on economic activity. Moreover, public infrastructure in- vestments are not randomly allocated in space and could be the result of the own interests of the government, such as political alignment or ethnic favoritism (Cadot et al., 2006; Brollo and Nannicini, 2012; Burgess et al., 2015). In our model, the economy comprises a fixed set of microregions in which three sectors— agriculture, manufacturing, and services—produce tradeable varieties of final consumption goods.1 The three sectors produce output using labor (which is freely mobile) and electric- ity as inputs.2 Total factor productivities (TFPs) are partly determined endogenously by the equilibrium values of the population and the quality of the electricity flow (see, e.g., Fisher- Vanden et al. (2015), Allcott et al. (2016) and Fried and Lagakos (2023)). This TFP impact of electricity indirectly affects the marginal cost of production and reduces the price at which firms in a particular location sell their products. Transportation improvements also reduce the cost of shipping goods. However, due to network spillovers, road improvements affect all locations that ship goods through affected routes. Additionally, infrastructure quality shapes the population distribution through its impact on wages. The quantitative analysis focuses on the Brazilian spatial distribution of economic activ- ity, public roads, and electricity supply quality in the year 2010. First, we ask how the spatial 1 The Brazilian Institute of Geography and Statistics (IBGE) defines a microregion as a set of municipalities that are comparable in terms of social characteristics, geography, and production structure. 2 Banerjee et al. (2020) argue that mobility of production factors is key for determining the benefits of infras- tructure investments. Nevertheless, free mobility is not crucial for our results, as Appendix H shows. 2 economic activity distribution can be improved by optimizing public infrastructure invest- ment. The counterfactual simulations indicate that improving spatial allocations of power infrastructure alone could increase consumer welfare by 7.5 percent and average income by 3.4 percent. In turn, improving spatial allocations of road infrastructure investments could in- crease consumer welfare by 11 percent and average income by 10.7 percent. We also find that improvements in the quality of the electrical grid and public roads accelerate structural trans- formation by increasing employment shares in both manufacturing and services and reducing the employment share in the primary sector. We then ask if the two types of public infrastructure complement each other in the process of economic development. On the one hand, concentrating infrastructure investments in space can be beneficial if there is some degree of complementarity between them. On the other, do- ing so could increase the degree of spatial inequality (Kanbur and Venables, 2005, Chapters 5 and 8). Estimating the complementarity degree is, therefore, of first-order importance from the policy-making perspective. We find that coordinating the optimal investment in both types of infrastructure (roads and electricity) improves welfare even when there is no fiscal align- ment across line ministries that have fixed separate budgets. In particular, gains in consumer welfare and average income due to this complementarity equal 1.26 and 2.2 percentage points, respectively. Third, we observe even greater welfare gains when fiscal aspects are introduced, allowing for a joint infrastructure budget and coordinating the spatial allocation of electric power and transportation networks. Allowing for fiscal complementarities increases consumer welfare and average income by an additional 2.3 and 5.9 percentage points, respectively. Even though our model predicts that areas with a higher quality of electricity should also have higher- quality roads, spatial inequality also matters since the optimal allocation ensures a minimum quality for all regions. Finally, we explore the political economy considerations of why the observed investment allocations are not fully in line with the model optimal allocations. Although many factors could influence these deviations, we find a positive relationship between overinvestment (rel- ative to the optimal allocations predicted by the model) and increased local support for the incumbent authority. We regress the log deviations between the infrastructure investment in the data and the one predicted by the model on the differences in votes for the incumbent Workers’ Party between the 2010 and 2002 elections. Since the Workers’ Party won the presi- dential elections in Brazil in 2002, 2006, and 2010, our explanatory variable measures increased support for the incumbent at the microregion level. Our results suggest that an increase of 1 percentage point in support for the incumbent is associated with an increase in the quality of electricity by 0.03 - 0.05 percentage points and the quality of roads by 0.015 - 0.02 percentage points. Our study is related to several recent papers that have explored the interactions between different types of infrastructure and economic development. Moneke (2019) studies how big 3 push infrastructure investments affect welfare through an exogenous positive TFP shock in Ethiopia in a spatial general equilibrium model. Vanden Eynde and Wren-Lewis (2021) and Abbasi et al. (2022) use a reduced-form econometric approach and find evidence of strong economic complementarities for roads and electricity in the context of rural India and the Sub- Saharan Africa region, respectively. None of these papers study the optimality of government infrastructure investment policies and their implications. Another strand of related literature endogenizes infrastructure investment in quantitative spatial economic models. Fajgelbaum and Schaal (2020) study transport networks to assess op- timal investments and inefficiencies in the road networks of European countries. Santamaria (2020) assesses how governments can adjust transportation infrastructure to unexpected eco- nomic changes. Both studies, however, exclusively focus on the transportation infrastructure.3 Arkolakis and Walsh (2023) provide a spatial theory in which a planner affects the production and demand of electricity by choosing the electricity price at each node, and apply it to assess the global impact of the rise of renewable energy. Their framework likewise does not account for investments in other infrastructure types. The remainder of the paper is organized as follows. Section 2 describes the expansion of the roads and electricity networks in Brazil and the relationship between infrastructure quality and structural change. Section 3 develops the model. Section 4 discusses its calibration. Section 5 shows the quantitative results and main counterfactuals. Section 6 investigates the relationship between optimal infrastructure and political preferences. Section 7 concludes. 2 Background and Data This section quantitatively describes the road and electricity network expansion in Brazil and the spatial structural transformation process that accompanied the increased infrastructure availability. Our data comes from several sources. The sectoral employment shares and elec- tricity access come from the Integrated Public Use Microdata Series (IPUMS-I, 2019), com- prising four waves of individual and household survey data from 1980-2010. We complement these data with the information from Brazil’s national electricity regulatory authority, ANEEL (Agência Nacional de Energia Elétrica), and from the Ministry of Infrastructure and Transport of Brazil, DNIT (Departamento Nacional de Infraestrutura de Transportes). 3 There is a growing literature on the economic impact of road quality improvements. Gallen and Winston (2021) have a dynamic general equilibrium model where transportation is treated as capital funded by taxpay- ers. Morten and Oliveira (2023) use a general equilibrium trade model and rich spatial data to study how road infrastructure facilitates the movement of traded goods. Trew (2020) and Allen and Arkolakis (2022) study how transportation improvements arise in decentralized settings. Balboni (2019) develops a dynamic spatial model to analyze whether transportation improvements should concentrate in coastal areas under the risk of rising sea levels. Alder (2023) builds a spatial model to analyze whether connecting large economic centers or medium- sized cities is more optimal. Reduced-form studies include Gertler et al. (2022), Asher and Novosad (2020), Bird and Straub (2020), and Jedwab and Storeygard (2021). 4 Figure 1 shows the relationship between sectoral employment shares and infrastructure availability (illustrated by electricity access) across municipalities and over time. The pattern of structural transformation and infrastructure availability is consistent with the literature on infrastructure, structural change, and economic development (Herrendorf et al., 2014; Perez- Sebastián and Steinbuks, 2017; Perez-Sebastián et al., 2019). As the share of people with access to electricity increases, the employment share in agriculture tends to decline, while manufac- turing and, especially, services tend to increase. These results are documented in the earlier literature demonstrating the role of infrastructure in the Brazilian structural transformation across time. Figure 1: Structural Transformation and Electrification Note: Data from IPUMS-I. Each point shows the municipality’s sectoral employment share and the percentage of people with access to electricity. Colors denote years. Infrastructure expansion in Brazil was slow in the first half of the twentieth century and significantly accelerated in the 1960s and 1970s. The impetus came when President Kubitschek (1956-1961) launched a major national development plan (Piano de Metas or “Targets Plan”) fo- cusing on public investment infrastructure to support industrialization and fulfill his electoral platform. This led to a major expansion of electricity and transportation systems and the creation of the new capital city, Brasilia (Ayres et al., 2019). The electrification process started from Brazil’s most developed southern municipalities, extended to the rapidly industrializing Center-West region, and finally, to the more agricultural- intensive North. By 1991, for most regions, the majority of inhabitants had full access to elec- tricity. In 2010, the electrification of Brazil was effectively complete, with only several remote regions in the North partially left without full coverage.4 Appendix Figure I.11 describes the spatial evolution of electricity access over time. Though electricity access is no longer a major obstacle in Brazil, its quality remains highly unreliable in some parts of the country, as the DEC and FEC indicators provided by ANEEL 4 These remaining regions were electrified fully in 2013 with the completion of Luz Para Todos rural electrifi- cation program. 5 confirm. The DEC and FEC indicators give the average length in hours and the average number of electricity outages per consumption unit in a given year, respectively. To illustrate their importance, Figure 2 shows the duration and frequency of power interruptions in 2010 relative to the national average for the 557 microregions that compose our sample.5 Power outages tend to be spatially correlated, and there is a clear divide between the Northwest (poor quality) and the Southeast (good quality) parts of the country. Figure 2: Duration and Frequency of Power Outages (2010) (a) Duration of Power Outages (DEC) (b) Frequency of Power Outages (FEC) Note: Data from ANEEL. The maps show the duration and frequency of power outages relative to the average value in 2010. Darker colors indicate higher values (worse outcomes). The DNIT georeferenced database provides information about the areas connected by roads, their distance, and quality (whether they are paved, have a duplicated or a single lane, and the federal status). Similar to electricity, while the construction of the road network of Brazil has been largely completed, there are significant differences in quality. Figure 3 shows the evolution of the road network from 1980 to 2021 (latest data available). We now use the DEC, FEC, and road quality indicators to examine the relationship be- tween the economic activity distribution, the population size, and the quality of infrastructure observed in the data. For that purpose, we use data for the years 2000 and 2010 and regress d ) on the log of the population ( L ), the sectoral employment shares in a region i at time t ( Li ,t i ,t the log of the demeaned indicator of the quality of electricity based on DEC and FEC (Gi,t , see ¯i,t , see footnote 20), an interaction term be- equation (32)), the log of demeaned road quality ( I 5 ANEEL provides these indicators at the municipality level. We aggregate them at the microregion by taking the average of the indicators across the municipalities that comprise a given microregion. 6 Figure 3: Spatial Evolution of Roads Access (1980-2021). (a) 1980 - 1990 (b) 1990 - 2000 (c) 2000-2010 (d) 2010 - 2017 Note: Data from DNIT. The map shows the evolution of the road network in Brazil from 1980 to 2021. 7 Table 1: Employment Shares and Quality of Infrastructure OLS GAM Agriculture Manufacturing Services Agriculture Manufacturing Services log( Li ) -0.063** 0.022** 0.041** -0.063*** 0.021*** 0.042*** (0.003) (0.002) (0.001) (0.004) (0.002) (0.003) log( Gi ) -0.027 0.016 0.011** -0.016*** 0.006** 0.011*** (0.006) (0.007) (0.001) (0.004) (0.002) (0.003) ¯i ) log( I 0.023* -0.006 -0.017*** 0.016*** -0.005*** -0.010*** (0.003) (0.003) (0.000) (0.003) (0.002) (0.002) ¯i ) log( Gi ) × log( I 0.002 0.003 -0.005* 0.007** -0.001 -0.005** (0.002) (0.001) (0.001) (0.003) (0.002) (0.002) Num.Obs. 1114 1114 1114 1114 1114 1114 R2 0.352 0.343 0.243 0.606 0.543 0.550 S (lat, lon) 0.000 0.000 0.000 Data: IPUMS-International for Brazil (2000-2010), DNIT and ANEEL. Geographic regions are consistent across years. Robust standard errors in parenthesis. * p < 0.1, ** p < 0.05, *** p < 0.01. Panel OLS controls for latitude and longitude. Panel GAM shows semiparametric regressions with thin-plate spline in latitude and longitude, S (lat, lon) shows the approximate p-value of the thin-plate spline. tween the quality of roads and electricity, year fixed effects (µt ), and the function S (lati , loni ) that controls for the possible spatial correlation of the errors. In particular, d ¯ Li ,t = α + β 1 log ( Li,t ) + β 2 log ( Gi,t ) + β 3 log ( Ii,t ) (1) ¯i,t )) + µt + S (lati , loni ) + ε i,t . + β 4 (log( Gi,t ) × log( I The first three columns in Table 1 show the OLS estimates of (1) in which the variables latitude and longitude enter the regression linearly. A second set of results in that table, labeled as GAM (Generalized Additive Model), follows Kelly et al. (2023) and considers a thin-plate spline specification of S (lati , loni ); see Appendix A for details. Table 1 shows that locations with larger shares of employment in manufacturing and ser- vices tend to have larger populations. Better quality of electricity is associated with smaller employment shares in agriculture and larger employment shares in manufacturing and ser- vices. Better road quality is associated with larger employment shares in agriculture and lower employment shares in manufacturing and services. The interaction term shows that the qual- ity of electricity (roads) raises the marginal effect of improving the quality of roads (electric- ity) for agriculture. For services, the quality of roads (electricity) raises the marginal effect of improving the quality of electricity (roads). These interaction results indicate possible com- plementarities between the transportation and electricity networks. The salient feature of our model is that the local electricity and road quality affects firms’ 8 decisions. Hence, we now look at how important public infrastructure quality is for Brazil’s firms and whether the degree of spatial heterogeneity in quality shown in Figures 2 and 3 is a relevant factor for them. For this, we rely on the World Bank’s Enterprise Survey for Brazil in 2007, which is the closest to our benchmark scenario of 2010. Table 2 shows descriptive statistics for selected variables in the Enterprise Survey for Brazil by firm size. Overall, 45.6% of firms experienced power outages over the fiscal year 2007, and small, medium, and large firms experienced them in a similar proportion. When asked how much electricity is an obstacle, 45.9% of firms identify it as a major or very severe one. For transport, this number equals 27.5%, but still, 31.8% find it a moderate obstacle. The survey also asks, if the establishment were to relocate, how much would roads and energy reliability be part of the decision? 70.8% and 76.0% of the firms find roads and energy reliability to be at least important for this decision, respectively. Finally, we see no significant differences across firm sizes when asked about the number and duration of power interruptions. These numbers suggest that (i) the quality of infrastructure matters for the firm’s location decision and (ii) firm size alone is insufficient to avoid power outages or obstacles derived from transport. 3 A Spatial Equilibrium Model with Endogenous Comple- mentary Infrastructure Networks 3.1 Model Outline This section outlines a spatial multisector equilibrium model with endogenous electric power and road transportation infrastructure networks. The model builds on the quantitative spatial modeling literature (Redding and Rossi-Hansberg, 2017) and is closely related to frameworks of Fajgelbaum and Schaal (2020) and Santamaria (2020), where the government chooses the quality of infrastructure to maximize social welfare. Our model extends these earlier studies by incorporating an additional infrastructure type (electricity grid), which generates location- specific productivity spillover effects. It also allows for different degrees of efficiency and coordination in infrastructure investment across government branches. We consider a country with a fixed set of N microregions (or locations). Consumers (or households) can move freely within the country. Each consumer supplies inelastically one unit of labor in a given location. The household consumes products from the agricultural, manufacturing, and services sectors. The electricity supply in each location is taken as given and can be consumed as a service by households or used as a production input by firms. The production and consumption activities take place in the center of the microregion. Throughout the paper, we use the index i for production locations and j for consumption locations. 9 Table 2: Descriptive Statistics Enterprise Survey Characteristic Overall, N = 1,8021 Small, N = 8141 Medium, N = 7381 Large, N = 2501 Experienced Power Outages Don’t know 0.3% 0.1% 0.1% 7.0% Yes 45.6% 45.2% 48.8% 40.7% No 54.1% 54.7% 51.2% 52.3% Obstacle: Electricity Don’t know 0.0% 0.0% 0.0% 0.0% Does not apply 0.3% 0.0% 1.5% 0.2% No Obstacle 30.0% 30.9% 26.0% 25.2% Minor Obstacle 10.5% 10.2% 13.5% 4.6% Moderate obstacle 13.3% 12.7% 17.3% 10.1% Major obstacle 21.5% 23.0% 13.0% 23.4% Very Severe Obstacle 24.4% 23.2% 28.6% 36.5% Obstacle: Transport Don’t know 0.0% 0.0% 0.0% 0.0% Does not apply 1.4% 1.6% 0.5% 0.8% No Obstacle 31.8% 31.5% 37.2% 9.8% Minor Obstacle 7.4% 6.8% 10.7% 7.5% Moderate obstacle 31.8% 32.9% 26.3% 32.1% Major obstacle 17.1% 17.3% 14.0% 29.2% Very Severe Obstacle 10.4% 9.9% 11.4% 20.7% Importance in Relocating: Roads Don’t know 0.0% 0.0% 0.0% 0.0% Does not apply 0.3% 0.1% 1.1% 0.9% Not important 15.7% 17.2% 10.1% 3.8% Slightly important 4.1% 4.2% 4.4% 2.1% Important 21.3% 21.9% 20.1% 9.2% Very important 26.7% 25.7% 27.9% 50.3% Extremely important 31.8% 30.8% 36.4% 33.7% Importance in Relocating: Energy Reliability Don’t know 0.0% 0.0% 0.0% 0.2% Does not apply 0.0% 0.0% 0.1% 0.9% Not important 16.5% 18.9% 6.1% 3.3% Slightly important 7.4% 7.6% 6.3% 9.7% Important 25.9% 28.0% 15.9% 20.3% Very important 25.2% 21.4% 43.8% 35.5% Extremely important 24.9% 24.2% 27.9% 30.1% Number of Power Outages 1.75 (4.76) 1.81 (4.75) 1.46 (4.84) 1.46 (4.58) Duration of Power Outages 1.74 (5.15) 1.92 (5.61) 0.93 (1.79) 0.96 (2.09) 1 %; Mean (SD). Observations weighted on firm eligibility. 10 3.2 Households The population size in a consumption location j equals L j . Consumer preferences favor hor- izontally differentiated varieties of products from agriculture, manufacturing, and services, denoted with subindices a, m, and s, respectively. Households also consume electricity ser- vices ( EUj ) at a price b j for home use and enjoy amenities (η j ) in the location they live. The amenities depend on spillovers from all the inhabitants in the location controlled by parameter ¯ j Lϑ ϑ (i.e., η j = η j ). The consumer’s utility function can be written as: 1− ν Uj = Cjν EUj ¯ j Lϑ η j (2) where ρ 1−1/ρ 1−1/ρ 1−1/ρ ρ −1 Cj = ωa Caj + ωm Cmj + ωs Csj (3) and σ σ −1 Cdj = ∑ c1 −1/σ dji . (4) i∈ N The bundle Cdj groups the consumption in location j of products from an industry d ∈ D ≡ { a, m, s} produced in any location i ∈ N . Those units produced in location i and con- sumed in j are denoted by cdji and sold at price pdji . Hence, the consumer’s budget constraint is given by wj = ∑ ∑ pdji cdji + bj EUj . (5) d∈ D i∈ N It sets total income from wages (w j ) equal to total expenditures in electricity and in products coming from all sectors, locations, and product varieties.6 Consumers maximize objective function (2) subject to (3) to (5). From the household prob- lem, we can show the usual within-sector and across-sector expenditure shares: 1− σ pdji cdji pdji = (6) Pdj Cdj Pdj and 1− ρ Pdj Cdj ρ Pdj = ωd ; (7) Pn Cn Pn where 1 1− σ Pdj = ∑ p1 −σ dji (8) i∈ N 6 The budget constraint implies that the marginal price of electricity is the same as its average price. In the real world, these prices are not the same since the electricity price is a two-part tariff with fixed and variable payment components. Including the fixed part would complicate derivations without significantly changing our results. 11 and 1 1− ρ ρ 1− ρ Pj = ∑ ωd Pdj (9) d∈ D are the exact price indices for consumption bundles Cdj and Cj , respectively. Because households can move freely across regions, consumers’ utility in equilibrium is the same across them. Given the Cobb-Douglas structure of preferences in equation (2), a constant share ν is spent on the final consumption of goods and services, and a share (1 − ν) ¯ , is given by on electricity. Consequently, the indirect utility function, U ¯ j Lϑ η j wj ¯ = ν ν (1 − ν )1− ν U , (10) −ν Pjν b1 j From the indirect utility function (10), electricity payments introduce a dispersion force that prevents the accumulation of people in the same location since their utility declines with the electricity price. 3.3 Firms In any production location i, firms can operate in three sectors: agriculture, manufacturing, and services. In each of these industries, firms produce competitively according to the follow- ing production function: β 1− β d ydi = Adi Ldid Edi . (11) That is, firms employ labor, Ldi , and electricity, Edi , with a unitary elasticity of substitution between the inputs. The labor share β d is assumed to be sector-specific, reflecting differences in the intensity of energy use.7 ¯ di ) and The sector-location specific productivities ( Adi ) have an exogenous component ( A a production externality that depends on the number of workers in the location and a measure of electrical-grid infrastructure quality (Gi ). More specifically, ¯ di Lαd G ιd . Adi = A (12) i i The parameters αd and ι d control for the size of the production externalities. We consider that αd ≶ 0. If firms use land intensively to produce output, an excess amount of workers can create a congestion effect leading to αd < 0. However, there are also agglomeration effects and knowledge spillovers (Carlino and Kerr, 2015), which may outweigh congestion and result in αd > 0. We assume that ι d > 0, which reflects possible spillovers that allow for the adoption of more efficient technologies, such as irrigation systems and information technologies that require a reliable supply of electricity (Andersen et al., 2012; Rud, 2012; Szerman et al., 2022). 7 Caliendo et al. (2018) assume that factor shares in value-added are equal across sectors but change across locations. We consider the opposite scenario due to the lack of sufficient data to calibrate factor shares at the regional level. 12 The total costs of a firm are given by wi Ldi + bi Edi . Firms choose the electricity and labor inputs to minimize those total costs subject to a given level of output, which in the optimum yields: Edi 1 − β d wi = . (13) Ldi β d bi Since firms produce in a perfectly competitive environment, the price equals the marginal cost. Therefore, the (free-on-board) price for a good produced in location i and sector d is: β 1− β d θ d w i d bi pdi = ; (14) Adi where βd 1− β d 1 1 θd = . βd 1 − βd 3.4 Gravity The price of a good in a sector d consumed in a location j and produced in location i is given by pdji = pdi Tji , where Tji denotes the iceberg cost of transporting the good from i to j. We can use (8), (12), and (14) to obtain:  1 1− σ   β 1− β θ d w i d bi d  1− σ Pdj = ∑ ( Tji )1−σ A¯ di Lαd G ιd . (15) i∈ N i i   Using the consumption optimality condition (6), multiplying by L j , and substituting in (15) delivers: β 1− β 1− σ θ d w i d bi d Lj ∑ pdji cdji = σ L j Pdj Cdj ∑ (Tji ) 1− σ A¯ di Lαd G ιd . i∈ N i∈ N i i It is straightforward to see from there that the proportion of sector-d goods consumed in j imported from i (πdji ) is given by the equation: β 1− β 1− σ θ d w i d bi d ( Tji )1−σ ¯ di Lαd G ιd A i i πdji = 1− σ (16) β 1− β θdn wn d bn d ∑ (Tjn )1−σ A¯ dn Lα d ιd n∈ N n Gn 1− σ Observe that the denominator of the import share is precisely Pdj . We can now express the sectoral location-specific price index in terms of the trade share with itself using that Tjj = 1 for all j ∈ N . 13 1  β 1− β  1 1− σ θdj w j d b j d Pdj =   (17) πdjj A¯ dj Lαd G ιd j j Using the definition of Pj in equation (9) and substituting in (17), we can derive the aggregate price index for location j:   1 β 1− β  1− ρ 1− ρ  1− ρ θdj w j d b j d Pj = ∑ (ωd )ρ (πdjj ) σ−1     (18)  A¯ dj L G α d ι d  d∈ D j j c ), Finally, to derive the consumer’s expenditure share that goes to sector d in region j ( Xdj we can combine condition (7), equation (15) and the fact that Pj Cj = ∑d∈ D Pdj Cdj to obtain: 1− ρ 1− σ  1 −σ β 1− β θ d w i d bi d ωd  ∑ ( Tji )1−σ ρ  A¯ di Lαd G ιd c Pdj Cdj i∈ N i i Xdj = =  1− ρ . (19) Pj Cj  1− σ  β 1− β θsi wi s bi s  1− σ ∑ ωs  ∑ (Tji )1−σ ρ A¯ si Lαs G ιs s∈ D i∈ N i i  Note that expenditure shares within sectors across locations (πdji ) and expenditure shares across sectors within locations ( Xdj ) are functions of {w j , b j , L j , Gj } j∈ N and of exogenous variables only. They do not depend on sector-specific endogenous variables. 3.5 Market Clearing We assume that the total supply of electricity in the region i (Ei ) is exogenous and cannot be stored. In equilibrium, the electricity price bi must be such that the electricity market clears. That is, the total residential and commercial consumption of electricity across regions must equal the total power supply in the country (E¯ ):8 ∑ L j EUj + ∑ ∑ Edj = E ¯. (20) j∈ N d∈ D j∈ N 8 More generally, we could consider that Ei is a composite of the available grid electricity supplied by the utilities and off-grid electricity generated by autoproducer firms as a backup generation when grid electricity is not available. Without market distortions in the power sector, the price of both grid and off-grid electricity, f g g f bi and bi , would equal the marginal cost of electricity service, with bi < bi due to economies of scale. Let Λi ∈ [0, 1] be the fraction of time when the grid electricity is available. Then, consumers and firms would end up paying g f bi = Λ i bi + ( 1 − Λ i ) bi . 14 Regarding the labor market, market clearing requires that the labor demand equals the labor supply: ∑ ∑ Ldj = ∑ Ln . (21) d∈ D j∈ N j∈ N We adopt an expenditure approach to derive the clearing condition for the goods market. Total production must be used to pay for the electricity and labor inputs. Total revenues from electricity sales must be equal to payments by consumers and firms. Because both the consumers’ utility (2) and the firms’ production functions (11) are of the Cobb-Douglas form, a constant share of consumers’ income and firms’ revenues is spent in electricity payments. Therefore, using bilateral trade shares (16) and expenditure shares (19), electricity payments need to satisfy bi Ei = (1 − ν)(wi Li + bi Ei ) + ∑ ∑ (1 − βd )πdji Xdj c ν(w j L j + b j Ej ) (22) d∈ D j∈ N Electricity Expenditure in i Share Revenues from Selling to Other Regions On the other hand, the Cobb-Douglas structure of the production function implies that a con- stant share of revenues β d is paid to workers in each region. Additionally, consumers in all locations devote a fraction ν of their income to purchasing consumption goods. Hence, the following condition must hold in location i for the labor payments: wi L i = ∑ ∑ βd πdji Xdj c ν ( w j L j + b j E j ). (23) d∈ D j∈ N Finally, adding up conditions (22) and (23) obtains an aggregate market-clearing condition for the goods market that implies that total income in a region i is equal to total expenditures: wi Li + bi Ei = ∑ ∑ πdji Xdj c ( w j L j + b j E j ). (24) d∈ D j∈ N 3.6 Transport and Electricity Networks Suppose that two adjacent locations i and j are joined by a road segment; and following Fajgel- baum and Schaal (2020) and Santamaria (2020), consider that the iceberg cost Tji is a function of the level of road investment Iji in that segment, that is, Tji = Tji Iji . If infrastructure improves, the transport costs decline: ∂ Tji Iji <0 ∂ Iji Furthermore, since not every region might be connected to every other region directly, we define the total iceberg costs for shipping a good from i to j as the sum of costs from shipping through intermediate regions: T δkn Tji Tji = ∑ ∑ Ikn ( Ikn )γ (25) n∈ N k∈ N 15 Tji where Ikn takes value one when the edge connecting n and k is in the least cost path between T captures purely geographic components such as i and j and zero otherwise. Parameter δkn ruggedness of the terrain, geographical distance, borders, or any other frictions affecting the unitary transport cost between adjacent regions k and n. Parameter γ controls the elasticity of the unitary transport cost to infrastructure. Moving now to the case of electricity, since Brazil was fully electrified by 2010, we focus on the quality of the electrical grid rather than on investments in grid expansion. That is, what is crucial is the grid transmission and distribution capacity. We assume that the government can improve the quality of the electricity infrastructure by carrying out investments ( Gi ) at the regional level. 3.7 The Decentralized Spatial Equilibrium Next, we define the decentralized spatial equilibrium and derive the population shares of each location. Definition 1 (Decentralized Equilibrium). Given the total population L ¯ , intrinsic sector-location ¯ di }d∈ D,i∈ N , intrinsic amenities η productivities { A ¯ i , electricity supply at each location { Ei }i∈ N , a T matrix of road-quality investments and shortest path indicators { Iij , Iijnk }∀i, j,k,n∈ N , and a vector of electricity quality { Gi }i∈ N , a competitive spatial equilibrium is a utility level U ¯ , a set of factor prices in each region {wi , bi }i∈ N , a set of sector-location labor and electricity allocations, and sector-location prices { Ldi , Edi , Pdi }d∈ D,i∈ N such that households maximize utility, firms maximize profits, all markets clear, intra-regional trade is balanced, and the utility is equalized across regions. Since a spatial equilibrium requires that the utility level is equalized across locations (at ¯ ), we can use (10) to get: level U +ϑ ¯ i wi L1 ¯ = 1 η U ¯ L ∑ν ν (1 − ν ) 1− ν i Piν bi1−ν , i∈ N which yields the following expression for the population shares:9 +ϑ ¯ j w j L1 η j −ν Lj Pjν b1 j ¯ = L ¯ i wi L1 η +ϑ . (26) ∑ i Piν bi1−ν i∈ N 9 Changes in relative factor prices are key for reallocating labor across locations and sectors. Propositions 3 and 4 in Appendices D.2 and D.3 show the effect of an increase in composite productivity and an increase in the quality of electricity on relative factor prices for a simplified version of the model that allows for closed-form solutions. They imply that the relative intensities in factor use and the relative size of the spillovers matter for the direction of the effect. 16 3.8 The Public Sector The central government (i.e., the line ministries) in this economy has an exogenously allocated budget, Z, to finance expenditures in the quality of public infrastructure. In particular, the government maintains the following balanced budget: Z= ∑ ξ iG Gi + ∑ ∑ ξ T jk I jk . (27) i∈ N k∈ N j∈ N The above equation deserves some additional comments. First, it does not include the generation and distribution of electricity services, provided exogenously by the utilities and reflected in the end-use electricity price bi . The parameter ξ iG captures the unitary variable electricity transmission costs to a given location (including technical losses and congestion costs) and maintaining grid electricity infrastructure.10 Second, the unitary variable cost of road access, ξ T jk , is also related to its maintenance. Knowing its exogenous budget constraint and the behavior of consumers and firms in the different markets, the government decides how much to invest in transport infrastructure ( Iji ) and how much to invest in quality of grid electricity ( Gi ). We proceed below to define formally the government’s problem. Definition 2 (Government’s Problem). The government solves the problem max ¯ U { Gi , Iji }i, j∈ N subject to: 1. Free labor mobility condition (10): +ϑ ¯ j w j L1 η j ¯ j = ν ν (1 − ν )1− ν UL ∀j ∈ N −ν Pjν b1 j 2. Electricity and labor markets clearing conditions (20) and (21): Li = ∑ Ldi ; L = ∑ Li ; Ei = EUi Li + ∑ ¯= Edi ; E ∑ Ei ∀i ∈ N ; ∀d ∈ D d∈ D i∈ N d∈ D i∈ N 3. Goods market clearing (24): ρ σ − ρ ρ −1 wi Li + bi Ei = ∑ ∑ ωd Pdj Pj ( Tji κdi )1−σ (w j L j + b j E j ) ∀i ∈ N d∈ D j∈ N 4. Transport costs (25): T Tji δkn Tji = ∑ ∑ Ikn ( Ikn )γ ∀i, j ∈ N n∈ N k∈ N 10 In practice, the state-owned transmission system operator charges these costs to electric utilities, which are then passed onto consumers. 17 5. Government’s budget constraint (27): Z= ∑ ξ iG Gi + ∑ ∑ ξ T jk I jk i∈ N k∈ N j∈ N 6. Previous underlying network investments: ¯ij < ∞ 0 ≤ I ij ≤ Iij ≤ I ¯i < ∞ 0 ≤ Gi ≤ Gi ≤ G In these constraints, Pdj is defined as in (8), Pj as in (9), κdi is given by β 1− β θ d w i d bi d κdi = (28) A¯ di Lαd G ιd i i and bi comes from equation (13), Li 1 − β d Ldi bi = w i ν Ei 1−ν+ ∑ β d Li . (29) d∈ D 4 Model Calibration We now give values to the model’s different parameters and exogenous variables. Except when noted, this calibration is performed using Brazilian data for the year 2010. Exogenously Calibrated Parameters. Table 3 shows the parameters taken from the lit- erature and their values. Let us start with the ones related to the household’s problem. The elasticity of substitution across sectors ρ is set to 1;11 otherwise, we cannot guarantee that the model equilibrium is unique (see Appendix D.1). The elasticity of substitution σ for goods within the sector is set to 5 so that the elasticity of sectoral trade flows with respect to trade costs equals 4 as estimated by Simonovska and Waugh (2014). The parameters of the utility function ωd are set to match the gross value added (GVA) share for each sector. Parameter ν is estimated from the Brazilian consumer expenditure survey (POF) for 2008-2009 as one minus the average expenditure share in electricity. Following Allen and Arkolakis (2014), we map the parameter ϑ to the expenditure share in housing, which equaled 25% for the POF 2002- 2003 wave and 21% for the 2008-2009 period. This implies a value of ϑ ∈ (−0.27, −0.33). Consequently, we take the intermediate value ϑ = −0.3.12 The production function parameter β d is obtained as one minus the share of energy con- sumption in production. The elasticity of trade costs to investment γ is another key parameter for our results. We choose an intermediate value among those estimated by previous literature and set it to 0.47.13 11 Notice that this implies that the country-wide sectoral GVA shares are constant but not the labor ones. Furthermore, policy in our model can change both the sectoral GVA and the sectoral labor shares across locations. 12 If κ is the expenditure share in housing, ϑ = − κ . 1−κ 13 Santamaria (2020) and Allen and Arkolakis (2022) estimate values of 0.84 and 0.4 for γ, respectively. Esti- 18 Table 3: Exogenously Calibrated Parameters Parameter Value Target/Reference σ 5 Simonovska and Waugh (2014) {βa, βm, βs } {0.92, 0.86, 0.95} Share of Energy in Production, 2010 {ωa , ωm , ωs } {0.204, 0.272, 0.524} Value Added Shares, 2010 ν 0.93 Expenditure Share in Electricity, 2008-2009 ϑ −0.30 Expenditure Share in Housing, 2002-2009 γ 0.47 Literature (see the text) Calibrated values correspond to Brazilian data, except when taken from the literature. Construction of the Road Network of Brazil. The road network is defined as R ≡ (E , V ) where E is a set of edges or roads, and V is a set of vertices or road junctions. Each microregion is associated with a point in the plane corresponding to its centroid, which we express as the set M. The set of road junctions V and the set of microregions centroids M do not need to coincide, so to construct an adjacency matrix of the network joining V and M we choose to shift points in M to the closest point in V that is still within the microregion. That is, for each mi ( xi , yi ) ∈ M, we associate a v j ( x j , y j ) ∈ V such that mi ( xi , yi ) ← argmin ( x i − x j )2 + ( y i − y j )2 . v j ∈V Definition of Microregion Centroids. Instead of using geographical centroids, we opt to choose a point of economic significance. To do so, we use data from the Brazilian Institute of Geography and Statistics, IBGE (Instituto Brasileiro de Geografia e Estatistica) on the registra- tion of selected locations, which offers georeferenced data on cities. A location is considered a city if it has a city hall. It consists of the urban area of the headquarters district and is bounded by the urban perimeter by municipal law. This dataset contains a total of 5,565 cities. If a mi- croregion has several cities, we choose the one with the largest population based on the IBGE Census data. If a microregion has no urban areas, we likewise select the rural center with the largest population. Figure 4a shows a zoomed-in view of a specific section of the observed network with the locations of the centroids. Figure 4b shows the discretized network with shifted centroids for the same section.14 mates in Couture et al. (2018) imply γ = 0.1, which is the one used by Fajgelbaum and Schaal (2020). We choose γ = 0.47, which is in the middle of the interval [0.1, 0.84]. 14 To compute to which region each centroid belongs to, we use the function inpoly2 that substantially reduces computation times (Engwirda, 2021). 19 Figure 4: Observed and discretized road networks in Brazil (a) Observed Network (b) Discretized Network Note: Data from IBGE and the World Bank. Dots represent the microregion centroids. For clarity, the map shows only the north of Brazil. Calibration of Roads’ Investment Costs. To calibrate (ξ T jk ), we rely on altitude data to construct an index of terrain ruggedness. In particular, we use data from the Global Land One- Kilometer Base Elevation (GLOBE) digital elevation model (DEM). This is a 30-arc-seconds degree grid, roughly corresponding to 1 km gridding at the Equator. The units represent ele- vation in meters above the mean sea level. We first construct a measure of average elevation in each location. After that, we choose the Euclidean difference in elevation across neighboring locations as a measure of ruggedness. That is, we compute ∑ 2 rugg j = elev j − elevk , (30) k∈ Nj where elev j corresponds to the average altitude in j, and Nj is the set of the neighboring locations that surround j. Finally, we choose the average difference in altitude associated with them as a measure of the bilateral geographical barriers between j and k: 1 rugged jk = (ruggk + rugg j ). 2 Following Fajgelbaum and Schaal (2020), we use estimates in Collier et al. (2015) and cali- brate the cost of investment between any two microregions as ξT jk T log = log ξ 0 − 0.11 × (dist jk > 50km) + 0.12 × log(rugged jk ) (31) dist jk 20 where dist jk is the distance in kilometers along the shortest path on the actual road network, T is calibrated (dist jk > 50km) takes on 1 if the inequality holds and on zero otherwise, and ξ 0 so that we match the share of the total government budget allocated to roads. Calibration of Transport Costs. Observed transport costs between two neighboring re- gions i and j are a function of the road quality investment Iij (derived from the DNIT geospatial T . The road quality along a given roads database) and the geographical-barriers parameter δij discretized segment is computed as the average speed at which products can travel (see Ap- T , in turn, we use the distance in kilometers taking into pendix E for details). To calibrate δij account that one can reach a location through the observed road network or, if there are no roads connecting i and j, by land. More specifically, we compute the geographical barriers between any two locations as the shortest path through the road network or by land weighted by the ruggedness index. We assume, following Allen and Arkolakis (2014), an exponential form for the normalized distance. Calibration of Electricity Investment Costs. To assign values to the marginal cost of ¯G / δ G , investing in quality of electricity, ξ G , we assume that ξ G can be expressed as ξ G = ξ i i i i ¯G is calibrated to match the share of investment in electricity (71.88%),15 and follow- where ξ ing Lipscomb et al. (2013), δiG is given by δiG = Φ ( β 1 log(Max Flow) + β 2 Average Slope in River + β 3 Max. Slope in River + β 4 Forest Cover) where Φ(·) denotes the CDF of the standard normal distribution, and the coefficients are taken from Table 1 in their paper. Their estimates provide the geographic potential for hydroelectric energy production. We assume that locations with smaller potential have a higher cost per unit of electricity investment because they are more dependent on the external electricity supply. Calibration of Available Grid Supply of Electricity. To calibrate Ei , we use the IBGE continuous cartographic database (Bases Cartográficas Contínuas), which provides data on the location and load of the main sources of energy generation. We use the load from thermal and hydroelectric stations to compute a distance-weighted average of the total available load for each region. We subtract the average electric power transmission losses.16 Finally, we normalize this index to be in the range [1, 100]. Calibration of Local Quality of Electricity. The observed quality of electricity Gi is cal- culated based on two collective indicators of continuity provided by ANEEL: the Equivalent 15 This number comes from the World Bank PPI database: ppi.worldbank.org. Other sources (Paiva, 2010; BNDES, 2014) suggest very similar values. 16 World Bank Data estimates that average electric power losses as a percentage of output from transmission and distribution are 14.6%. The series identifier is EG.ELC.LOSS.ZS. 21 Figure 5: Distribution of Power Generation Stations and Substations in Brazil. Interruption Duration per Consumer Unit (DEC) and the Equivalent Interruption Frequency per Consumer Unit (FEC). In particular, H Git = √ . (32) DECit × FECit where H is a positive constant equal to the total number of hours in a year. We normalize Gi so that its mean equals 1. With this normalization, Gi captures the quality of electricity in a location relative to the average quality. Notice that the denominator of the last formulation equals the square root of the number of hours without access to electricity per year and household. Therefore, equation (32) implicitly implies an increasing marginal cost of reducing the time affected by outages. Calibration of Composite Productivities and Amenities. We calibrate composite sec- toral productivities, Adi , using the sectoral version of market clearing condition (23): wi Ldi = β d ∑ πdji Xdj c ν ( w j L j + b j E j ). (33) j∈ N Substituting now equations (16) and (7) in the last expression, and imposing that ρ = 1 (then, c = ω ), we get e.g., Xdj d 1− β d 1− σ ∑ ωd −1 σ −1 β wi Ldi = β d Tji θd wi d bi Aσ di Pdj ν ( w j L j + b j E j ). j∈ N 22 Note further that the latter equation, together with the expression for the price index (8) form a system of 2 × N × D equations for 2 × N × D unknowns { Adi , Pdi }. We can express this system as 1− β d 1− σ 1− σ ∑ (Tij )1−σ θd w j d bj −1 β Pdi = Aσ dj (34a) j∈ N βd 1− β d 1− σ A1 −σ ∑ ωd σ −1 β di = Tji θd wi d bi Pdj ν ( w j L j + b j E j ), (34b) wi Ldi j∈ N and we can now show the following proposition. Proposition 1. Given observed wages {wi }, employment in each sector { Ldi }, total population { Li }, total electricity supply in each location { Ei }, and trade costs { Tji }, there exists a unique (up to scale) solution for the system (34a)-(34b). Proof. The proof is in Appendix D.1. For amenities, we can invert equation (26) to obtain normalized amenities. More specifi- cally, the following can be shown. Proposition 2. Given observed wages {wi }, employment in each sector { Ldi }, total population { Li }, total electricity supply in each location { Ei }, trade costs { Tji }, and composite sectoral productivities { Adi }, there exists a unique (up to scale) value for amenities at each location η ¯i that satisfies the free population mobility condition (26). Proof. The proof is similar to Proposition 2 in Michaels et al. (2011). Estimation of Productivity Spillovers. We estimate productivity spillovers with the Sim- ulated Method of Moments (SMM). To do so, we need to obtain the composite productivities and amenities employing the methodology explained previously. This provides empirical dis- ¯ j .17 tributions for Adi in each industry and for η We have six parameters to estimate, {αd , ι d }, and SMM proceeds in three steps. First, we draw S matrices of size N × D of sectoral composite productivities and of size N × 1 of amenities and investment in infrastructure Ijk and Gi for synthetic locations from the esti- mated distributions and compute the decentralized spatial equilibrium. Second, with these draws, we compute average moments from the simulated economies m ˜ , Θ), i.e., ˆ (x 1 S s∑ m ˜ |Θ) = ˆ (x ˜ s | Θ ), m( x ∈S where Θ are the model parameters and x ˜ s the model implied values for each simulation s. Finally, the SMM estimator is the set of parameters Θˆ SMM that solves ˆ SMM = arg min Θ (m ˜ , Θ) − m( x ))′ W(m ˆ (x ˜ , Θ) − m( x )), ˆ (x Θ 17 Figure I.12 in Appendix I shows the empirical and fitted cumulative distribution functions for each of the three sectors. 23 where W is a weighting matrix. We choose W as a diagonal matrix with entries given by the ith moment 1/mi ( x )2 . When drawing amenities and sectoral productivities, we use a normal copula to take into account that the model will generate negative correlations between amenities and productiv- ities. To generate the actual draws, we rely on inverse transform sampling. (See Appendix F for additional details.) We employ 11 moment conditions to identify the 6 spillover parameters: • The correlation between population and sectoral employment shares. • The correlation between the quality of electricity and sectoral employment shares. • The 10th , 25th , 50th , 75th , and 90th percentiles of the distribution of population. Table 4: TFP Spillovers Parameter Description Agriculture Manufacturing Services αd Population 0.0010 0.0513 0.0780 ιd Quality of Electricity 0.0101 0.0827 0.1135 The estimation results are given in Table 4. They suggest that agglomeration externalities and electricity quality spillovers are the largest in the service sector, then manufacturing, and finally agriculture. The TFP elasticity with respect to the quality of electricity is sizable in all sectors, which is in line with arguments by Andersen et al. (2012) that the reliability of the electric power is key for IT adoption. Model Fit. To assess how the model fits the data, we plot the model equilibrium values against the actual data. Figure 6 shows that the model matches exactly the population in each location. This is not surprising since the population is one of the calibration targets. It also fits very accurately factor prices and employment shares. The main reason why the model does not match the employment shares exactly is that we do not allow for differences in wages across sectors. 5 Quantitative Results This section employs the model to quantitatively assess the impact of infrastructure on spa- tial development and how optimal are the observed infrastructure allocations in Brazil. The model calibration reproduces the observed data, taking the infrastructure investment as given. Using the estimated and calibrated parameters, we can now obtain the optimal infrastructure 24 Figure 6: Model Fit. Note: The orange straight line is a 45◦ line. Each data point corresponds to a microregion. investment by solving the government’s problem for a given reference year. To be consistent with the model calibration, the chosen reference year is 2010. We present results for four versions of the government’s problem to account for different scenarios of government agency in infrastructure investment. The “Fully Optimal” version solves the problem exactly as in Definition 2. The “Efficiency + Coordination” version considers the case when different government agencies (e.g., line ministries) have separate budgets for infrastructure investment and make decisions optimally coordinating their choices. That is, location-specific investment choices in both electricity and road infrastructure are still optimal, but the aggregate share of government spending that can go to roads relative to the electricity network is forced to match the actual data. Finally, the “Efficiency-Electricity” and “Efficiency- Roads” problems force the government to make optimal investment decisions in only one type of infrastructure (electricity and roads, respectively), taking the investment in the other type as exogenous and equal to the one observed in the data. The last two scenarios amount to the lowest optimization level in infrastructure spending. In the model, infrastructure is captured by the flow of services it provides. However, in the real world, those services result from the available stock of public capital. We imposed a lower bound on the local infrastructure investments in 2010 to account for this. The lower bound results from applying a depreciation rate to the value of the roads and electricity quality (or investment, in our framework) in 2000 over the following ten years. In particular, we assume that roads and electricity quality depreciate at 5% and 4% per annum, respectively.18 We also impose an upper bound that is 50% larger than the maximum investments observed in the 18 These numbers come from Fajgelbaum and Schaal (2020) and authors’ inquiries with the World Bank elec- tricity planning team in Brazil. 25 data in 2010.19 Table 5: Aggregate Employment Shares and Investment Shares Agriculture Manufacturing Services Electricity Share Data 21.45 23.74 54.81 71.88 Fully Optimal 35.34 21.07 43.59 61.11 Efficiency + Coordination 34.84 21.32 43.84 71.88 Efficiency-Roads 32.84 21.46 45.70 71.88 Efficiency-Electricity 30.94 22.38 46.68 71.88 Correlations with the Data in Shares of Investment Roads Electricity Fully Optimal 0.71 0.31 Efficiency + Coordination 0.79 0.23 Efficiency-Roads 1.00 0.22 Efficiency-Electricity 0.80 1.00 5.1 Data versus Predictions We start by comparing the model’s optimal sectoral employment and investment allocations to the actual data. The top panel in Table 5 shows the data in the first row and the solutions of the Fully Optimal and the Efficiency + Coordination versions of the government’s problem in the second and third rows, respectively. They indicate that optimal government investment policies result in a substantial reallocation across sectors. We can see that the agricultural employment share increases relative to the data while one of the non-primary activities falls. Furthermore, the share of investment in electricity declines in the fully optimal model version, which suggests underinvestment in roads in the data. The employment share rises in agriculture because there is a larger positive impact of the infrastructure investment improvement on the productivity of the non-primary sectors, and then, either consumers purchase more from those sectors or labor must be pushed towards the primary activity. However, a unitary elasticity of substitution between consumption products implies that the sectoral consumption shares are constant, and hence, what dominates at the national level is the reallocation of labor towards agriculture. The effect at the local level is different. As will become clear in Section 5.4, additional investment in infrastructure in a given microregion does not increase the labor share of agriculture in that location. 19 This is necessary to reduce the numerical complexity of finding the optimal allocation. Alternative upper bounds provide similar results. 26 The bottom panel in Table 5 shows the correlations between the investment shares pre- dicted by the model and the ones in the data. They are positive and substantially stronger for roads than for electricity. This, in principle, implies a larger degree of misallocation in the electricity infrastructure investments conditional on the available budget. Looking at roads in more detail, many locations in the data are almost isolated (i.e., have few low-quality connections). When the government connects these locations via additional road segments, trade costs for all locations trading through that road segment fall. Figure 7a ¯i ) for the Fully Optimal scenario against the data.20 We see that shows the degree centrality ( I the correlation is positive, but the optimal solution requires substantial reallocation. Because the share of the government budget allocated to roads increases, most locations are above the 45° line. This suggests that the planner reduces investment in roads with less central roles. Figure 7: Equilibrium Allocations 35 -3 30 -4 25 -5 20 Model Model -6 15 -7 10 -8 5 -9 5 10 15 20 25 30 35 -9 -8 -7 -6 -5 -4 -3 Data Data ¯i (a) Quality of Roads I (b) Log of Population 70 60 50 Model 40 30 20 10 10 20 30 40 50 60 70 Data (c) Aggregate Price Index The equilibrium investment allocations induce movements in the population. The rela- tionship between the population predicted by the model and the data is shown in Figure 7b. 20 ¯i = ∑ j∈ N Iij . ¯i measures the quality of the roads connected to location i as I Degree centrality I i 27 If there were no mobility of labor, each point would lie on the 45° line. What we see is that there is a substantial reallocation of labor. Actually, this reallocation generates a greater con- centration of people in the more service-intensive locations. Finally, Figure 7c shows that the equilibrium allocations imply a substantial decline in aggregate prices across all locations, mainly due to the reduction in transportation costs. Table 6: Investment Allocations Fundamentals and Investment in Electricity ¯ ai ), log( Gi )) Corr (log( A ¯ mi ), log( Gi )) Corr (log( A ¯ si ), log( Gi )) Corr (log( A Data −0.05 0.32 0.20 Model (Fully Optimal) 0.20 0.59 0.59 Model (Efficiency + Coordination) 0.25 0.61 0.63 Model (Efficiency-Electricity) 0.30 0.60 0.62 Model (Efficiency-Roads) −0.05 0.32 0.20 Fundamentals and Investment in Roads ¯ ai ), log( I Corr (log( A ¯i )) ¯ mi ), log( I Corr (log( A ¯i )) ¯ si ), log( I Corr (log( A ¯i )) Data 0.06 0.11 0.05 Model (Fully Optimal) 0.09 0.18 0.11 Model (Efficiency + Coordination) 0.08 0.27 0.21 Model (Efficiency-Electricity) 0.06 0.11 0.05 Model (Efficiency-Roads) 0.08 0.27 0.21 5.2 Sources of the Misallocation The above findings provide evidence of a substantial spatial misallocation of public capital. In our model, the decision of the government is affected by amenities, the exogenous compo- nent of productivities, the geographical barriers, or the complementarities in infrastructure investment. We now consider some of these potential sources in more detail. 5.2.1 Differences in Productivity We first look at how investment decisions relate to the exogenous component of the produc- tivities. This will allow us to understand better the guiding policies of the government and assess misallocation. The top panel in Table 6 shows the correlation observed in the data (first row) and the one predicted by the model (second to fourth rows) between the log of sectoral exogenous ¯ di )) and the log of the electricity infrastructure quality (log( Gi )). In productivities (log( A all cases, the allocations predicted by the model have a much larger correlation with the pro- ductivities than in the data. The correlations are largest in the Efficiency + Coordination and 28 the Efficiency-Electricity scenarios. This occurs because, when we do not allow for shared budgets, the government can get less advantage of the infrastructure complementarities, and then the weight of the exogenous productivities in its decision becomes larger. The correlations between the log of sectoral productivities and the log of road-quality investment are given by the bottom panel of Table 6. Except for agriculture, the sign and relative magnitude of the correlations are similar to the ones in the top panel but smaller in absolute value. This occurs because any improvement in the quality of roads that directly affects two locations can potentially change the least-cost path of other locations, which makes the correlation with its own productivity less meaningful. In the case of agriculture, the model- solution correlations are smaller than in the data, signaling that other fundamentals, such as geographical barriers and the degree of infrastructure complementarities, can be playing against isolated locations in rural areas. Taken together, Table 6 says that, in most cases, the quality of infrastructure moves more closely with the exogenous productivities in the model solutions than in the data. Hence, a source of the misallocation is the lack of a correct internalization of the productivity effects. 5.2.2 Infrastructure Investment Complementarities Next, we turn to another important aspect behind the public capital misallocation: the com- plementarity of investments. We first ask the question: if the government optimally allocates one type of investment, keeping the other type of infrastructure as observed in the data, what would be the relative gains? The last two columns in Table 7 show the results of this counter- factual experiment. If the government can only reallocate electricity or road investments, the welfare gains are large, 7.52% and 10.97% relative to the benchmark (given by the observed infrastructure data), respectively. We can see also that the average income increases by 3.4% and 10.7%, while for median income, the gains are slightly smaller, 2.85% and 10.25%. When the government can decide on both types of investments, the gains are substantially larger. This is shown in the first two columns of Table 7. When budgets are kept separated, the Efficiency + Coordination allocation generates median income, average income, and welfare gains of 14.88, 15.85, and 19.75 percent, respectively. If we also allow for a joint budget, the gains are even larger. The Fully Optimal solution produces welfare gains of around 22%, comparable in magnitude to the gains in average and median income. In the Fully Optimal counterfactual, it is important to note that, besides reassigning investments to different road segments, the government also increases the share of total expenditure on roads. Table 7 provides clear evidence that the government benefits from the complementarities of investments in both electricity and road infrastructures. When comparing the Efficiency + Coordination scenario to the Efficiency-Roads and Efficiency-Electricity cases, we see an increase of 1.26 percentage points in the welfare gains due to a larger degree of efficiency in the allocation within each infrastructure budget. Moreover, comparing the Fully Optimal and 29 Table 7: Welfare and Income Gains Relative to the Data Efficiency Efficiency Efficiency Fully Optimal and Coordination (Electricity) (Roads) Income (median) 21.15 14.88 2.85 10.25 Income (mean) 21.75 15.85 3.40 10.70 Welfare 22.01 19.75 7.52 10.97 the Efficiency + Coordination solutions, we find that the between-budgets reallocation allows for an additional welfare gain of 2.26 percentage points.21 To get a better idea of the importance of infrastructure complementarities, we turn to the question of what is the correlation between the optimal investments in the quality of electricity and road infrastructure. Figure 8 shows the quality of roads against the quality of electricity in the four scenarios we run. The correlation is positive in the four cases, but the largest correlation is found in the Efficiency + Coordination scenario (0.36). For reference, the correlation in the data between the two types of infrastructure is 0.16. This suggests that keeping the budget for each type of infrastructure as in the data, the optimal allocation would imply a stronger correlation between both types of infrastructure, thus implying an important role of the investment complementarities. However, when either the quality of electricity or roads is kept as in the data (bottom charts), the optimal allocation does not imply a stronger correlation. This suggests that the efficiency gains might be larger than complementarity gains (as suggested by the welfare differences in Table 7). Both conclusions are also supported by the Fully Optimal solution that shows a correlation of 0.26, which is larger than in the data but lower than in the Efficiency + Coordination scenario. The reason is that the budget reallocation raises the average quality of roads in low electricity quality locations by more than in high-quality ones.22 Finally, to know how the efficiency and complementarity aspects reshape the spatial distri- bution of infrastructure, Figure 9 shows the optimal allocation of electricity infrastructure in each location relative to the data for the Efficiency + Coordination (left panels) and the Fully Optimal (right panels) versions. As in Figure 2, the quality of the electricity infrastructure has a strong spatial component. The model predicts that the largest increments (not levels) are in the regions with relatively low quality of electricity. The two regions with the largest incre- ments are Macapá, and Manaus. Macapá is an economic hub of the state of Amapá, and many of the minerals and oil obtained in the interior of the state pass through this region. Manaus is the capital city of the state of Amazonas and the largest city in the north of Brazil. The city 21 To get these numbers, 1.26 = 19.75 − (7.52 + 10.97) while 2.26 = 22.01 − 19.75. 22 That is, the constant of the regression line in the Fully Optimal scenario is larger than in the Efficiency + Coordination scenario. 30 Figure 8: Density of Investments for Roads and Electricity has burgeoning rubber, petrochemical, and oil refining industries. With respect to roads, Figures 9c and 9d suggest that the priority of the planner is to connect locations along the eastern coast and to the north while also connecting the south to Brasilia. In the fully optimal scenario, the planner also connects Brasilia to Manaus in line with the increases in the quality of electricity suggested in Figures 9a and 9b. 5.3 Sources of the Welfare Gains We now decompose the welfare changes into their different components. In particular, from equation (10), welfare changes relative to the benchmark can be written as: ¯ ) = ∆ log(wi / Pi ) + ϑ ∆ log( Li ) − (1 − ν)∆ log(bi / Pi ) ∆ log(U (35) which allows us to decompose them into changes in real factor prices and changes in popula- tion. To do that, we divide each of the summands on the right-hand side of the equation (35) by the term on its other side. Table 8 shows the average values across locations from this exercise for different versions of the Fully Optimal solution. The first column gives the Fully Optimal scenario considered previously, and the “No TFP” and “No Trade Costs” columns consider the case in which the effect of the electricity and road quality investments are neutralized, respectively. More specif- ically, the “No TFP” case solves the problem assuming that ι d = 0 in all sectors, which implies that the TFPs do not react to improvements in the quality of electricity. In turn, the “No Trade 31 Figure 9: Optimal relative to Observed Quality of Infrastructure (a) Optimal Gi (Efficiency + Coordination) (b) Optimal Gi (Fully Optimal) (c) Optimal Ijk (Efficiency + Coordination) (d) Optimal Ijk (Fully Optimal) Note: Maps ( a) and (b) show the optimal Gi divided by the observed Gi for the Efficiency + Coordination and the Fully Optimal scenarios, respectively. Darker colors represent smaller outcomes. Maps (c) and (d) show the optimal road network relative to the data for the same counterfactuals. Green links denote links with improved infrastructure relative to the data. Red links show road segments with lower infrastructure relative to the data. More intense colors denote a larger size of the difference. 32 Costs” one assumes that the government allocates both infrastructure types, but actual trade costs remain as in the benchmark. Table 8: Welfare Decomposition Fully No No No TFP and Optimal TFP Trade Costs No Trade Costs Real Wages 97.11 100.98 44.27 208.57 Labor 8.07 4.95 48.67 −76.07 Real Electricity bill −5.18 −5.93 7.05 −32.50 Welfare Gain (%) 22.01 11.30 7.41 0.50 We see that most of the gains come from improvements in real wages. The exception is the third column. In the “No Trade Costs” case, trade costs remain the same, and then, goods prices and wages in real terms change by less. Labor reallocations (directly linked to the reaction in wages) are also a source of welfare improvement. Although smaller in absolute terms, there is also a contribution from the real price of electricity, negative in the first two columns and positive in the third one. This occurs because the reduction in the prices of goods is larger (smaller) than the electricity prices when trade costs fall (remain constant). Finally, the fourth row of the table says that the “No TFP” case achieves a welfare gain of 11.3%, while the “No Trade Costs” scenario increases welfare by 7.41%. This suggests that welfare gains are better transmitted through improvements in trade costs, which is in line with the bilateral nature of road investments and the increase in the budget directed to roads implied by the Fully Optimal scenario. 5.4 Infrastructure Complementarities and Structural Transformation Model simulations indicate the presence of complementarities of investing in both types of infrastructure at the same time. Now, we turn to the question of whether investment com- plementarities significantly affect the spatial allocation of sectoral economic activity. To do d) so, we regress the changes, relative to the data, in sector-d location-i employment shares ( Li predicted by the model on the predicted changes in infrastructure types, population, and the interaction between the two types of investment. In particular, we estimate the following regression: d ∆ Li ¯i + β 4 (∆ Gi × ∆ I = α + β 1 ∆ Li + β 2 ∆ Gi + β 3 ∆ I ¯i ) + S (lati , loni ) + υi (36) where ∆ xi ≡ log( xiM / xiD ), and xiM and xiD denote values of variable x predicted by the model and obtained in the data, respectively. The equation incorporates a function S (lati , loni ) of latitude and longitude that absorbs the spatial component of the regression and adopts the same specifications as in Section 2. 33 Table 9: Infrastructure Complementarities and Structural Change OLS GAM Agriculture Manufacturing Services Agriculture Manufacturing Services ∆ Li -0.011 -0.769*** -0.735*** -0.005 -0.694*** -0.491*** (0.052) (0.074) (0.071) (0.055) (0.071) (0.070) ∆ Gi -0.110*** 0.214*** 0.274*** -0.131*** 0.187*** 0.283*** (0.010) (0.015) (0.014) (0.012) (0.015) (0.015) ∆I ¯i -0.046*** 0.242*** 0.203*** -0.041*** 0.187*** 0.145*** (0.014) (0.020) (0.020) (0.015) (0.019) (0.019) ∆ Gi × ∆ I ¯i -0.065*** -0.057*** -0.070*** -0.066*** -0.050*** -0.046*** (0.009) (0.013) (0.013) (0.009) (0.012) (0.012) Num.Obs. 557 557 557 557 557 557 R2 0.433 0.569 0.535 0.475 0.677 0.652 Std.Errors IID IID IID S (lat, lon) 0.000 0.000 0.000 Robust standard errors in parenthesis. * p < 0.1, ** p < 0.05, *** p < 0.01. Panel OLS controls for latitude and longitude. Panel GAM shows semiparametric regressions with thin-plate spline in latitude and longitude, S (lat, lon) shows the approximate p-value of the thin-plate spline. We use the allocations of the Efficiency + Coordination counterfactual to isolate the ef- fect of complementarities since the fully optimal model has an additional effect through the ¯i , we de- changes in aggregate budgets. To ease the interpretation, before interacting Gi and I ¯i , respectively. mean them so that coefficients β 2 and β 3 denote the average effect of Gi and I We find that increases in the quality of electricity or roads are associated with reductions in the employment share in agriculture and increases in manufacturing and services. In fact, focusing on the GAM estimates, for a 1 percentage point increase in the quality of electricity (roads), the employment share in agriculture declines by 0.13 (0.04) percentage points while manufacturing and services employment shares increase by 0.19 and 0.28 (0.19 and 0.15) percentage points, respectively. This result is consistent with Szerman et al. (2022), who find strong gains along the intensive margin in Brazilian agriculture when electricity access is improved. The interacted term is always negative and significant in all sectors. For agriculture, this suggests that the complementarities in infrastructure investment help reduce even further the agricultural employment shares.23 For manufacturing and services, the estimated coefficient 23 Through the lens of our model, this complementarity acts through a labor push mechanism (Alvarez- Cuadrado and Poschke, 2011), as increased productivity in agriculture reduces the employment share in the primary sector. It is also consistent with Vanden Eynde and Wren-Lewis (2021), who find that roads and electric- ity are complements in dry season agricultural production and suggest the effect is through productivity, which is in line with our results. 34 in Table 9 implies that the complementarities tend to weaken the reallocation of labor towards these sectors induced by both infrastructure investments. 6 Political Preferences and Infrastructure Investment We have found evidence of misallocation of infrastructure investment. In this section, we focus on one possible source—increased political support— which has recently received substantial interest in the quantitative spatial economics literature (Glaeser and Ponzetto, 2018; Bordeu, 2023; Fajgelbaum et al., 2023). It is well established in the political economy literature that government preferences may diverge from the welfare maximization principle and may reflect the electoral incentives of the incumbent politicians. Furthermore, this is especially relevant for Brazil, where the importance of clientelist politics in securing local government transfers is well established (Brollo and Nannicini, 2012; Brollo et al., 2013, 2020). This section seeks to determine how significant these political effects are in the spatial misallocation of public infrastructure investment. ¯i ) Specifically, we regress the log changes between data and model in infrastructure (Gi or I on the change in votes between 2010 and 2002, controlling for the population change. To take into account the spatial structure of both votes (see Figure 10) and infrastructure, we control for latitude and longitude through S (lati , loni ). In particular, we estimate the equation XiData LiData log = α + ∆Vote10−02 + S (lati , loni ) + log + νi (37) XiModel LiModel where XiData denotes Gi or Ri in the data while XiModel denotes the model counterpart; ∆Vote10−02 denotes the difference in vote shares between 2010 and 2002 for the winner of the presidential elections. To control for the spatial structure, we first let S (lati , loni ) = lati + loni , and later, also be a thin-plate spline. Finally, νi denotes the error term. The Workers’ Party won the presidential elections in 2010, 2006, and 2002. Our model is calibrated to 2010, so we take the differences in votes between the 2010 and 2002 elections as a change in support for the Workers’ Party (the incumbent party). A shorter time interval may not allow infrastructure projects to be finalized. The share of people who voted for the incumbent party candidate, Luiz Inácio Lula da Silva, comes from the publicly available data provided by Tribunal Superior Eleitoral (Superior Electoral Court).24 Figure 10 shows the geo- graphical distribution of votes in the 2006 elections. The incumbent party is supported by the Northeast, North, and East regions, while the opposition candidate prevails in the Central and South regions. Table 10 gives the result of estimating equation (37). Models (1) and (4) correspond to the Fully Optimal allocations, models (2) and (5) correspond to the efficiency and coordination 24 The data can be accessed through their open data portal https://dadosabertos.tse.jus.br/. Numbers are at the municipality level, which we aggregate to the microregion level. 35 Figure 10: Percentage of people who voted for Luiz Inácio Lula da Silva (2006) scenarios, while models (3) and (6) correspond to the efficiency-only electricity and roads sce- narios, respectively. The estimates suggest that increased support for the winning candidate increases the quality of infrastructure in the data relative to the optimal allocation suggested by the model. In particular, if the vote share for the winner increased by 1 percentage point with respect to the 2002 elections, the quality of electricity in the data increases relative to the model by 0.035 − 0.048 percentage points, depending on the specification. The quality of roads increases by 0.016 − 0.019 percentage points depending on the specification. This suggests that the degree of misallocation measured as deviations from the optimal allocation given by the model is associated with increased support for the incumbent’s party. These re- sults align with the earlier reduced-form evidence (Brollo and Nannicini, 2012) that politically aligned regions receive larger discretionary transfers in preelection years in Brazil. These results also suggest that increased political support affects the misallocation of the quality of electricity more than the quality of roads. This could be due to the electricity sys- tem being fragmented, with most of the regional electricity consumption being sourced from nearby plants (Lipscomb et al., 2013), while the Brazilian highway system is administered, constructed, managed, and maintained by two central authorities (DNIT and the Ministry of Infrastructure). This could mean local governments have more authority in matters of quality of electricity than roads. 36 Table 10: Model with 2010 - 2002 Difference Gi ¯i I (1) (2) (3) (4) (5) (6) Panel A: OLS Estimates ∆Votes (2010-2002) 0.037*** 0.046*** 0.045*** 0.016*** 0.019*** 0.018*** (0.005) (0.006) (0.006) (0.002) (0.002) (0.002) Num.Obs. 557 557 557 557 557 557 R2 0.215 0.216 0.214 0.459 0.312 0.287 Panel B: GAM Estimates ∆Votes (2010-2002) 0.035*** 0.045*** 0.048*** 0.019*** 0.018*** 0.019*** (0.005) (0.006) (0.006) (0.002) (0.003) (0.003) Num.Obs. 557 557 557 557 557 557 R2 0.396 0.408 0.421 0.568 0.451 0.443 S (lat, lon) 0.000 0.000 0.000 0.000 0.000 0.000 Note: * p < 0.1, ** p < 0.05, *** p < 0.01. Columns (1) and (4) correspond to the Fully Optimal scenario, columns (2) and (5) to the Efficiency+Coordination scenario, column (3) to the Efficiency (Electricity) scenario, and column (6) to the Efficiency (Roads) scenario. All regressions control for the change in population. Panel A controls for latitude and longitude. Panel B shows semiparametric regressions with thin-plate spline in latitude and longitude, S (lat, lon) shows the approximate p- value of the thin-plate spline. 37 7 Conclusions This paper highlights the importance of complementary public infrastructure assets in the spatial distribution of economic activity, focusing specifically on the quality of the public roads and electrical grid networks. In doing so, it develops a multi-sector spatial equilibrium model with endogenous government investment in these two types of infrastructure and calibrates it to the Brazilian economy in 2010. A novel feature of the model is its ability to characterize spillovers from population and public infrastructure on location-specific productivities in the agriculture, industry, and services sectors. We have used the full structure of the model to estimate these spillovers by the simulated method of moments. The paper’s key insight is the evidence of significant distortions in investment in both infrastructure types and substantial gains from reallocating these investments. Owing to the existence of economic complementarities, the government can substantially increase aggre- gate welfare by jointly planning investment in both types of public goods. On the other hand, welfare gains diminish if fiscal restrictions or political preferences affect the infrastructure choices. All in all, our results underscore the significance of coordinated technical planning of welfare-enhancing infrastructure investments. These efforts can, however, be obscured by complex political economy reasons. There are many interesting, related issues that we have not addressed. They include the analysis of other sources of public capital misallocation, like the regional equalization policies, fiscal decentralization, the artificial creation of Brasilia, and deforestation prevention policies. 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Henning (2022): “A Dynamic Baseline Calibration Procedure for CGE models,” Computational Economics. 42 A Details on Generalized Additive Models Since we focus on the spatial dimension throughout the paper, it is important to account for the spatial structure of the data. The standard approach is to adjust standard errors for a spatial structure but, as Kelly et al. (2023) show, estimates differ significantly across methods. We follow Kelly et al. (2023) and estimate a Generalized Additive Model (GAM) with a thin-plate spline to optimally absorb the spatial structure. GAMs are flexible models that have an additive structure with a general form y i = f 1 ( X1 ) + f 2 ( X2 ) + · · · + f m ( X m ) where f j are different functional forms and X j are different set of explanatory variables. The type of regressions we estimate is yi = Xi β + S (lati , loni ) + ε i where Xi are a set of controls and S (lati , loni ) is an unknown function. Thin-plate spline smoothing estimates S (·) by minimizing the distance between the data and S (lati , loni ) pe- nalizing overfitting. B Households Indirect Utility: The FOCs with respect to Cn and EUn yield ν 1− ν λ Pn Cn = νCn EUn ηn (B.38) ν 1− ν λbn EUn = (1 − ν)Cn EUn ηn (B.39) Households earn labor income through wages wn and spend all their income on consumption or electricity services (i.e., wn = Pn Cn + bn EUn ). Then, equations (B.38) and (B.39) imply ν 1− ν λ ( Pn Cn + bn EUn ) = Cn EUn ηn Which can be used, along with (B.38) and (B.39), to get optimal demands wn λ Pn Cn = νλwn ⇒Cn = ν Pn wn λbn EUn = (1 − ν)λwn ⇒ EUn = (1 − ν) bn Substituting optimal demands into the utility function we get ¯ = ηn ν ν (1 − ν )1− ν wn U ν b 1− ν Pn n which yields equation (10) in the main text. 43 Price Indices FOC with respect to cdni ( j) is 1 −1 −σ ν −1 1− ν Cdn ρ cdni ( j) λ pdni ( j) = νCn EUn ηn ωd (B.40) Cn Cdn Using the ratio of (B.40) for varieties j and j′ , integrating, and taking the sum we get 1 1 ∑ pdni ( j)cdni ( j)dj = ∑ pdni ( j′ )σ cdni ( j′ ) pdni ( j)1−σ dj (B.41) i∈ N 0 i∈ N 0 From (B.40), integrating and summing across locations we also get 1 −1 Cdn ρ λ ∑ pdni ( j)cdni ( j)dj = ν −1 1− ν ωd νηn Cn EUn Cn Cdn i∈ N 0 1 Note that ∑i∈ N 0 pdni ( j)cdni ( j)dj is the total expenditure in sector d which we denote as Pdn Cdn . Substituting and simplifying −1 ν −1 1− ν Cdn ρ λ Pdn = ωd νηn Cn EUn (B.42) Cn which we can substitute into (B.40) to get σ Pdn Cdn = pdni ( j)σ cdni ( j) (B.43) Which directly yields (6) in the main text. Finally, we substitute last expression into (B.41) to obtain 1 1 1− σ Pdn = ∑ pdni ( j)1−σ dj i∈ N 0 which is equation (8) in the main text. To obtain the aggregate price index, we raise equation (B.42) to the power of 1 − ρ and sum across sectors to get 1 1− ρ ρ 1− ρ λ ∑ ωd Pdn ν −1 1− ν = νηn Cn EUn d∈ D ν E1−ν which then shows that using (B.42) we can also show that λ Pn Cn = νηn Cn Un 1 1− ρ ρ 1− ρ Pn = ∑ ωd Pdn d∈ D which is equation (9) in the main text. ν−1 E1−ν to get To get (7) in the main text, we can use (B.42) and substitute λ Pn = νηn Cn Un ρ ρ Pdn Cdn = ω ρ Pn Cn which, rearranging immediately yields (7). 44 C The Firm’s Cost Problem The total costs of a firm are given by wi ℓdi ( j) + bi edi ( j); (C.44) and firms minimize costs. Before minimizing costs, we can solve for electricity as a function of labor and output: 1 ydi ( j) 1− β di β di e( j) = ℓdi ( j) βdi −1 . (C.45) Adi Substituting (C.45) into the objective function of the cost minimization problem and taking the first order condition, the optimal choice of labor subject to a certain output level is given by: β di −1 ydi ( j) 1 − β di wi ℓdi ( j) = . (C.46) Adi β di bi We can substitute now (C.46) into (C.45) and get the optimal choice of electricity given an output level as: β di ydi ( j) 1 − β di wi edi ( j) = . (C.47) Adi β di bi Combining (C.46) and (C.47), we can get the following expression of the optimal electricity- to-labor ratio for all varieties within a sector-location pair (d, n): edn ( j) 1 − β dn wn = . (C.48) ℓdn ( j) β dn bn Aggregating across varieties 1 Ldn = ℓdn ( j)dj 0 1 Edn = edn ( j)dj 0 condition (C.48) can be expressed as β dn bn Edn = (1 − β dn )wn Ldn which is equation (13) in the main text. Substituting (C.46) and (C.47) into the total cost func- tion (C.44), writing it as a function of ydi ( j), and taking the first derivative yields the (free-on- board) price for a good produced in location i and sector d as: β 1− β di θdi wi di bi pdi ( j) = ; Adi where β di 1− β di 1 1 θdi = β di 1 − β di which is equation (14) in the main text. 45 D Propositions and Proofs D.1 Proof of Proposition 1 Proof. Note that given employment shares, total population, wages, and the total electricity supply, the electricity bill rate is given by (29). Then, we can re-write the system (34a)-(34b) as the type of systems analyzed by Allen et al. (2020). In their notation: ∏ xhihh ∑ Kijh ∏ γ ′ β ′ = x jhhh ′ h∈H j∈ N ′ h ∈H where H = 2, x1i = Pdi , x2i = Adi , β dj 1− β dj 1−σ Kij1 = ( Tij )1−σ θdj w j b j β di β 1− β 1− σ Kij2 = ωd Tji θdi wi di bi di ν(w j L j + b j Ej ) wi Ldi and the matrices Γ and B are given by 1−σ 0 0 σ−1 Γ= B= 0 1−σ σ−1 0 Allen et al. (2020) show that if the spectral radius of matrix BΓ−1 is lower than or equal to 1, the system has a unique (up to scale) solution. In our case, the spectral radius is equal to one. We can also show that the theorem does not apply if ρ ∈ (0, 1) by first writing the system as β dj 1− β dj 1−σ 1− σ Pdi = ∑ (Tij )1−σ θdj w j b j Aσ dj −1 (D.49a) j∈ N ρ −1 β di 1− β di 1− σ σ−ρ ¯ j w j Lϑ η j ν 1− ρ A1 −σ ∑ ωd ¯ ρ β di = Tji θdi wi di bi Pdj −ν U ν ν(w j L j + b j Ej ) wi Ldi j∈ N b1 j (D.49b) Matrix Γ remains the same, but matrix B becomes 0 σ−1 B= σ−ρ 0 which implies that the spectral radius of the matrix (λ(BΓ−1 )) is given by σ−ρ λ= σ−1 which is strictly larger than one for any ρ < 1 and σ > 1. This implies that the system has multiple solutions that are up-to-scale different depending on Kijh . 46 D.2 Proposition 3 ¯ di for a simplified version of the We now show the effect of an improvement in productivity A model. Since the full three-sector general equilibrium model does not allow for closed-form solutions, we restrict our model to comprise only two sectors and assume that only one region receives the productivity improvement. Moreover, we do not allow for labor mobility and con- sider the region that receives the improvement as a small open economy. These assumptions imply Proposition 3, which shows how relative factor prices are affected by changes in the exogenous part of sectoral productivity. Intuitively, if sector d is more intensive in electricity use ( β d < β s ), an improvement in the productivity of sector d induces an increase in bi /wi . Proposition 3. Consider a simplified version of the model with two sectors d and s and no labor mobility. Suppose that region i is a small open economy and receives an improvement in productivity in sector d. Let 1 − β d Ldi 1 − β s Lsi νb E ϱi = 1 − ν + + = i i β d Li β s Li wi L i then (σ − 1)ldi (1 − ldi ) (1 − β d ) β s − β d (1 − β s ) ∂ log(ϱi ) ϱi βd βs = ∂ log( Adi ) (σ − 1)ldi (1 − ldi ) (1 − β d ) β s − β d (1 − β s ) 1+ (βs − βd ) ϱi βd βs which is positive if and only if β s > β d , negative if β s < β d , and zero if β s = β d . Proof. Suppose the two sectors are d and s and let ldi ≡ Ldi / Li . The market clearing condition implies ∂ldi ∂l 0= + si ∂ Adi ∂ Adi which can be rewritten as: ∂ log(lsi ) l ∂ log(ldi ) = − di (D.50) ∂ log( Adi ) lsi ∂ log( Adi ) 1− β si Denote ϱi ≡ 1 − ν + ∑s∈ D β si lsi then ∂ log(ϱi ) A 1 − β si ∂lsi ∂ log( Adi ) = di ϱi ∑ β si ∂ Adi s∈ D Simplifying the previous expression and using equation (D.50), it can be shown that ∂ log(ϱi ) l ∂ log(ldi ) 1 − β di 1 − β si = di − (D.51) ∂ log( Adi ) ϱi ∂ log( Adi ) β di β si Since there is no population mobility, the ratio of the total population to electricity supply is constant, so we denote hi ≡ Ei / Li and write the free on board price (14) in terms of hi and ϱi using equation (29) as − β di − β di θ bi θ b ρi pdi = di bi = di i . Adi wi Adi ν hi 47 We can use the previous expression to obtain the sectoral market clearing condition as: 1− σ − β di (σ−1) θdi bi ρi wi Ldi = β di ωd Adi ν hi ∑ Tji 1− σ σ −1 Pdj ν(w j L j + b j Ej ). j∈ N We assume that region i is a small open economy. This implies that it cannot affect sectoral price indices of other regions j. So, for region i, Pdj is exogenous. Moreover, the fact we do not allow for labor mobility implies that we can treat the sum in the previous expression as a con- stant exogenously given to location i; let us denote this by Γdi . Using the previous expression, the small open economy assumption, and aggregating across sectors, we can express sectoral employment shares in location i as σ −1 λdi Adi (ϱi / hi ) β di Γdi ldi = σ −1 (D.52) ∑s∈ D λsi Asi (ϱi / hi ) β si Γsi 1− σ where λdi is a constant term equal to ωd β di θdi ν β di . We can now use equation (D.52) to show after some algebra that ∂ log(ldi ) ∂ log(ϱi ) = (σ − 1)(1 − ldi ) 1 + ( β di − β si ) (D.53) ∂ log( Adi ) ∂ log( Adi ) Substituting (D.53) into (D.51) obtains (σ − 1)ldi (1 − ldi ) (1 − β di ) β si − β di (1 − β si ) ∂ log(ϱi ) ϱi β di β si = (D.54) ∂ log( Adi ) (σ − 1)ldi (1 − ldi ) (1 − β di ) β si − β di (1 − β si ) 1+ ( β si − β di ) ϱi β di β si which is the expression shown in Proposition 3. The sign of the derivative is going to be determined by whether β di ≷ β si . Note that the denominator is always positive and larger than 1. Then, the sign of the numerator is determined by (1 − β di ) β si − β di (1 − β si ) > 0 ⇔ β si > β di . So, if β si > β di , the numerator is positive and so is the derivative. D.3 Proposition 4 We now show the effect of an increase in the quality of electricity on relative factor prices. The intuition is very similar to the one behind Proposition 3: relative intensities in factor use matter for the direction of the effect, and so do the relative spillovers. In addition, since all sectoral productivities increase after an improvement in the quality of electricity, which sectoral productivity increases the most is also important. 48 Proposition 4. Consider a simplified version of the model with two sectors d and s and no labor mobility. Suppose that region i is a small open economy and receives an improvement in the quality of electricity Gi . Let 1 − β d Ldi 1 − β s Lsi νb E ϱi = 1 − ν + + = i i β d Li β s Li wi L i then (σ − 1)ldi (1 − ldi ) (1 − β d ) β s − β d (1 − β s ) (ι d − ι s ) ∂ log(ϱi ) ϱi βd βs = ∂ log( Gi ) (σ − 1)ldi (1 − ldi ) (1 − β d ) β s − β d (1 − β s ) 1+ (βs − βd ) ϱi βd βs The sign of the previous derivative depends on the parameters as follows: 1. If β s > β d and ι d > ι s the derivative is positive. 2. If β s > β d and ι d < ι s the derivative is negative. 3. If β s < β d and ι d > ι s the derivative is negative. 4. If β s < β d and ι d < ι s the derivative is positive. Proof. The proof follows the same structure as the proof of Proposition 3 in Appendix D.2 except that now equation (D.53) becomes equation (D.55) and (D.51) becomes (D.56). ∂ log(ldi ) ∂ log(ϱi ) = (σ − 1)(1 − ldi ) ι d − ι s + ( β di − β si ) (D.55) ∂ log( Gi ) ∂ log( Gi ) ∂ log(ϱi ) l ∂ log(ldi ) 1 − β di 1 − β si = di − (D.56) ∂ log( Gi ) ϱi ∂ log( Gi ) β di β si Substituting (D.55) into (D.56) yields the derivative of the proposition. Note that in terms of β di ≷ β si the effects are the same, but the sign of the numerator also depends on ι d ≷ ι s , which controls which sectoral productivity increases the most. Note that by substituting the derivative of the proposition into (D.55) we directly obtain how the employment share in sector d and region i changes after an increase in the quality of electricity in region i. In particular, ∂ log(ldi ) (σ − 1)(1 − ldi )(ι d − ι s ) = ∂ log( Gi ) (σ − 1)ldi (1 − ldi ) (1 − β di ) β si − β di (1 − β si ) 1+ ϱi β di β si 49 E Construction of the Road Quality Measure Each discretized road segment that links two neighboring locations in our model results from aggregating a subset of actual segments. To perform this aggregation, we first use the DNIT traffic data to construct a congestion index in direction l ∈ { a, d}, Ql (s), for each actual road segment s as     1.0 if x (s) = A   0.8 if x (s) = B       0.6 if x (s) = C  l Q (s) =    0.4 if x (s) = D   if x (s) = E     0.2    0.1 if x (s) = F where the ranking x (s) ranges between A (low congestion) and F (high congestion) depending on the capacity of the road. Then, we define the congestion index in an actual segment s as Q(s) = 0.5 × [ Q a (s) + Qd (s)]. The idea is that higher congestion of traffic increases effective distance. Once we have Q(s), we create the following network weight, W (s): d(s) (1− P(s)) W (s) = χP × K ( s ) (1− χ K ) ; Q(s) where d(s) is the length of the road segment; χ P and χK and road user-cost mark-ups for gravel relative to paved roads and secondary roads relative to other national roads, respectively; P(s) is the road pavement status; K (s) is the road category (available only for federal roads). Following Fajgelbaum and Schaal (2020) and Combes et al. (2005), we set χ P = 1.35, χK = 1.07, and P(s) = 2 for paved roads and zero otherwise. Following the DNIT classification, we distinguish between the following categories for federal roads: double-lane paved road, single-lane paved road in the process of conversion to double-lane road, single-lane paved road, single-lane unpaved road in the process of conver- sion to paved road, unpaved gravel road, and unpaved dirt road. We assume that improving road coverage by each level reduces the cost of passing through this road by 10%. Having this information, we define the quality of a road connecting neighbors j and k as Ijk = ∑ o jk (s)C (s) s∈P ( j,k) where  d(s) s ∈ P ( j, k )  ∑ d(s′ )    o jk (s) = s′ ∈P ( j,k)   s ̸ ∈ P ( j, k )  0 50 ˜ (s) and P ( j, k ) denotes the least cost path through a network obtained using the weights W as the costs of going through segment s. F Computational Details of the Spillovers Estimation To draw the amenity and productivity shocks, we rely on an inverse transform sampling us- ing a normal copula to take into account that the model will generate negative correlations between them. We proceed in three steps for each simulation s. 1. Define the empirical set Z = ( Z1 , Z2 , Z3 , Z4 ) ∼ N (0, C ) where C is the variance- covariance matrix, and Z1 to Z4 are N × 1 vectors of agriculture, manufacturing and services productivities and amenities, respectively. 2. Define Vk = Φ( Zk ) for k = {1, 2, 3, 4} where Φ(·) is the standard normal CDF. Now Vk ∼ U (0, 1), but they are dependent. 3. Finally, draw the shocks performing inverse transform sampling from the empirical −1 CDFs as variables Yk = Fk (Vk ), where Yk and Vk are N × 1 vectors. To estimate the spillover parameters, we solve the following optimization problem: min(m ˜ , Θ) − m( x ))′ W (m ˆ (x ˜ , Θ) − m( x )) ˆ (x Θ ˜ , Θ) across S simulations. This is a com- ˆ (x This problem involves computing the moments m putationally expensive function since it first needs to solve for S spatial equilibria, compute the required moments, and then compute the average. Even more, the fact that this function is a result of simulating the model with a set of random sequences of productivities, amenities, and infrastructure makes it extremely hard to rely on methods that use derivatives. Because of this, we rely on surrogate optimization. That is, we use a probabilistic surrogate model to infer a probability distribution over the true objective function. Surrogate optimization methods are global optimization routines that rely on approximat- ing the true function with another simpler function.25 The basic procedure is split in three ˜ , Θi ); (ii ) build a surrogate model ˆ (x steps: (i ) sample points to evaluate the true function m by approximating the function around those points; (iii ) perform surrogate optimization. We follow Urquhart et al. (2020) to perform the whole optimization routine. To get our sample points, we use Latin Hypercube Sampling (LHS). This sampling plan divides each pa- rameter into M equally sized intervals where the parameter takes unique values. To ensure 25 Chapters 14-16 in Kochenderfer and Wheeler (2019) provide an introduction to surrogate models, probabilis- tic surrogate models, and surrogate optimization. A recent example in economics is Ziesmer et al. (2022) who use a surrogate model to replace their CGE model and reduce the computational time. 51 that we maximize the space explored in the sampling, Urquhart et al. (2020) provide an LHS routine that optimizes the choice of these points by minimizing P P 1 min ∑ ∑ L2 , p =1 q = p +1 pq where L2 pq is the L2−norm between two sample points. To build the surrogate model, we use Gaussian radial basis functions. In general, radial basis functions depend on a set of center points, which are the sampled points, and take as argument the Euclidean distance between the interpolated points and the sampled points. Urquhart et al. (2020) provide a method in which the width factor that determines the size of the radial basis function changes across points. Finally, the optimization is done with a differential evolution algorithm. These methods do not have stopping criteria. Typically, the user sets a maximum number of iterations or a maximum amount of time. So, while within the computational budget, when new points are found through the optimization routine, we sample new points and repeat the whole process. We allow for 150 iterations in total and limit the space of the function to (0, 0.3] for the six spillover parameters. After the surrogate optimization routine is done, we re-optimize using an adaptive dif- ferential evolution algorithm in the full function to improve upon the surrogate optimization solution. We limit the function space to be within ±25% of the previous solution found. We simulate 400 economies of 100 regions each and use 45 points for the LHS sampling. F.1 Assessing Identification of Moments One potential problem is that the number of simulations S with N locations might not be sufficient to evaluate the moments properly. To partially assess this, we simulate M batches of S × N simulations with different shocks and compute the moments and confidence intervals. This will tell us the variation across the M batches of simulations of each moment conditional on a set of parameter values exogenously posed. Table F.11 shows the mean and the 95% confidence interval for each moment across the M simulations. It shows that the confidence intervals are small for all moments, which suggests all moments are computed with relatively high accuracy or that the number of simulations is enough to identify them. For this experiment, we set M = 150 with S = 400 and N = 100 as in the estimation. G Algorithm to Find the Equilibrium We solve for the decentralized equilibrium using the following algorithm. 1. We start by guessing Ξ ≡ { L0 0 0 i , wi , bi }i ∈ N and pick tolerance level ε. 52 Table F.11: Variation of Moments Across Batches Moment Mean 95% CI 1 -0.028 -0.034 -0.021 2 -0.128 -0.136 -0.121 3 0.619 0.614 0.624 4 -0.013 -0.022 -0.004 5 0.010 0.001 0.019 6 0.031 0.021 0.043 7 0.002 -0.007 0.010 8 -0.026 -0.034 -0.017 9 -0.033 -0.042 -0.024 10 0.360 0.355 0.364 11 0.059 0.057 0.061 12 0.060 0.058 0.062 13 0.065 0.063 0.067 14 0.456 0.451 0.461 15 0.476 0.471 0.482 16 0.027 0.025 0.028 17 0.044 0.042 0.046 18 0.041 0.039 0.043 19 0.412 0.407 0.417 53 2. Given Ξ, obtain {wi 1} i ∈ N from solving (23) and recover { bi }i ∈ N from (29). Iterate on 1 , b1 } these equations until we find {wi i i ∈ N that satisfy the market clearing conditions for a guess of { L0 i }i ∈ N . 1 , b1 } 3. With our new {wi 1 i i ∈ N solve for { Li }i ∈ N using (26). 4. Update Ξ ≡ { L1 1 1 1 0 i , wi , bi }i ∈ N and repeat from Step 2 until ∥ Li − Li ∥ < ε. H Fixed Labor Mobility We solve a version of the model with fixed labor. The new assumption implies that utility is not equalized across regions. This exercise is useful to assess the impact of migration in the model. We find that the free mobility assumption is not as restrictive as could be thought. Ta- ble H.12 shows the correlation between infrastructure investments in the free mobility and the fixed labor mobility models for the Fully Optimal and the Efficiency + Coordination scenarios. The correlations are very large, indicating that the planner’s decisions are not substantially changed because of this assumption. Table H.13, in turn, compares the predicted welfare and income gains. They are again similar; for example, the largest difference in welfare is 2 per- centage points. Table H.12: Fixed vs Free Labor Mobility Allocations Fully Optimal Efficiency + Coordination Quality of Electricity 0.995 0.988 Quality of Roads 0.925 0.965 Table H.13: Fixed vs Free Labor Mobility Allocations Free Mobility Fixed Mobility Fully Optimal Efficiency + Coordination Fully Optimal Efficiency + Coordination Income (median) 21.15 14.88 20.20 12.04 Income (mean) 21.75 15.85 21.51 13.47 Welfare 22.01 19.75 21.34 18.40 I Supplemental Figures 54 (a) 1980 (b) 1991 (c) 2000 (d) 2010 Figure I.11: Spatial Evolution of Electricity Access (1980-2010) Note: Data from IPUMS-I. The maps show the percentage of people with access to electricity for each municipality over time. 55 (a) Agriculture (b) Manufacturing (c) Services Figure I.12: Empirical CDFs 56