metadata: authors: 0: author: Pérez-Sebastián,Fidel
1: author: Serrano Quintero,Rafael
2: author: Steinbuks,Jevgenijs
count: Brazil
docna: 0: docna: Spatial Misallocation of Complementary
            Infrastructure Investment : Evidence from Brazil
docty: Policy Research Working Paper
abstracts: cdata!: How does the misallocation of
            complementary public capital affect the spatial organization
            of economic activity? To answer this question, this paper
            endogenizes the government's decision to invest in the
            transport and electricity networks. A novel multi-sector
            quantitative spatial equilibrium model incorporates the
            quality of the electric power and the road transportation
            infrastructure networks, which determine sectoral
            productivities and trade costs. Simulation results for the
            Brazilian economy point to significant welfare gains from
            reallocating infrastructure investment. Spatial and fiscal
            complementarities in heterogeneous infrastructure provision
            determine a sizeable part of those gains. Misallocation of
            both infrastructure investments is positively associated
            with local political support for the incumbent authority.
display_title: Spatial Misallocation of
            Complementary Infrastructure Investment : Evidence from Brazil
listing_relative_url: /research/2023/12/34218347/spatial-misallocation-complementary-infrastructure-investment-evidence-brazil
topicv3: Infrastructure and Growth m1326244 1083,Public Investment Management Index
            m1326537 2067,Roads and Highways m1327398 168,Electricity Access m1503452 3922
publishtoextweb_dt: 2023-12-18T00:00:00Z
dois: 10.1596/1813-9450-10650
origu: Off of Sr VP Dev Econ/Chief Econ (DECVP)
no_of_pages: 58
author_keys: 0: author_key: NA
1: author_key: 1900186
2: author_key: 666646
model_extractions: 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 capital affect the spatial organization of economic activity? To answer this question, this paper endogenizes the government’s decision to invest in the transport and electricity networks. A novel multi-sector quantitative spatial equilibrium model incorporates the quality of the electric power and the road transportation infrastructure networks, which determine sectoral productivities and trade costs. Simulation results for the Brazilian economy point to significant welfare gains from reallocating infrastructure investment. Spatial and fiscal complementarities in heterogeneous infrastructure provision determine a sizeable part of those gains. Misallocation of both infrastructure investments is positively associated with local political support for the incumbent authority.

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**<sup>∗</sup>

Fidel Pérez-Sebastián<sup>1</sup> , Rafael Serrano-Quintero<sup>2</sup> , and Jevgenijs Steinbuks<sup>3</sup>

> 1Department of Economics (FAE), University of Alicante

> 2Department of Economics, University of Barcelona

> 3Office of the Chief Economist, Infrastructure Practice, The World Bank

**_Keywords:_** Brazil; electrical grid; infrastructure investment; public roads; spatial misallocation

**_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 1<sup>st</sup> University of Barcelona Macro Workshop, the Monthly Seminars of the Armenian Economic Association, the 1<sup>st</sup> International Workshop on Economic Growth and Macroeconomic Dynamics, the 16<sup>th</sup> North American Meeting of the Urban Economics Association, the 11<sup>th</sup> European Meeting of the Urban Economics Association, the Internal Seminar of the University of Barcelona, the 4<sup>th</sup> Workshop of the Spanish Macroeconomics Network, the 2<sup>nd</sup> LAUrban Annual Meeting, and the World Bank InfraXchange Workshop for their helpful comments 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 constrained governments improve economic welfare by investing across different infrastructure types? How large are the economic gains of coordinating infrastructure investments in different 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 investments 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.<sup>1</sup> The three sectors produce output using labor (which is freely mobile) and electricity as inputs.<sup>2</sup> 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., FisherVanden 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 activity, public roads, and electricity supply quality in the year 2010. First, we ask how the spatial

> 1The 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.

> 2Banerjee et al. (2020) argue that mobility of production factors is key for determining the benefits of infrastructure 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 investment. 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 increase 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 transformation 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, doing 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 alignment 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 higherquality 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 (relative 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 presidential 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 SubSaharan 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 optimal investments and inefficiencies in the road networks of European countries. Santamaria (2020) assesses how governments can adjust transportation infrastructure to unexpected economic changes. Both studies, however, exclusively focus on the transportation infrastructure.<sup>3</sup> 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 electricity access come from the Integrated Public Use Microdata Series (IPUMS-I, 2019), comprising 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_ ).

> 3There 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 taxpayers. 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 mediumsized 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
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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)|
|log( <sup> ̄</sup>_Ii_)|0.023*|-0.006|-0.017***|0.016***|-0.005***|-0.010***|
||(0.003)|(0.003)|(0.000)|(0.003)|(0.002)|(0.002)|
|log(_Gi_)_×_log( <sup> ̄</sup>_Ii_)|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|
|R<sup>2</sup>|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_ (lat _i_ , lon _i_ ) that controls for the possible spatial correlation of the errors. In particular, 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_ (lat _i_ , lon _i_ ); see Appendix A for details.

Table 1 shows that locations with larger shares of employment in manufacturing and services tend to have larger populations. Better quality of electricity is associated with smaller employment shares in agriculture and larger employment shares in manufacturing and services. 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 quality of electricity (roads) raises the marginal effect of improving the quality of roads (electricity) 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 complementarities 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 Complementary 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 locationspecific 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 = 1,802<sup>_1_</sup>|**Small**, N = 814<sup>_1_</sup>|**Medium**, N = 738<sup>_1_</sup>|**Large**, N = 250<sup>_1_</sup>|
|---|---|---|---|---|
|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 _Lj_ . Consumer preferences favor horizontally differentiated varieties of products from agriculture, manufacturing, and services, denoted with subindices _a_ , _m_ , and _s_ , respectively. Households also consume electricity services ( _EUj_ ) at a price _bj_ for home use and enjoy amenities ( _ηj_ ) in the location they live. The amenities depend on spillovers from all the inhabitants in the location controlled by parameter _θ_ (i.e., _ηj_ = _η_ ̄ _jL_<sup>_θ_</sup> _j_<sup>). The consumer’s utility function can be written as:</sup> where and 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 consumed in _j_ are denoted by _cdji_ and sold at price _pdji_ . Hence, the consumer’s budget constraint is given by It sets total income from wages ( _wj_ ) equal to total expenditures in electricity and in products coming from all sectors, locations, and product varieties.<sup>6</sup> Consumers maximize objective function (2) subject to (3) to (5). From the household problem, we can show the usual within-sector and across-sector expenditure shares: and where

> 6The 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 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_ ) on electricity. Consequently, the indirect utility function, _U_<sup> ̄</sup> , is given by 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 following production function: That is, firms employ labor, _Ldi_ , and electricity, _Edi_ , with a unitary elasticity of substitution between the inputs. The labor share _βd_ is assumed to be sector-specific, reflecting differences in the intensity of energy use.<sup>7</sup> The sector-location specific productivities ( _Adi_ ) have an exogenous component ( _A_<sup> ̄</sup> _di_ ) and a production externality that depends on the number of workers in the location and a measure of electrical-grid infrastructure quality ( _Gi_ ). More specifically, The parameters _αd_ and _ιd_ 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 _ιd >_ 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).

> 7Caliendo 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
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Using the definition of _Pj_ in equation (9) and substituting in (17), we can derive the aggregate : price index for location _j_ Finally, to derive the consumer’s expenditure share that goes to sector _d_ in region _j_ ( _X_<sup>_c_</sup> _dj_<sup>),</sup> we can combine condition (7), equation (15) and the fact that _PjCj_ = ∑ _d∈D PdjCdj_ to obtain: Note that expenditure shares within sectors across locations ( _πdji_ ) and expenditure shares across sectors within locations ( _Xdj_ ) are functions of _{wj_ , _bj_ , _Lj_ , _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_<sup> ̄</sup> ):<sup>8</sup>

> 8More 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, _bi_<sup>_f_and</sup><sup>_b_</sup> _i_<sup>_g_,wouldequalthemarginalcostofelectricityservice,with</sup><sup>_b_</sup> _i_<sup>_g<b f_</sup> _i_<sup>duetoeconomiesofscale.Let</sup> Λ _i ∈_ [0, 1] be the fraction of time when the grid electricity is available. Then, consumers and firms would end up paying

14
Regarding the labor market, market clearing requires that the labor demand equals the labor supply: 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 On the other hand, the Cobb-Douglas structure of the production function implies that a constant share of revenues _βd_ 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: 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:

# **3.6 Transport and Electricity Networks**

Suppose that two adjacent locations _i_ and _j_ are joined by a road segment; and following Fajgelbaum 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: 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: 15
_T_ where **I** _knji_<sup>takes value one when the edge connecting</sup><sup>_n_and</sup><sup>_k_is in the least cost path between</sup> _i_ and _j_ and zero otherwise. Parameter _δkn_<sup>_T_capturespurelygeographiccomponentssuchas</sup> 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_<sup> ̄</sup> _, intrinsic sector-location productivities {A_<sup> ̄</sup> _di}d∈D_ , _i∈N, intrinsic amenities η_  ̄ _i, electricity supply at each location {Ei}i∈N, a matrix of road-quality investments and shortest path indicators {Iij_ , **I** _ij_<sup>_Tnk}∀i_,</sup><sup>_j_,</sup><sup>_k_,</sup><sup>_n∈N, and a vector_</sup> _of electricity quality {Gi}i∈N, a competitive spatial equilibrium is a utility level U_<sup> ̄</sup> _, 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 level _U_<sup> ̄</sup> ), we can use (10) to get: which yields the following expression for the population shares:<sup>9</sup>

> 9Changes 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: 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 _ξi_<sup>_G_capturestheunitaryvariable</sup> electricity transmission costs to a given location (including technical losses and congestion costs) and maintaining grid electricity infrastructure.<sup>10</sup> Second, the unitary variable cost of road access, _ξ_<sup>_T_</sup> _jk_<sup>, is also related to its maintenance.</sup> 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_

_subject to:_ _1. Free labor mobility condition_ (10) _:_ _2. Electricity and labor markets clearing conditions_ (20) _and_ (21) _:_ _3. Goods market clearing_ (24) _:_ _4. Transport costs_ (25) _:_ 10In 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) _:_ _6. Previous underlying network investments:_ _In these constraints, Pdj is defined as in_ (8) _, Pj as in_ (9) _, κdi is given by_ _and bi comes from equation_ (13) _,_

# **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 literature 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;<sup>11</sup> 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 20022003 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.<sup>12</sup>

The production function parameter _βd_ is obtained as one minus the share of energy consumption 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.<sup>13</sup>

> 11Notice 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 <u>κ</u> If κ is the expenditure share in housing, _θ_ = _−_ 1 _−_ κ<sup>.</sup>

> 13Santamaria (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 _vj_ ( _xj_ , _yj_ ) _∈V_ such that

**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 registration 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 microregion 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.<sup>14</sup>

> 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].

> 14To compute to which region each centroid belongs to, we use the function `inpoly2` that substantially reduces computation times (Engwirda, 2021).

19
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| I

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where _distjk_ is the distance in kilometers along the shortest path on the actual road network, ( _distjk >_ 50 _km_ ) takes on 1 if the inequality holds and on zero otherwise, and _ξ_ 0<sup>_T_is calibrated</sup> so that we match the share of the total government budget allocated to roads.

**Calibration of Transport Costs.** Observed transport costs between two neighboring regions _i_ and _j_ are a function of the road quality investment _Iij_ (derived from the DNIT geospatial roads database) and the geographical-barriers parameter _δ_<sup>_T_The road quality along a given</sup> _ij_<sup>.</sup> discretized segment is computed as the average speed at which products can travel (see Appendix E for details). To calibrate _δ_<sup>_T_inturn,weusethedistanceinkilometerstakinginto</sup> _ij_<sup>,</sup> 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 investing in quality of electricity, _ξi_<sup>_G_,we assume that</sup><sup>_ξ_</sup> _i_<sup>_G_can be expressed as</sup><sup>_ξ_</sup> _i_<sup>_G_=</sup><sup>_ξ_ ̄</sup><sup>_G_/</sup><sup>_δ_</sup> _i_<sup>_G_,</sup> where _ξ_<sup> ̄</sup><sup>_G_</sup> is calibrated to match the share of investment in electricity (71.88%),<sup>15</sup> and following Lipscomb et al. (2013), _δi_<sup>_G_is given by</sup>

_δ_<sup>_G_Φ (</sup><sup>_β_1 log(Max Flow) +</sup><sup>_β_2Average Slope in River +</sup><sup>_β_3Max. Slope in River +</sup><sup>_β_4Forest Cover)</sup> _i_<sup>=</sup> 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.<sup>16</sup> 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 calculated based on two collective indicators of continuity provided by ANEEL: the Equivalent

> 15This number comes from the World Bank PPI database: `ppi.worldbank.org` . Other sources (Paiva, 2010; BNDES, 2014) suggest very similar values.

> 16World 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
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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 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 specifically, 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 Simulated Method of Moments (SMM). To do so, we need to obtain the composite productivities and amenities employing the methodology explained previously. This provides empirical distributions for _Adi_ in each industry and for _η_  ̄ _j_ .<sup>17</sup>

We have six parameters to estimate, _{αd_ , _ιd}_ , 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 estimated distributions and compute the decentralized spatial equilibrium. Second, with these draws, we compute average moments from the simulated economies _m_ ˆ ( ̃ _x_ , Θ), i.e., 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 Θ<sup>ˆ</sup> _SMM_ that solves

> 17Figure 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 _i_ th moment 1/ _mi_ ( _x_ )<sup>2</sup> .

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 productivities. 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 10<sup>th</sup> , 25<sup>th</sup> , 50<sup>th</sup> , 75<sup>th</sup> , and 90<sup>th</sup> 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 spatial 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
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data in 2010.<sup>19</sup> 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|
|Efciency + Coordination|34.84|21.32|43.84|71.88|
|Efciency-Roads|32.84|21.46|45.70|71.88|
|Efciency-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|
|Efciency + Coordination|0.79|0.23|
|Efciency-Roads|1.00|0.22|
|Efciency-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.

> 19This 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 predicted 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 shows the degree centrality ( _I_<sup> ̄</sup> _i_ ) for the Fully Optimal scenario against the data.<sup>20</sup> We see that 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 <!-- Start of picture text --> 35<br>-3<br>30<br>-4<br>25<br>-5<br>20<br>-6<br>15<br>-7<br>10<br>-8<br>5<br>-9<br>5 10 15 20 25 30 35 -9 -8 -7 -6 -5 -4 -3<br>Data Data<br>(a) Quality of Roads I ̄ i (b) Log of Population<br>Model Model<br><!-- End of picture text --> <!-- Start of picture text --> 70<br>60<br>50<br>40<br>30<br>20<br>10<br>10 20 30 40 50 60 70<br>Data<br>(c) Aggregate Price Index<br>Model<br><!-- End of picture text --> The equilibrium investment allocations induce movements in the population. The relationship between the population predicted by the model and the data is shown in Figure 7b. 20 Degree centrality _I_ ̄ _i_ measures the quality of the roads connected to location _i_ as _I_ ̄ _i_ = ∑ _j∈Ni Iij_ .

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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 concentration 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

||Fundam|entals and Investment in El|ectricity|
|---|---|---|---|
||_Corr_(log( <sup> ̄</sup>_Aai_), log(_Gi_))|_Corr_(log( <sup> ̄</sup>_Ami_), log(_Gi_))|_Corr_(log( <sup> ̄</sup>_Asi_), log(_Gi_))|
|Data|_−_0.05|0.32|0.20|
|Model (Fully Optimal)|0.20|0.59|0.59|
|Model (Efciency + Coordination)|0.25|0.61|0.63|
|Model (Efciency-Electricity)|0.30|0.60|0.62|
|Model (Efciency-Roads)|_−_0.05|0.32|0.20|
||Funda|mentals and Investment in|Roads|
||_Corr_(log( <sup> ̄</sup>_Aai_), log( <sup> ̄</sup>_Ii_))|_Corr_(log( <sup> ̄</sup>_Ami_), log( <sup> ̄</sup>_Ii_))|_Corr_(log( <sup> ̄</sup>_Asi_), log( <sup> ̄</sup>_Ii_))|
|Data|0.06|0.11|0.05|
|Model (Fully Optimal)|0.09|0.18|0.11|
|Model (Efciency + Coordination)|0.08|0.27|0.21|
|Model (Efciency-Electricity)|0.06|0.11|0.05|
|Model (Efciency-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 component 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 productivities. 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 productivities (log( _A_<sup> ̄</sup> _di_ )) and the log of the electricity infrastructure quality (log( _Gi_ )). In all cases, the allocations predicted by the model have a much larger correlation with the productivities 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 modelsolution 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 complementarity 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 counterfactual 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

||Fully Optimal|Efciency<br>and Coordination|Efciency<br>(Electricity)|Efciency<br>(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.<sup>21</sup> 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.<sup>22</sup> Finally, to know how the efficiency and complementarity aspects reshape the spatial distribution 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 increments 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.

> 22That is, the constant of the regression line in the Fully Optimal scenario is larger than in the Efficiency + Coordination scenario.

30
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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 complementarities significantly affect the spatial allocation of sectoral economic activity. To do so, we regress the changes, relative to the data, in sector- _d_ location- _i_ employment shares ( _Li_<sup>_d_)</sup> 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: where ∆ _xi ≡_ log( _xi_<sup>_M_/</sup><sup>_x_</sup> _i_<sup>_D_), and</sup><sup>_x_</sup> _i_<sup>_M_and</sup><sup>_x_</sup> _i_<sup>_D_denote values of variable</sup><sup>_x_predicted by the model</sup> and obtained in the data, respectively. The equation incorporates a function _S_ (lat _i_ , lon _i_ ) of latitude and longitude that absorbs the spatial component of the regression and adopts the same specifications as in Section 2.

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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)|
|∆<sup> ̄</sup>_Ii_|-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 ×_∆<sup> ̄</sup>_Ii_|-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 effect of complementarities since the fully optimal model has an additional effect through the changes in aggregate budgets. To ease the interpretation, before interacting _Gi_ and _I_<sup> ̄</sup> _i_ , we demean them so that coefficients _β_ 2 and _β_ 3 denote the average effect of _Gi_ and _I_<sup> ̄</sup> _i_ , respectively.

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.<sup>23</sup> For manufacturing and services, the estimated coefficient

> 23Through the lens of our model, this complementarity acts through a labor push mechanism (AlvarezCuadrado 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 electricity 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.

Specifically, we regress the log changes between data and model in infrastructure ( _Gi_ or _I_<sup> ̄</sup> _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_ (lat _i_ , lon _i_ ). In particular, we estimate the equation where _Xi_<sup>_Data_</sup> denotes _Gi_ or _Ri_ in the data while _Xi_<sup>_Model_</sup> 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_ (lat _i_ , lon _i_ ) = lat _i_ + lon _i_ , 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).<sup>24</sup> Figure 10 shows the geographical 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.

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<!-- Start of picture text --> 4 pt az eee 4 Votes for Winner (%)<br>re, 30<br>Se<br>€ 0ef wee‘wtatTM |a:20<br>x) 1?<br>a #<br>)<br><!-- End of picture text -->
Table 10: Model with 2010 - 2002 Difference

|||_Gi_||| ̄_Ii_||
|---|---|---|---|---|---|---|
||(1)|(2)|(3)|(4)|(5)|(6)|
|Panel A: OLS Estima|tes||||||
|∆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|
|R<sup>2</sup>|0.215|0.216|0.214|0.459|0.312|0.287|
|Panel B: GAM Estim|ates||||||
|∆Votes (2010-2002)|0.035***<br>(0.005)|0.045***<br>(0.006)|0.048***<br>(0.006)|0.019***<br>(0.002)|0.018***<br>(0.003)|0.019***<br>(0.003)|
|Num.Obs.|557|557|557|557|557|557|
|R<sup>2</sup>|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.

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# **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 aggregate 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. All of them deserve further investigation. Another important problem is understanding the impact of optimal infrastructure investment on spatial inequality. The incorporation of inputoutput linkages into our model could be another interesting avenue. We leave them to future research.

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# **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 where _fj_ are different functional forms and _Xj_ are different set of explanatory variables. The type of regressions we estimate is where _Xi_ are a set of controls and _S_ (lat _i_ , lon _i_ ) is an unknown function. Thin-plate spline smoothing estimates _S_ ( _·_ ) by minimizing the distance between the data and _S_ (lat _i_ , lon _i_ ) penalizing overfitting.

# **B Households**

**Indirect Utility:** The FOCs with respect to _Cn_ and _EUn_ yield

Households earn labor income through wages _wn_ and spend all their income on consumption or electricity services (i.e., _wn_ = _PnCn_ + _bnEUn_ ). Then, equations (B.38) and (B.39) imply Which can be used, along with (B.38) and (B.39), to get optimal demands Substituting optimal demands into the utility function we get which yields equation (10) in the main text.

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~~-)~~ ~~<u>(-)</u>~~ | | ~~(—~~ <u>)</u> ~~(—~~ <u>)</u> pf | ~~| |~~
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# **D.2 Proposition 3**

We now show the effect of an improvement in productivity _A_<sup> ̄</sup> _di_ for a simplified version of the 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 consider 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 ( _βd < βs_ ), 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_

_then_ _which is positive if and only if βs > βd, negative if βs < βd, and zero if βs_ = _βd._ _Proof._ Suppose the two sectors are _d_ and _s_ and let _ldi ≡ Ldi_ / _Li_ . The market clearing condition implies which can be rewritten as: Simplifying the previous expression and using equation (D.50), it can be shown that 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 47
We can use the previous expression to obtain the sectoral market clearing condition as: 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 constant exogenously given to location _i_ ; let us denote this by Γ _di_ . Using the previous expression, the small open economy assumption, and aggregating across sectors, we can express sectoral employment shares in location _i_ as where _λdi_ is a constant term equal to _ωdβdi_ � _θdiν_<sup>_βdi_�1</sup><sup>_−σ_</sup> . We can now use equation (D.52) to show after some algebra that Substituting (D.53) into (D.51) obtains which is the expression shown in Proposition 3. The sign of the derivative is going to be determined by whether _βdi_ ≷ _βsi_ . Note that the denominator is always positive and larger than 1. Then, the sign of the numerator is determined by So, if _βsi > βdi_ , 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.

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**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_

_then_ _The sign of the previous derivative depends on the parameters as follows:_ _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).

Substituting (D.55) into (D.56) yields the derivative of the proposition. Note that in terms of _βdi_ ≷ _βsi_ the effects are the same, but the sign of the numerator also depends on _ιd_ ≷ _ιs_ , 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, 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}_ , _Q_<sup>_l_</sup> ( _s_ ), for each actual road segment _s_ as 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_<sup>_a_</sup> ( _s_ ) + _Q_<sup>_d_</sup> ( _s_ )]. The idea is that higher congestion of traffic increases effective distance.

Once we have _Q_ ( _s_ ), we create the following network weight, _W_<sup>�</sup> ( _s_ ): where _d_ ( _s_ ) is the length of the road segment; _χP_ 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 _χP_ = 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 conversion 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 where 50
and _P_ ( _j_ , _k_ ) denotes the least cost path through a network obtained using the weights _W_<sup> ̃</sup> ( _s_ ) 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 using 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_ = ( _Z_ 1, _Z_ 2, _Z_ 3, _Z_ 4) ∼ _N_ (0, _C_ ) where _C_ is the variancecovariance matrix, and _Z_ 1 to _Z_ 4 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 CDFs as variables _Yk_ = _Fk_<sup>_−_1(</sup><sup>_Vk_), where</sup><sup>_Yk_and</sup><sup>_Vk_are</sup><sup>_N ×_1 vectors.</sup> To estimate the spillover parameters, we solve the following optimization problem: This problem involves computing the moments _m_ ˆ ( ̃ _x_ , Θ) across _S_ simulations. This is a computationally 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 approximating the true function with another simpler function.<sup>25</sup> The basic procedure is split in three steps: ( _i_ ) sample points to evaluate the true function _m_ ˆ ( ̃ _x_ , Θ _i_ ); ( _ii_ ) build a surrogate model 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 parameter into _M_ equally sized intervals where the parameter takes unique values. To ensure 25Chapters 14-16 in Kochenderfer and Wheeler (2019) provide an introduction to surrogate models, probabilistic 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 where _L_<sup>2</sup> _pq_<sup>is the</sup><sup>_L_2</sup><sup>_−_norm between two sample points. To build the surrogate model, we use</sup> 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 differential 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 _≡{L_<sup>0</sup> _i_<sup>,</sup><sup>_w_</sup> _i_<sup>0,</sup><sup>_b_</sup> _i_<sup>0</sup><sup>_}i∈N_and pick tolerance level</sup><sup>_ε_.</sup>

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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|

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2. Given Ξ, obtain _{wi_<sup>1</sup><sup>_}i∈N_fromsolving(23)andrecover</sup><sup>_{bi}i∈N_from(29).Iterateon</sup> these equations until we find _{wi_<sup>1,</sup><sup>_b_</sup> _i_<sup>1</sup><sup>_}i∈N_that satisfy the market clearing conditions for</sup> a guess of _{L_<sup>0</sup> _i_<sup>_}i∈N_.</sup> 3. With our new _{wi_<sup>1,</sup><sup>_b_</sup> _i_<sup>1</sup><sup>_}i∈N_solve for</sup><sup>_{L_1</sup> _i_<sup>_}i∈N_using (26).</sup> 4. Update Ξ _≡{L_<sup>1</sup> _i_<sup>,</sup><sup>_w_</sup> _i_<sup>1,</sup><sup>_b_</sup> _i_<sup>1</sup><sup>_}i∈N_and repeat from Step 2 until</sup><sup>_∥L_1</sup> _i_<sup>_−L_0</sup> _i_<sup>_∥<ε_.</sup>

# **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. Table 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 percentage points.

Table H.12: Fixed vs Free Labor Mobility Allocations

||Fully Optimal|Efciency + Coordination|
|---|---|---|
|Quality of Electricity|0.995|0.988|
|Quality of Roads|0.925|0.965|

Table H.13: Fixed vs Free Labor Mobility Allocations

||F|ree Mobility|Fi|xed Mobility|
|---|---|---|---|---|
||Fully Optimal|Efciency + Coordination|Fully Optimal|Efciency + 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**

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