metadata: authors: 0: author: Krantz,Sebastian Martin
count: Africa
docna: 0: docna: Optimal Investments in Africa’s Road Network
docty: Policy Research Working Paper
abstracts: cdata!: This paper characterizes economically
            optimal investments in Africa&#x2019;s road network in
            partial and general equilibrium &#x2014;based on a
            detailed topography of the network, road con-struction
            costs, frictions in cross-border trading, and economic
            geography. Drawing from data on 144 million
            trans-continental routes, it first assesses local and global
            network efficiency and market access. It then derives a
            large network connecting 447 cities and 52 ports along the
            fastest routes, devises an algorithm to propose new links,
            analyzes the quality of existing links, and estimates
            link-level construction/upgrading costs. Subsequently, it
            computes market-access-maximizing investments in partial
            equilibrium and conducts cost-benefit analysis for
            individual links and several investment packages. Using a
            spatial economic model and global optimization over the
            space of networks, it finally elicits welfare-maximizing
            investments in spatial equilibrium. Findings imply that
            cross-border frictions and trade elasticities signifi-cantly
            shape optimal road investments. Reducing frictions yields
            the greatest benefits, followed by road upgrades and new
            construction. Sequencing matters, as reduced frictions
            generally increase investment returns. Returns to upgrading
            key links are large, even under frictions.
display_title: Optimal Investments in Africa’s Road Network
listing_relative_url: /research/2024/09/34386132/optimal-investments-africa’s-road-network
topicv3: Transport and Economic Geography m1326235 1813,Transport m1326661 135,Trade Facilitation and Logistics m1327174 772,Transport Infrastructure m1327632 1803
publishtoextweb_dt: 2024-09-04T00:00:00Z
dois: 10.1596/1813-9450-10893
origu: Off of Sr VP Dev Econ/Chief Econ (DECVP)
no_of_pages: 88
author_keys: 0: author_key: 2417327
model_extractions: Policy Research Working Paper 10893 Optimal Investments in Africa’s Road Network _Sebastian Krantz_ Africa Region Office of the Chief Economist September 2024
Policy Research Working Paper 10893

# **Abstract**

This paper characterizes economically optimal investments in Africa’s road network in partial and general equilibrium —based on a detailed topography of the network, road con-struction costs, frictions in cross-border trading, and economic geography. Drawing from data on 144 million trans-continental routes, it first assesses local and global network efficiency and market access. It then derives a large network connecting 447 cities and 52 ports along the fastest routes, devises an algorithm to propose new links, analyzes the quality of existing links, and estimates link-level construction/upgrading costs. Subsequently, it computes market-access-maximizing investments in partial equilibrium and conducts cost-benefit analysis for individual links and several investment packages. Using a spatial economic model and global optimization over the space of networks, it finally elicits welfare-maximizing investments in spatial equilibrium. Findings imply that cross-border frictions and trade elasticities signifi-cantly shape optimal road investments. Reducing frictions yields the greatest benefits, followed by road upgrades and new construction. Sequencing matters, as reduced frictions generally increase investment returns. Returns to upgrading key links are large, even under frictions.

This paper is a product of the Office of the Chief Economist, Africa Region and the Kiel Institute. 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 author may be contacted at skrantz@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
# **Optimal Investments in Africa’s Road Network**

Sebastian Krantz<sup>*</sup> Originally published in the Policy Research Working Paper Series in _September 2024_ . This version is updated in _November 2025_ . To obtain the originally published version, please email prwp@worldbank.org.

**Keywords:** African roads, spatially optimal investments, big data, PE and GE analysis **JEL Classification:** O18; R42; R10; O10

## **1 Introduction**

For several decades, development economists and policymakers have highlighted insufficient or inefficient transport infrastructure, coupled with border frictions and high trading costs, as a major obstacle to African economic development and regional integration. A World Bank report from 2010 states that Sub-Saharan Africa (SSA) has 31 paved roads _km_ per 100 _km_<sup>2</sup> of land, compared to 134 _km_ in other low-income countries, estimates the annual infrastructure gap at US $ 93 billion and urges SSA countries to spend 1 percent of GDP on roads (Foster & Brice ̃noGarmendia, 2010). This is echoed by the African Development Report 2010, attesting a financing gap at 5 percent of GDP to overhaul the infrastructure sector (ADB, 2010). More extensive and detailed work by Teravaninthorn & Raballand (2009) and Lamarque & Nugent (2022) unveils the nature of transportation and trucking in Africa, the determinants of transport costs and frictions, and discusses potentials and policy options for development along specific transport corridors. Yet, still little is known about the economic optimality of such (large-scale) transport investments.

This paper thus presents a comprehensive, data-driven, and quantitatively rigorous assessment of Africa’s road network and optimal investments into it at the trans-African scale. It (1) conducts a detailed empirical stocktake of Africa’s present road network; (2) characterizes global economic optimality in network investments from a market access and welfare perspective; (3) investigates the effects of cross-border frictions on optimal spatial allocations; and (4) presents cost-benefit analyses and evidence on the macroeconomic feasibility of large scale (optimal) investments. This agenda mirrors questions of international policymakers who, in theory, can invest in roads anywhere on the continent and may ask: (1) is it still important to invest in African roads?; (2) where exactly should we invest in African roads?; (3) are investments sensible with/without trade facilitation?; and (4) are they macroeconomically feasible? The paper engages this ambitious agenda by building realistic economic topographies informed by routing engines, geospatial big data, and surveys, and analyzing investments on them using tools from graph theory and spatial economic modeling.

> * Kiel Institute for the World Economy and World Bank

> _E-mail:_ sebastian.krantz@kielinstitut.de or skrantz@worldbank.org

Latest version at sebastiankrantz.com/research, data and code at github.com/SebKrantz/OptimalAfricanRoads
Several significant contributions have already been made in economic literature. Atkin & Donaldson (2015) show that the effect of log distance on trade costs within Ethiopia or Nigeria is 4-5 times larger than in the US, and _≤_ 30% of this effect is explained by the availability and quality of roads. Using 10 years of disaggregated monthly price data, Porteous (2019) estimates a dynamic model of agricultural storage and trade comprising 6 grains traded across 230 regional markets in 42 SSA countries, connected in a network with 413 links to 30 international ports. He finds median trade costs 5x higher than elsewhere in the world. Reducing trade costs to the world average would yield a 46% reduction in grain prices and a welfare gain equivalent to 2.17% of GDP. He further shows that 88% of the welfare gain can be achieved by lowering trade costs through ports and links representing just 18% of the trade network, supporting corridor-based approaches.

Graff (2024) studies spatial inefficiency in African national road networks using a regular 0.5<sup>_◦_</sup> grid-network parameterized with Open Street Map (OSM) routes between adjacent cells. Employing data on population, nightlights (productivity) and ruggedness (Nunn & Puga, 2012), and the quantitative framework of Fajgelbaum & Schaal (2020), he generates ideal national networks, and estimates that Africa would gain 1.3% of total welfare from better organizing its national road systems (with national welfare gains ranging between 6.6% for Somalia and 0.5% for South Africa), and 0.8% from a 10% optimal expansion. He also documents that colonial infrastructure projects skewed trade networks towards a sub-optimal equilibrium and that regional favoritism and inefficient aid provision hinder optimal network investments (Graff, 2024). The latter is consistent with findings by Jedwab & Moradi (2016); Jedwab et al. (2017), Burgess et al. (2015), and Bonfatti et al. (2019). Similarly, Gorton & Ianchovichina (2021) analyze trade network inefficiencies and optimal expansions for Latin America closely following Fajgelbaum & Schaal (2020). They find average welfare gains of 1.6% from optimal reallocation and 1.6%/1% from a 50%/10% optimal expansion of national road networks. They also consider transnational networks in the MERCOSUR and Andean Community trade blocs. Disregarding borders, they find that optimal expansions better connect major cities across countries at welfare gains of 1.5%-1.9%. Throughout, they find that optimal network investments reduce spatial wealth disparities.

An emerging policy literature evaluates the effects of planned road interventions in Africa. Notably, Fontagn ́e et al. (2023) model the effects of planned trade (AfCFTA) and transport (PIDA) integration and find that AfCFTA alone would raise exports by 3.4% and GDP by 0.6%, whereas concurrent completion of PIDA priority road projects up to 2040 would raise exports by 11.5% and GDP by 2%. The novel analysis incorporates data on border frictions and trade through ports. Further similar works investigate West-Africa (Lebrand, 2023) and the Horn of Africa (Herrera Dappe & Lebrand, 2023) regions. Lebrand (2023) finds that upgrading the DakarLagos road corridor yields benefits 3x the estimated costs, and 6x with reduced border delays. Herrera Dappe & Lebrand (2023) find a 1% welfare gain from road investments along 4 corridors, which is raised to 4.3% if border delays are reduced by 50%, and to 7.8% with added electrification.

Many further works such as Jedwab & Storeygard (2022), Peng & Chen (2021), Abbasi et al. (2022), Fiorini et al. (2021), and Moneke (2020) study the economic effects of past road interventions in Africa. The latter two find that road network expansion in Ethiopia enhanced the impact of trade liberalization/electrification on local firm productivity/industrialization, respectively. A vast further literature studies the effects of road interventions in other world regions, particularly in China (Faber, 2014; Baum-Snow et al., 2020) and India (Asher & Novosad, 2020; Alder, 2016).

This paper does not examine planned or past interventions but instead characterizes optimal investments.<sup>1</sup> Apart from the partial equilibrium simulations by Porteous (2019) on a heuristic agricultural trade network, and the general equilibrium work by Graff (2024) examining hypothetical optimal reshuffling and 10% expansion of national road networks on a regular 0.5<sup>_◦_</sup> grid parameterized in a way that has limited resemblance to optimal routing (see Section 4), the literature has not yet (to my knowledge) seen a detailed quantitative attempt to compute what should optimally be done to improve or extend Africa’s present road network. This paper attempts to fill this void.

Starting from a precise continental grid of 12,016 cells with centroids exactly 50.28km apart, I

> 1Eyeballing the results of this paper against PIDA investments (Fontagn ́e et al., 2023) suggests that they resemble optimal general equilibrium trans-African road network investments under (unrealistically) low trade elasticities.

2
obtain 144 million OSM routes connecting all points and compute distance-weighted local measures of network quality and continental market access (MA). Using data on trading across borders, which I translate into road distance/travel time, I characterize how cross-border trade frictions impede cell-MA. A trade across the median African border is equivalent to _∼_ 1000km of national roads or 3.2 road days. My results imply that particularly cells across the border from major markets such as Nigeria suffer _≤_ 80% MA losses due to frictions.

In the second part of the paper, I identify 447 large cities with populations above 100,000 and 52 international ports with a quarterly outflow above 100,000 TEU and obtain routes between them. I process these routes into a realistic representation of the African transport network comprising 1379 nodes and 2344 edges representing 315,000km of road segments. The network connects 444 million people or 86% of the African urban population. I parameterize the network with fresh routing information along all links and develop an algorithm to find 481 new links (104,000km), yielding route efficiency gains of _≥_ 50%. Using this network graph, I simulate partial equilibrium MA returns from extending and/or upgrading the entire network or individual links. I find that continental MA, denominated in travel time, can be improved by _≤_ 78% through eliminating crossborder frictions. Upgrading the entire network to allow travel speeds of 100km/h yields a 42% frictionless MA gain and a 27% MA gain under current frictions. MA gains from new roads are lower and also decrease under frictions. Spatially, including frictions favors investments in national links, significantly reducing the value of transnational and remote links. I also find sizeable differences in the marginal values and cost-benefit ratios of links and propose three investment packages containing high-return links with and without border frictions. These packages cost between 17 and 61 billion 2015 USD and generate MA gains between 20% and 38%. They are macroeconomically feasible if they can raise the aggregate economic growth rate by _≤_ 0.54%.

The final part of the paper moves this to a general equilibrium setting following Fajgelbaum & Schaal (2020) and Graff (2024). I let cities trade with each other and with port cities, computing optimal investments into the full continental scale transport network. To simulate optimal investments in trans-African roads, I then reduce the network to links connecting the 47 largest (port-)cities. On both networks, I let welfare maximizing social planners spend budgets covering _∼_ 1/3 of all work—15 and 35 billion USD’15, respectively. I simulate cases with/without frictions, increasing returns to infrastructure, and inequality aversion in the planner’s objective.

The $ 35B planner on the full network optimally connects large cities with ports and surrounding cities. Investments are redistributional as in Gorton & Ianchovichina (2021). Under increasing returns to infrastructure, this planner upgrades many more roads and expands into remote regions, yielding higher and more equitable welfare gains. Similarly, the trans-African $ 15B planner primarily upgrades road segments within different populated regions, enhancing connectivity between major cities and with ports. An inequality-averse planner also upgrades an entire corridor crossing southern central Africa from Luanda/Kinshasa to Mombasa/Dar es Salaam. Lowering the elasticity of substitution between different goods to a value of 1.5 creates incentives to also build corridors spanning northern central Africa, better connecting Rwanda/Bukavu and Khartoum to Western Africa. Under current border frictions, investments in connections through Congo and into Nigeria are reduced. Prioritizing the central African infrastructure gap thus requires inequality-averse planners, unrealistic trans-African trade elasticities, and/or efforts to reduce border frictions.

Methodologically, in this paper, I develop a rigorously quantitative approach to characterize economically optimal investments into large road networks. I demonstrate the use of geospatial big data and routing engines to algorithmically create large and accurate network graphs, including potential new roads, and simulate investments on them via established quantitative frameworks. I am also the first to account for cross-border frictions and trade through ports to study their implications for optimal spatial allocations. Detailed estimation of road construction/upgrading costs facilitates interpretation and enables link-level cost-benefit analysis. A corollary of the work is an easily accessible Julia implementation<sup>2</sup> of Fajgelbaum & Schaal (2020)’s MATLAB toolbox.<sup>3</sup>

> 2https://github.com/OptimalTransportNetworks/OptimalTransportNetworks.jl

> 3https://github.com/OptimalTransportNetworks/OptimalTransportNetworkToolbox

3
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# **2.1 Routing Data**

Plotting the road network, while informative about its extent, does not permit a rigorous assessment of its efficiency. Routing engines<sup>4</sup> provide more precise information on how cars can use the network to get from A to B. In my case, A and B could be anywhere on the continent. Two approaches invite themselves to choose points: (1) using a uniform spatial grid and taking the cell centroids, or (2) selecting the centers of important cities/places. In this paper, I employ both approaches, but for spatial analysis, the grid approach is more suitable. Hence, I begin with it.

Accurate spatial analysis at the continental scale warrants using an equally spaced grid. Discrete Global Grid Systems (DGGS) provide these via hierarchical tessellation of cells partitioning the globe. Sahr et al. (2003) propose the Icosahedral Snyder Equal Area Aperture 3 Hexagon (ISEA3H) as a good general-purpose Geodesic DGGS. The grid is implemented as part of the DGGRID C++ library (Sahr, 2022), accessed via the ’dggridR’ R package (Barnes, 2020). To limit complexity, I choose a resolution 9 ISEA3H grid, with a cell area of 2591.4km<sup>2</sup> and a centroid spacing of 50.28km. Partitioning the African mainland and Madagascar with this grid yields 12,016 cells.

Figure 3: Grid Cell Centroids and OSRM Route Starting and End Points Grid Centroids (Inaccessible Cells in Grey) Actua ~~l~~ Route Starting/Destination Points _<u>Notes:</u>_ <u>Figure shows centroids of 50.28km ISEA3H grid</u> (LHS) <u>and the nearest road segment to each centroid (RHS).</u> To compute a large number of routes, I configure a server with the Open Source Routing Machine (OSRM)<sup>5</sup> and a copy of the Africa OSM from the 5th of June 2023, and compute fastest car routes between all cell centroids. This amounts to 12,016<sup>2</sup> _−_ 12,016 = 144,372,240 routes, for which I obtain both the road distance in _m_ and the estimated travel time in minutes. OSRM ’snaps’ cell centroids to the nearest road, yielding divergent route start/endpoints. Figure 3 shows the outcome of this process. For some cells in the Sahara, the Congo Basin, and Madagascar, colored in light grey on the left-hand side (LHS), OSRM cannot find the nearest road and returns ’NA’. For many further cells, the nearest road snapped to is outside of the cell, as shown on the RHS.

I clean the data by first mapping inaccessible cells to the nearest route start/endpoints. I then follow Graff (2024) in adding the spherical centroid-start/end distance to all routes. This increases the length of routes by 0.53% on average. I also adjust travel time estimates by assuming a travel speed of 10km/h on these straight passages, increasing travel time by 3.4% on average.<sup>6</sup>

> 4Large computer programs used for navigating (routing) through digitized transport networks.

> 5A high-performance open routing engine in C++ optimized for OSM. Website: https://project-osrm.org/.

> 6In this regard I differ from Graff (2024), which assumes walking speed of 4km/h. My main reason is that the centroid of such cells is unlikely of economic significance, and assuming a travel speed of 4km/h increases travel time by 8.5% on average, whereas assuming 10km/h only increases travel time by 3.4% on average. Furthermore, since most cells where the distance between the cell centroid and the route stating/endpoints is large are in the Sahara, I believe that motorized transport with specialized vehicles (4x4) is possible.

6
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route start/end points using population weights.<sup>8</sup> To approximate this, I compute a global weighted average across routes with weights equal to the product of start/end cell populations divided by their spherical distance.<sup>9</sup> This yields NS of 64.2, RE of 0.68, and TE of 43.9, reflecting greater efficiency on longer routes (Wolfram, 2021). But even the most favorable RE estimate of 0.68 is far below Wolfram (2021)’s estimates of 0.843/0.767 for US/European road networks.

## **2.3 Market Access**

Whereas route and time efficiency are important from a transportation standpoint, the spatial economics literature such as Donaldson & Hornbeck (2016), Jedwab & Storeygard (2022) or Peng & Chen (2021) has used market access (MA), typically defined as an inverse-distance weighted sum of population, or a proxy therefore such as built-up in the case of Peng & Chen (2021), to evaluate economic gains from transport investments. The MA measure for cell _i_ is where _Lj_ is a measure of population and _τ_ a measure of distance, travel time, or transport cost, weighted by an elasticity _θ_ measuring how trade volumes fall as distance/time/costs increase. The value of _θ_ is contentious, for example Jedwab & Storeygard (2022) use _θ_ = 3 _._ 8 whereas Peng & Chen (2021) use _θ_ = 1. In both cases, _τij_ measures travel time.

To appropriately measure continental MA implied by the road network, I incorporate spatial differences in purchasing power into _Lj_ . Kummu et al. (2018) provide a spatial estimate of GDP in 2015 USD PPP terms by distributing national GDP estimates from the World Development Indicators at 5 arc-min (0.0833 degrees or 9.3km at the equator) resolution using subnational value-added estimates from Gennaioli et al. (2013) and population from the HYDE 3.2 database.

Using Eq. 4, I compute MA maps using both measures of economic mass ( _Lj_ ), and trade elasticities of _θ_ = 1 (Peng & Chen, 2021) or _θ_ = 3 _._ 8 (Jedwab & Storeygard, 2022). As Eq. 4 excludes cell _i_ itself, the centers of several large cities have a lower MA measure than adjacent cells. To remedy this, I add _i_ ’s GDP/wealth to the MA measure using a distance _δij_ = 25km/ _κi_ , where _κi_ is cell _i_ ’s RE/TE measure from Figure 4. Figure 5 shows the resulting maps with _σ_ = 1 and Appendix Figure A4 with _σ_ = 3 _._ 8.

Evidently, _θ_ = 3 _._ 8 yields a much stronger concentration of MA around major cities and a high correlation of the MA measure with cell GDP. Both _θ_ values elicit northern Egypt, Nigeria, and, to a lesser extent, greater Johannesburg as high MA areas. Cairo has the greatest MA.

# **3 Barriers to Cross-Border Trade**

Africa has ’thick borders,’ and any evaluation of continental roads should at least be mindful of them. Trade barriers can broadly be distinguished into traditional trade tariffs and regulatory obstacles and frictions at the border. This paper will concentrate on the latter as tariffs and other non-tariff barriers tend to be product- and quantity-specific. But, for the sake of completeness, I begin with a brief overview of tariffs applied by African countries in international comparison.

## **3.1 Trade Tariffs**

The World Development Indicators provide aggregate estimates of the extensive and intensive margins of trade tariffs applied by each country. Table 2 summarizes these indicators for different country groups. While SSA countries only apply specific (non-ad-valorem) rates on 0.1% of tariff lines, 33.9% of African tariffs exceed 15%. When weighted by the product import shares

> 8Concretely, Wolfram (2021) computes the route efficiency of the US road network using OSRM-generated routes between 2000 population-weighted random points in the US and finds a median route efficiency of 0.843 with a standard deviation of 0.045 across 4 million routes.

> 9Following Newton’s Law of Gravity as as operationalized in the empirical trade literature (Head & Mayer, 2014), which finds a distance elasticity around _θ_ = _−_ 1. Inter-city migration gravity models such as Zipf (1946) or Poot et al. (2016) also suggest that migration varies according to the distance rather than the squared distance.

8
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# **3.3 Border Frictions Adjusted Routes**

It is difficult to take tariffs and NTMs into account when evaluating transport networks and market access because their effects depend on the value, type, and quantity of products being traded. In addition, AfCFTA has set clear aims to eliminate most inner-African tariffs in the foreseeable future. Regulatory and border frictions on the other hand are more uniformly applicable.

A critical question then is how the estimates from Table 3 can be sensibly added to road distance and travel time estimates. Surely time spent preparing documents or waiting at the border is different from driving time. To better understand the relative cost of these times, I collect data on domestic transport costs from the 2019 DB survey for a sample of 23 (mostly landlocked) African economies where exporting/importing involved domestic transportation of 100km or more. This information is not standardized across countries and thus not published in the DB indicators. For each country surveyed, the distance in km to the border, the time in hours to the border, and the cost in USD of exporting/importing a representative product are recorded.

To estimate the average cost of an hour spent on the road, I regress the cost of the trip on the travel time in hours while controlling for the log of the velocity and a dummy indicating whether the transport was import related.<sup>12</sup> I do the same for the distance traveled. Since I am more interested in converting times into each other rather than converting cost to time, I also regress the cost of border and documentary compliance from Table 3 on the respective time required using data for all 53 available African economies. Table 4 reports the results.

Table 4: Relative Cost of Exporting/Importing Times

|Cost in USD:|Tran|sport|Border|Documentary|
|---|---|---|---|---|
|Model:|(1)|(2)|(3)|(4)|
|Constant|-711.6<sup>_∗∗_</sup>|276.7|217.8<sup>_∗∗∗_</sup>|51.83<sup>_∗_</sup>|
||(285.9)|(317.4)|(65.28)|(29.63)|
|Travel Time (Hours)|15.79<sup>_∗∗∗_</sup><br>(2.074)||||
|Travel Distance (Km)||1.560<sup>_∗∗∗_</sup><br>(0.2987)|||
|Border Time (Hours)|||3.988<sup>_∗∗∗_</sup><br>(0.6781)||
|Documentary Time (Hours)||||1.655<sup>_∗∗∗_</sup><br>(0.3954)|
|Log Velocity (Km/Hour)|346.3<sup>_∗∗∗_</sup><br>(85.84)|-43.95<br>(102.9)|||
|Import Dummy|106.6|107.7|-44.92|78.76<sup>_∗∗∗_</sup>|
||(133.3)|(135.9)|(66.38)|(25.79)|
|Observations|46|46|106|106|
|R<sup>2</sup>|0.448|0.415|0.471|0.388|
|Adjusted R<sup>2</sup>|0.408|0.373|0.461|0.376|

_Heteroskedasticity-robust standard-errors in parentheses. Signif. Codes: ***: 0.01, **: 0.05, *: 0.1_ Evidently, time spent on the road, valued at 15.8 USD per hour in the average African country, is more costly than time spent at the border (4 $ /h) or filing documents (1.7 $ /h). The estimates imply that in the average African country, road time is 15 _._ 79 _/_ 3 _._ 988 = 3 _._ 959 _≈_ 4 times more costly than border time, and 15 _._ 79 _/_ 1 _._ 655 = 9 _._ 540785 _≈_ 10 times more costly than documentary time. To create the NTB-adjusted travel time matrix, I divide border compliance time by 4 and documentary compliance time by 10 before and add them to the OSRM estimates. Using the coefficient on distance indicating an average cost of 1.6 $ /km, I also obtain an adjusted road distance matrix by dividing documentary and border compliance costs by 1.6. These estimates are

> 12I also test the interaction of travel time and the import dummy, but the term is insignificant; thus export and import time do not differ in terms of cost.

11
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in MA shrinks to 1.13%-1.19% in all cases, implying that with a high trade elasticity, cross-border frictions become largely irrelevant. Evidence from inter-city migration (Zipf, 1946; Poot et al., 2016), international trade (Head & Mayer, 2014), and by Peng & Chen (2021), suggests that a lower elasticity is more plausible.

Cell-level MA losses due to border frictions range between -6.6% and -81.1% in GDP/min terms with _θ_ = 1. Figure 8 shows their spatial distribution. Large parts of central Africa incur 50-80% time- and 20-50% distance-denominated losses. Cells across the border from major markets also incur very large losses. Under _θ_ = 3 _._ 8, losses are large in some cells close to borders but negligible in most populated cells. Appendix Figure A6 shows the changes under _σ_ = 3 _._ 8.

<!-- Start of picture text --> Figure 8: Market Access Loss (%) from Border Frictions under θ = 1<br><!-- End of picture text --> <!-- Start of picture text --> Travel Time Road Distance<br>0<br>−10<br>−20<br>−30<br>−40<br>−50<br>−60<br>−70<br>−80<br><!-- End of picture text --> _Notes:_ Figure shows local MA %-loss from frictions (using Eq. 4 where _Lj_ is cell-GDP from Kummu et al. (2018)).

In summary, assuming _θ ≪_ 3 _._ 8, many regions could gain significant MA from lower border frictions. In the following sections, I show that these frictions also reduce the benefits of (transAfrican) road investments. Thus, their reduction should be a top policy priority. To simulate such investments, however, I first need to create suitable graph representations of Africa’s road network.

# **4 Optimal Network Representations and Extensions**

The central aim of this paper is to provide quantitative evidence on which roads should be enhanced or built. This requires accurate representations of the road network via graphs to simulate the effects of network investments. Most research creates these graphs in an ad-hoc fashion by either connecting adjacent cells in a gridded representation and parameterizing them by the connectivity along these links, or by choosing the largest roads and their intersections as nodes/edges of the graph. This section instead uses a routing engine (OSRM) to create graphs that optimally represent the road network as interpreted by the engine, implying that the optimal network graph produces road distance/travel time estimates that closely approximate those of the routing engine. I then extend this graph with high-value new links connecting previously unconnected nodes.

## **4.1 An Optimal Graph of Africa’s Transport Network**

Previous research such as Graff (2024) has created a graph representation of Africa’s road network by dividing the continent into cells and computing routes connecting adjacent cells. However, cell centroids may not be of economic interest or well-connected at all. I thus generate a more realistic network representation by sampling important cities and ports and letting OSRM compute routes between them, which I then process into an accurate graph. To sample cities, I use the updated 13
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should be no direct AB link because these points should be connected via C or F. The same process is repeated taking point B as a vantage point, such that if the algorithm finds either AB or BA intercepted by other route-efficient points, this link is dropped from the undirected graph. Figure 11 shows the thus generated US-grade hypothetical network connecting the 453 (port-)cities. It has 1459 edges, some of which intersect and should be broken up further.

<!-- Start of picture text --> Figure 11: Demonstration of Network Finding Algorithm<br>543 (Port-)City Sizes (Population) US-Grade Hypothetical Network<br><!-- End of picture text --> _Notes:_ The LHS shows the 453 (port-)cities; circles reflect population. The RHS shows the result of applying the network finding algorithm described above, requiring US route efficiency of 0.834 (Wolfram, 2021) and a maximum 20<sup>_◦_</sup> deviation.

To propose optimal network extensions, I apply the algorithm to a complete transport network graph where all 1379 nodes are fully connected. Using the US network efficiency and 20<sup>_◦_</sup> angle criterion as illustrated above yields a staggering 2,031 new edges, shown in Appendix Figure A8. Since this is close to the number of existing edges, and many of the new links overlap, I adjust the parameters of the algorithm to generate fewer potential links. Concretely, I increase the maximum angle for intercepting nodes to _α_ = 45<sup>_◦_</sup> , and use the European route efficiency estimate of 0.767 by Wolfram (2021), which is lower than the US value of 0.843 partly due to Europe’s more fragmented (water-bound) geography. In addition, I let OSRM compute a full road distance matrix between all 1,379 nodes and remove connections that already have a route efficiency of 2/3 or higher. The 516 links retained by this modified algorithm, if straight, thus reduce travel distance by _≥_ 50%.

I then verify whether these links can be constructed and remove 46 links significantly intercepted by waterbodies or protected areas. Some proposed links across the closed Morocco-Algeria border already exist. I keep them and reduce their construction costs in the network optimization process. Finally, I manually add 11 links that I consider important for improved transport flow. In total, I retain 481 new links. Figure 12 draws them as thin green lines. Exempting one crossing in northern Mali, where adding a node would add no value for routing purposes, and a similar case in central Kenya, these links don’t intersect and thus imply no additional nodes to the graph.

In reality, roads constructed along these links will not be perfectly straight. I assume that new links have the same average route efficiency as existing links, which is 0.83.<sup>20</sup> Thus, I divide the spherical distance spanned by new links by 0.83 to estimate their road distance (edge weight), yielding 104,000km in total. In the remainder of this paper, I show simplified graphs with straight edges in all smaller plots to make them more legible. All computations, however, use edge weights corresponding to actual road distances and travel times, with existing network as in Figure 12.

> 20The route efficiency across individual links of 0.83 is larger than the overall network route efficiency of 0.68 because the latter considers full trans-African routes comprising multiple links.

16
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they find that terrain ruggedness and population density increase unit costs, whereas long roads ( _>_ 50km) decrease it. Furthermore, conflict, corruption, the business environment, and the origin of construction firms affects unit costs. ROCKS version 2.3 used by Collier et al. (2016) runs up to 2008. In 2018, the World Bank’s transport unit released an update including only projects with complete information (Bosio et al., 2018). Unfortunately, most projects in this database are road upgrades, but a few new road projects are included as well and summarized in Table 5.

Table 5: Road Building Costs: Road Costs Knowledge System (ROCKS) - 2018 Update

|Work Type|N|Length|(km)|Cost (M$’15)|Cost/km|
|---|---|---|---|---|---|
|_Continental Africa_||||||
|New 1L Road|1||99.0|21.15|0.214|
|New 2L Highway|4||122.0|59.53|0.611|
|_All Low and Lower-M_|_iddle _|_Income _|_Countr_|_ies_||
|New 1L Road|6||59.5|7.65|0.135|
|New 2L Highway|6||84.5|44.09|0.611|
|New 4L Expressway|1||84.8|328.08|3.869|
|_All Countries_||||||
|New 2L Highway|7||50.0|43.03|0.682|
|New 4L Expressway|24||74.9|213.01|5.152|

_Notes:_ Statistics are aggregated across projects using the median. Costs are in millions of 2015 USD. The raw costs in current USD were deflated at their project end date using the US GDP deflator with base year 2015.

According to the final column of Table 5, which provides median costs per km in millions of constant 2015 USD, the median new African road in the 2-lane highway category (implying one lane in each direction) costs $ 611K/km. This estimate, though derived from only four projects, is consistent with other estimates: a 2014 report by the African Development Bank (African Development Bank, 2014) analyzes 172 Projects contracted between 2000 and 2008 and reports a median cost of 227,800 2006 USD per lane-km. This figure includes road upgrades. Multiplying by 2 for a 2-lane highway and inflating by 16% to 2015 USD yields 528,800 USD per lane-km.

Following Collier et al. (2016), I allow for heterogeneity in road construction costs. Since it is difficult to measure (ever-changing) political conditions across the continent, I follow Fajgelbaum & Schaal (2020) and Graff (2024) and only consider average returns to ruggedness and distance, as well as population. My preferred specification is The first part is taken from Eq. (21) of Fajgelbaum & Schaal (2020). The 120,000 USD/km constant makes the mean cost/km across all existing links approximately equal to $ 611K/km. The population coefficient is averaged across Tables 4 and 5 of Collier et al. (2016), following Fajgelbaum & Schaal (2020), which obtain the other coefficient in this way. The tables suggest that Collier et al. (2016) regressed the log of cost on the population density (in 100 people/ _km_<sup>2</sup> ), but applying the coefficient to the levels density yields peak construction costs above $ 10M/km near large cities, which I regard as unrealistic. Thus, I use the natural log of the population density and add 1 to avoid negative values in remote regions. This effectively reduces the relevance of this term and remains close to Fajgelbaum & Schaal (2020). I extract ruggedness from the original 30 arc-second layer by Nunn & Puga (2012) within a 3km buffer around existing and proposed links, and do the same for population from the WorldPop 2020 population layer at 1km resolution.<sup>22</sup> Figure 17 shows ruggedness and population density thus estimated for each link.

> 22It is possible to track the terrain around existing roads more closely, but being more realistic with existing links complicates comparison with potential new links. Thus, I opt for a simpler approach of considering terrain within a broad 3km buffer in the direction of travel for both existing and potential new links.

21
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Table 6: Road Upgrading Costs: Road Costs Knowledge System (ROCKS) - 2018 Update

|Work Type|N|Length (km)|Weight|Cost (M$’15)|Cost/km|
|---|---|---|---|---|---|
|_Continental Africa_||||||
|Upgrading|28|103.5|2898|49.73|0.513|
|Reconstruction|27|33.0|891|20.20|0.116|
|Asphalt Mix Resurfacing|14|133.8|1874|16.93|0.143|
|Strengthening|11|110.0|1210|42.97|0.316|
|Weighted Average||104.0||35.77|0.326|
|_All Low and Lower-Middle _|_Inco_|_me Countries_||||
|Upgrading|68|61.8|4199|20.63|0.320|
|Reconstruction|53|73.0|3869|20.20|0.162|
|Asphalt Mix Resurfacing|35|143.9|5036|15.50|0.090|
|Strengthening|26|92.0|2392|20.89|0.316|
|Weighted Average||95.9||18.90|0.205|

_Notes:_ Statistics are aggregated across projects using the median. The ’Weight’ column is the number of projects (N) times the median length (Length). This is used to compute a weighted average across different work types. Costs are in millions of 2015 USD. The raw costs in current USD were deflated at their project end date using the US GDP deflator with base year 2015.

These estimates are consistent with African Development Bank (2014), who report an average cost of 180,300 USD’06/lane-km for a category called ’Rehabilitation,’ yielding 418,500 USD’15/km for 2-lane highways. Their 1st and 3rd quartile estimates translate to 254,900 and 673,100 USD’15/km, respectively. As mentioned earlier, African Development Bank (2014) lump together ’Upgrading’ with new construction at a median price of 528,800 USD’15/km. Their estimate is thus between the ’Upgrading’ estimate of $ 513K in Table 6 and new construction at $ 611K from Table 5.

Upgrading a link to allow speeds of 100km/h should require different types of road work depending on the existing condition of the road. It is unrealistic to assume that this is a continuous function of the distance of travel speed from 100km/h. For example, two roads with travel speeds of 85 and 90km/h likely both require a full asphalt resurfacing to bring them to 100km/h. Since the distribution of speeds in Figure 19 is trimodal, I propose the following three categories: (1) roads with a travel speed below 60km/h require a full ’Upgrade’ at an average cost of $ 513K/km; (2) roads with travel speeds of 60-79km/h may require mixed works at an average cost of $ 326K/km; (3) roads with travel speeds of 80km/h and higher require an ’Asphalt Mix Resurfacing’ at $ 143K/km on average. To introduce heterogeneity in these cost estimates, I use Eq. 6 and modify the intercept such that the median (across all routes, regardless of their upgrade type) is close to the respective category medians: For category (1), the intercept is ln(101,600), for category (2) it is ln(64,600), and for category (3), it is ln(28,400). In addition, I clip estimates at the 2.5th and 97.5th percentile to guard against outliers, such that, e.g., ’Asphalt Mix Resurfacing’ costs remain between $ 94K/km and $ 206K/km. Figure 23 reports the classification and estimated unit costs.

The entire central-African network and trans-Saharan connections via Mali and Niger, as well as most of the network around the horn of Africa, is classified as requiring a full upgrade, at costs around $ 400K/km due to mostly flat terrain and small population density. The total cost of upgrading the entire 315K km transport network is thus estimated at $ 106B (in 2015 USD), comprising $ 57.8B full road upgrades, $ 35.6B mixed works, and $ 12.3B mixed asphalt resurfacing.

# **5.6 Cost-Benefit Analysis: Road Upgrading**

Comparing this cost to the total MA gain of 42% or $ 701B/min implies a return of $ 6.6/min/ $ . With border frictions, the MA gain is reduced to 27% or between 214 and 253 billion USD/min in the two scenarios, implying a 2/2.4 $ /min/ $ return in the cumulative/simple frictions scenarios, respectively. Thus, road upgrades yield significantly higher average returns than fresh construction.

It remains to gauge the returns to upgrading individual links. Figure A11 shows estimates following the same schema as Figure 18, signifying that upgrading links in populated areas and critical trans-continental connections yields significant returns. The consensus package again 27
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The medium-term economic returns to increased MA are very difficult to estimate, especially on such a large scale. The only large-scale analysis I am aware of is Donaldson & Hornbeck (2016). The authors regress changes on the log of land values across 2,327 US counties between 1870 and 1890 on changes in the log of MA, and find an elasticity of 0.5, which is quite robust to different choices of _θ_ . This suggests that in the medium run MA translates into economic activity at a ratio of 2:1. However, present-day Africa may be very different from the historical US.

I thus also run some regression similar to Donaldson & Hornbeck (2016), examining the effect of log(MA) on GDP per capita (Kummu et al., 2018) and the International Wealth Index (Lee & Braithwaite, 2022) using the dataset of 1379 nodes/cities. I control for country fixed effects and also for the log of port outflows in 2020Q1, adding one so that the log is zero in non-port cities. Since MA is endogenous to GDP, I apply the network finding algorithm from Section 4.2 (with EU parameters but not checking against OSM) to generate a hypothetical network with 2869 edges, shown in Appendix Figure A14, and use the MA measure based on this network as an instrument for MA. Table 7 reports the results for MA denominated in both road distance and travel time.

Table 7: Returns to Market Access in Africa ( _N_ = 1379 Nodes)

|Dependent Variable:|log(GDP|/Capita)|I|WI|log(|IWI)|
|---|---|---|---|---|---|---|
|Estimator:|OLS|IV|OLS|IV|OLS|IV|
|_Variables_|||||||
|log(MA): Distance|0.3221<sup>_∗∗_</sup>|0.3250<sup>_∗∗_</sup>|10.44<sup>_∗∗∗_</sup>|10.53<sup>_∗∗∗_</sup>|0.3450<sup>_∗∗∗_</sup>|0.3445<sup>_∗∗∗_</sup>|
||(0.1432)|(0.1451)|(2.258)|(2.425)|(0.0918)|(0.0934)|
|log(Outfows + 1)|0.0103<sup>_∗∗_</sup>|0.0103<sup>_∗∗_</sup>|1.390<sup>_∗∗∗_</sup>|1.390<sup>_∗∗∗_</sup>|0.0370<sup>_∗∗∗_</sup>|0.0370<sup>_∗∗∗_</sup>|
||(0.0042)|(0.0042)|(0.1451)|(0.1456)|(0.0048)|(0.0048)|
|_Fit statistics_|||||||
|R<sup>2</sup>|0.9709|0.9709|0.8079|0.8079|0.7602|0.7602|
|Within R<sup>2</sup>|0.1399|0.1399|0.2465|0.2465|0.1676|0.1676|
|Wu-Hausman, p-value||0.5409||0.6842||0.9508|
|Wald (1st stage), p-value, log(MA)||_<_0_._01||_<_0_._01||_<_0_._01|
|_Variables_|||||||
|log(MA): Travel Time|0.3216<sup>_∗∗_</sup>|0.3147<sup>_∗∗_</sup>|11.33<sup>_∗∗∗_</sup>|10.01<sup>_∗∗∗_</sup>|0.3891<sup>_∗∗∗_</sup>|0.3277<sup>_∗∗∗_</sup>|
||(0.1278)|(0.1379)|(2.044)|(2.177)|(0.0856)|(0.0835)|
|log(Outfows + 1)|0.0100<sup>_∗∗_</sup><br>|0.0100<sup>_∗∗_</sup><br>|1.383<sup>_∗∗∗_</sup><br>|1.378<sup>_∗∗∗_</sup><br>|0.0369<sup>_∗∗∗_</sup><br>|0.0366<sup>_∗∗∗_</sup><br>|
||(0.0041)|(0.0041)|(0.1427)|(0.1421)|(0.0047)|(0.0046)|
|_Fit statistics_|||||||
|R<sup>2</sup>|0.9714|0.9714|0.8140|0.8137|0.7684|0.7678|
|Within R<sup>2</sup>|0.1541|0.1540|0.2703|0.2691|0.1962|0.1940|
|Wu-Hausman, p-value||0.3023||_<_0_._01||_<_0_._01|
|Wald (1st stage), p-value, log(MA)||_<_0_._01||_<_0_._01||_<_0_._01|
|_Fixed-efects_|||||||
|iso3c (49)|Yes|Yes|Yes|Yes|Yes|Yes|

_Clustered (iso3c) standard-errors in parentheses Signif. Codes: ***: 0.01, **: 0.05, *: 0.1_ The coefficients in both MA denominations suggest that in Africa MA translates to GDP/Capita or household wealth with an elasticity of _∼_ 0.33, which is slightly lower than in the US. Using the IWI in level suggests that a doubling of MA raises the IWI [0, 100] by 10 points—a sizeable return.

The failure of the Wu-Hausman test of endogeneity in most cases indicates that the instrument is too strong and possibly itself endogenous. Since the network was constructed by intersecting routes between 453 (port-)cities, 1379 _−_ 453 = 926 nodes/cities were found by intersecting the optimal routes between these cities, and thus already reflect the existing road network.

However, repeating the exercise using just the original 453 (port-)cities and a hypothetical network between them gives the same gdp elasticity (around 0.3), with Wu-Hausman p-values all greater than 0.1, and partly insignificant IWI elasticities around 0.15-0.3. Thus, the cross-sectional IV approach using a hypothetical network in any case does not add much to the exercise.

29
The results are still useful in suggesting that in Africa the returns to MA may be up to 50% lower than in the US following Donaldson & Hornbeck (2016). A reason for this may be the prevalence of other market inefficiencies, often of a regulatory nature (Herrera Dappe et al., 2024). A prudent assumption is that in Africa MA gains translates into economic activity at a ratio of 4:1.

In the following, instead of computing exact returns, I determine the returns needed in order to break even. This will take the form of a reverse macroeconomic present discounted value (PDV) calculation. African GDP in 2022 was 2.81 trillion 2015 USD. The median growth rate in 2000-2022 was 4.1%. If we are pessimistic, let it be 3% going forward. I discount the future by 10%. This is a reasonable rate given that stock market returns are generally below this—the MSCI World has a long-term return rate of 8-9%. Emerging market bonds yield even lower returns, around 6%, according to the EMBI index by JP Morgan. So at the macroeconomic scale, it is very difficult to invest with more than 10% returns. I also assume that the horizon of policymakers is 30 years and disregard benefits accruing after 30 years. With these assumptions, I calculate the change in the macroeconomic growth rate of Africa needed to break even in PDV terms for each of the three scenarios proposed above as well as the full/consensus construction and upgrade scenarios considered in Sections 5.4 and 5.7. Table 8 reports these calculations.

Table 8: Break Even Changes in the Macroeconomic Growth Rate for Different Investments

|**Work Package**|**Cost**|**%**∆**MA**|**BEG** (4.1%)|**BEG** (3%)|
|---|---|---|---|---|
|Full Extension (_↑_RE)|$56.1B|3_._9_−_5_._5|0.31% _→_4.113|0.50% _→_3.015|
|Consensus Extension (_↑_RE)|$12.1B|2_._4_−_4_._0|0.07% _→_4.103|0.11% _→_3.003|
|Full Upgrade (_↑_TE)|$105.7B|27_._0_−_42_._1|0.58% _→_4.124|0.94% _→_3.028|
|Consensus Upgrade (_↑_TE)|$45.0B|21_._2_−_33_._1|0.25% _→_4.110|0.40% _→_3.012|
|All Links _>_$1/min/$|$60.9B|26_._8_−_38_._1|0.33% _→_4.114|0.54% _→_3.016|
|All Links _>_$2/min/$|$36.6B|24_._0_−_30_._6|0.20% _→_4.108|0.32% _→_3.010|
|All Links _>_$4/min/$|$17.0B|19_._5_−_20_._4|0.09% _→_4.104|0.15% _→_3.005|

_Notes:_ Table shows the cost in 2015 USD PPP of different investment packages, their MA %-gain under the worst frictions and frictionless scenarios, and the change in Africa’s macroeconomic growth rate needed to break even after 30 years under realistic (4.1%) and pessimistic (3%) baseline growth.

I illustrate their interpretation using the largest consensus package containing all links with _>_ $ 1/min/ $ marginal returns. It costs $ 60.9B and yields a MA gain between 26.8% (worst frictions scenario) and 38.1% (no frictions). Under the 4.1% baseline growth scenario, the growth rate needs to change by 0.33% to 4.114% to yield a return of $ 60.9B in PDV terms after 30 years. Comparing the 26.8% lower bound to 0.33% yields a ratio of 81:1. Thus, if changes in MA translate to changes in the growth rate at 81:1 or higher, the package is macroeconomically feasible.

This appears very plausible given the 0.5 elasticity of Donaldson & Hornbeck (2016) and the _∼_ 0.33 elasticity of Table 7 suggesting a ratio of 3:1 (prudent estimate is 4:1). However, many things can go wrong. First, growth could be slower at 3%, requiring a 0.54% increase in the growth rate or a return ratio above 50:1. Then, the package could end up being more expensive for various reasons related to terrain, weather, corruption or delays. Assuming it turns out twice as expensive, this would require a ratio above 25:1. Finally, if the investment is financed through debt, it could adversely affect both the total costs and the macroeconomic growth rate by limiting fiscal space and worsening credit ratings. Such forces could lower the required return ratio to as low as 10:1. Nevertheless, this is still above 4:1. Thus, overall, these back-of-the-envelope calculations suggest that large-scale _optimal_ investments in Africa’s road network are macroeconomically feasible.

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# **6 Optimal Network Investments in General Equilibrium**

Having simulated many scenarios in partial equilibrium by modifying one link at a time or jointly adding/improving a selection of (high marginal value) links and computing MA returns, this section goes a bold step further and considers welfare-maximizing network investments in general equilibrium (GE). In particular, using a quantitative framework established by Fajgelbaum & Schaal (2020), I optimize over the space of networks after setting an infrastructure budget. The framework accommodates trade in multiple goods and requires exogenous productivities for the production of each good in each location. Cities must have different productive endowments to create incentives for trade. For my application to the African network, this is a strong limitation, firstly because such city-level productivity data is unavailable, and secondly because global optimization with many goods traded on a large network is computationally infeasible.

To remedy the lack of detailed productivity data, I follow Armington (1969), Fajgelbaum & Schaal (2020), and Graff (2024) and assume that products are differentiated by city of origin and each city produces just one good. To overcome the dimensionality problem, I create 22 traded products differentiated by origin, of which 17 are produced by selected mega-cities<sup>26</sup> , and the remaining 5 are divided among large port-cities (population _>_ 1M and outflows _>_ 1M TEU), large cities (population _>_ 2M), ports (outflows _>_ 0), cities (population _>_ 200K), and smaller cities/towns or unpopulated nodes (which do not produce traded goods).

In my model specification and parameterization, I largely follow Graff (2024). I set the existing level of infrastructure along each link from city _j_ to city _k_ , I _jk_ , to the average link speed in km/h I _jk_ is bounded from below by the present speed (I<sup>min</sup> _jk_ = I _jk_ ), which is zero for proposed links. It is also bounded from above by the greater of 100km/h and the present speed, denoted by I<sup>max</sup> _jk_ = max(I<sup>max</sup> _jk_<sup>_,_100).Ithencalculatethe(perkm/h)infrastructurebuildingcost</sup><sup>_δ_</sup> _jk_<sup>Ias</sup> where upgrade cost _jk_ corresponds to the cost per km of building each link or upgrading it to 100km/h from Figure A12, multiplied by the link length in km. Since in the model of Fajgelbaum & Schaal (2020) the lower bound on I _jk_ is simply a constraint on how a given infrastructure budget may be spent, I set the baseline budget equal to _δjk_<sup>I</sup><sup>_×_I</sup><sup>_jk_.Ithenconsidersocialplannerswith$20</sup> billion, $ 35 billion, and $ 50 billion USD’15 in addition to the baseline budget, which is well below the theoretical total of $ 162 billion cost of all proposed work (upgrading and construction). The planner can freely choose which links to build/upgrade at cost _δjk_<sup>I.</sup> In reality, most road upgrade/construction decisions are discrete, but the quantitative framework does not allow a discretization of optimal infrastructure investments. The results may instead be interpreted as ’partial’ upgrades/completions. For example, suppose a road at 80km/h has a speed of 90km/h after the optimization. In that case, we may conclude that the planner decided to upgrade (90-80)/(100-80) = 50% of the road.<sup>27</sup> The model also features iceberg trade costs such that traders need to send 1+ _τjk_<sup>_n_units of a good for one unit to arrive.Following Graff (2024),</sup> who averages estimates from Atkin & Donaldson (2015) for Nigeria and Ethiopia, I set

> 26Dakar, Casablanca, Abidjan, Kumasi, Algiers, Lagos, Kano, Yaounde, Luanda, Kinshasa, Cape Town, Johannesburg, Cairo, Khartoum, Nairobi, Addis Ababa, and Dar es Salaam.

> 27While this interpretation is plausible, the cost per km of building/upgrading segments decreases for roads longer than 50km (Eq. 6, following Collier et al. (2016)), and it is thus not realistic that the planner considers prices _δjk_<sup>Iinelastictothequantity(length)constructed.Inaddition,iftheplannerpartiallybuildsanewlinks,wemust</sup> assume that the remainder corresponds to a dirt road that can be constructed free of cost. Thus, the framework may slightly overstate the amount of infrastructure that can be constructed on a given budget.

31
The trade cost positively depends on the flow of good _n_ across link _j → k_ , _Q_<sup>_n_</sup> _jk_<sup>,subjecttoan</sup> elasticity _β_ , an effect Fajgelbaum & Schaal (2020) describe as ’congestion forces’. It also depends negatively on the level of infrastructure _Ijk_ , subject to an elasticity _γ_ . Following Graff (2024), I set _γ_ = 0 _._ 946.<sup>28</sup> I would also follow him and set _β_ = 1 _._ 177, however, due to the large size of the network, only the dual solution to the model which casts all variables as function of prices is viable, stipulating a maximum of _β_ = 1 for strong duality (Fajgelbaum & Schaal, 2020). As also discussed at length by Fajgelbaum & Schaal (2020), the lagrangian objective of the model is only convex if _β > γ_ , in which case a unique global solution to the network design problem can be computed. If _β < γ_ , the problem becomes non-convex and yields a local optimum, which Fajgelbaum & Schaal (2020) further refine using simulated annealing methods. They denote this case as ’increasing returns to infrastructure’ (IR) and show that the optimal network is more tree-like, with larger investments in fewer links. Graff (2024)’s parameterization implies _β > γ_ . This may, however, not hold for large transnational networks; thus, I also examine the IR case assuming _γ_<sup>_′_</sup> = 1 _._ 2.<sup>29</sup> Fajgelbaum & Schaal (2020)’s model has a normalized inequality-averse Cobb-Douglas utility function giving the utility of a single worker in location _j_ in terms of consumption and housing where _ρ_ controls the level of inequality aversion. Following Graff (2024), I assume _α_ = 0 _._ 7, which is based on estimates by Porteous (2022), and _ρ_ = 0 (no inequality aversion), but I also explore _ρ_ = 2 (moderate inequality aversion). Following Graff (2024), I set _hj_ = 1 _− α_ , i.e., the local housing availability is proportional to the current population. Total consumption in location _j_ is _Cj_ = _Ljcj_ , which in turn is a CES aggregate over _N_ traded goods with elasticity of substitution _σ_ In line with the international trade literature, Fajgelbaum & Schaal (2020) and Graff (2024) use _σ_ = 5, following recommendations by Atkin & Donaldson (2022). Both assume that the largest _N − k_ cities produce their own good and mostly conduct single-country simulations.<sup>30</sup> Since I have few cities per country, it is also plausible to interpret _σ_ as an Armington (1969) elasticity, and here a meta-study by Bajzik et al. (2020) suggests a median value of 3.8. Since the network is very large, even _σ_ = 3 _._ 8 yields equilibria with limited inter-city trade where optimal investments are quite localized around population centers. I thus also run simulations with _σ_ = 2, which is at the low end of empirical estimates (Bajzik et al., 2020). Production of good _n_ in node _j_ is given by Due to lack of better measures,<sup>31</sup> I set city productivity _Zj_<sup>_n_equaltothepredictedInternational</sup> Wealth Index of Lee & Braithwaite (2022) (Figure A2). Since the original estimate is only available

> 28The value for _γ_ is informed by estimates of the costs of road delays to African truckers by Teravaninthorn & Raballand (2009). The value of _β_ is informed by estimates on the cost of traffic congestion by Wang et al. (2011). Fajgelbaum & Schaal (2020) use _β_ = 0 _._ 13 and _γ_ = 0 _._ 1, but their characterization of infrastructure ( _Ijk_ ) in terms of road-lane km yields significantly larger values than the formulation in terms of km/h adopted from Graff (2024).

> 29Fajgelbaum & Schaal (2020) page 1439 simulate an IRS case by inverting the ratio of _β_ and _γ_ keeping _β_ fixed, which, for _β_ = 1, yields _γ_<sup>_′_</sup> = 1 _._ 057. This, however, yields very similar solutions to _γ_ = 0 _._ 946, thus I set _γ_ = 1 _._ 2 to induce stronger effects. There are, as far as I am aware, no studies or empirically accessible data on road congestion in inter-city or international transport.

> 30Graff (2024) sets _N_ = 6 _, k_ = 2 (with one international good produced outside the country and one agricultural good produced by smaller cities) and conducts simulations for every African country. Fajgelbaum & Schaal (2020) set _N_ = 10 _, k_ = 1 and conduct simulations for France, Spain, and Western Europe.

> 31Graff (2024) uses nightlights divided by city population to form the productivity measure. While this is a viable approach, the task of measuring differences in city productivity across countries encourages the use of richer data sources. Lee & Braithwaite (2022) combine daylight satellite imagery and OSM data in an iterative ML framework that has proven able to predict wealth within and across SSA countries reliably. As an attempt to verify, taking a city-population weighted average of the IWI by countries yields a correlation of 0.62 with the 2020 World Bank national GDP/capita estimate and a correlation of 0.73 with the log of GDP/capita. Using NASA’s Black Marble nightlights, on the other hand, only yields a correlation of 0.15 between city-population weighted nightlights/capita and national GDP/capita. However, using the log of nightlights/capita yields a 0.58 correlation with GDP/capita and a 0.65 correlation with the log of GDP/capita. While this appears more promising, a visual comparison, provided in Appendix Figure A10 for the interested reader, shows that the IWI is higher in larger cities in richer countries - in line with much scientific evidence. The nightlights measure appears quite noisy and imprecise.

32
for SSA, I use the imputed version of Krantz (2023) (Figure A3) and compute the productivity of each city as the inverse-distance-weighted average of cells within 30km of the city centroid. I assume that average household wealth, aggregated at the city level, approximates worker productivity. Footnote 31 argues that national average urban wealth strongly correlates with GDP per capita.

I also calibrate the productivity of international ports so that overall production matches national accounts data. Averaging estimates from the UN National Accounts Main Aggregates database between 2020 and 2022 yields that imports were equal to 26.5% of African GDP. Since my network does not include foreign locations,<sup>32</sup> the imported goods will have to be produced by port cities. Assuming that port outflows in 2020Q1 are proportional to the inflows, multiplying outflows from the 52 largest ports by a factor of 37 yields (precisely) an imports measure summing to 26.5% of the total production volume using the IWI as productivity measure. These imports need to be divided by port city population to yield productivities. Since the framework of Fajgelbaum & Schaal (2020) endogenously allocates labor to different tasks, letting port cities produce both a domestic and international variety would obscure this calibration. Thus, I compound both tasks so that port cities use all labour to produce a mixed international variety at productivity The median intl. productivity is 27 IWI points, but the mean is much higher at 2748, indicating high import volumes in some port cities with small populations. Top of the list is Egypt’s Sokhna Port (towards the Red Sea), with a population of 1000 and intl. productivity of 134,656, followed by Egypt’s Port Said, with a population of 894,850 (within a 30km radius) and intl. productivity of 1,264. Other ports with intl. productivity above 500 include Tanger Med and Walvis Bay. Since utility functions are mildly concave ( _α_ = 0 _._ 7) and the planner maximizes utility, the comparatively small populations of these port cities downweight their importance for network optimization. The resulting increase in purchasing power also implies that a part of the equilibrium consumption of port cities is exported. For simplicity, I assume the export share is equal to the share of international productivity in total city productivity and adjust consumption and welfare accordingly.

Table 9 summarizes productivity and population by city type, indicating that the 17 megacities together account for 35.7% of population (159 million) and 46.5% of total domestic production.

Table 9: Productivity and Population by City Type

|City Type|City/Node|City_>_200K|Port|City_>_2M|Port-City|Megacity|
|---|---|---|---|---|---|---|
|Count|1,022|286|32|13|9|17|
|_Population_|||||||
|Sum|48.5M|154.6M|20.7M|38.6M|23.2M|158.7M|
|Mean|47.4K|540.5K|645.4K|2.97M|2.58M|9.3M|
|Median|21.2K|390.7K|590.7K|2.69M|1.89M|6.9M|
|_Production_|||||||
|Total|1.50B|5.86B|3.32B|1.84B|3.88B|10.80B|
|Total Domestic|1.50B|5.86B|0.97B|1.79B|1.38B|10.01B|
|_Productivity_|||||||
|Domestic Mean|27.72|36.15|47.34|46.58|60.83|60.89|
|Domestic Median|23.74|32.41|47.88|46.92|60.37|59.65|
|International Mean|0|0|4,376|1.59|203.6|7.1|
|International Median|0|0|28.4|0|80.88|0|

_Notes:_ Table shows population and productivity (IWI) summarized for cities of different sizes and port-cities. International productivity (for imports) is the second term on the RHS of Eq. 14.

> 32It would be difficult to add foreign markets to an already large network. In theory, this could be done by adding additional nodes for ports and separating the city from the port in this way, but for any goods to flow to these locations, they need to have populations. Since flows are an equilibrium outcome, they would have to be calibrated to match the data by reverse engineering the model, which is quite complicated solving for populations of multiple ports. The framework may also have difficulties computing optimal flows along zero-cost links.

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# **7 Summary and Conclusion**

Using detailed routing data, large and accurate network graphs, and a rich economic geography including population, wealth/gdp, trade through ports, border frictions, road construction and upgrading costs, this paper rigorously examines Africa’s full continental road network and characterizes optimal investments into it in partial and general equilibrium (PE and GE).

Presently, Africa has 1.6 million km of non-residential roads, around 30% of which are paved. This fraction differs significantly by country and road type. Particularly in Central Africa, the Sahara, and the Horn of Africa region, most roads are unpaved and rough, severely limiting trans-African connectivity. The network’s overall route efficiency (RE) of 0.68 is acceptable but considerably lower than that of the US network at 0.84. The time efficiency and average travel speed along many roads is significantly worse than in other world regions. Particularly the aforementioned regions can only be traversed at speeds around 40km/h. In addition, cross-border frictions substantially impede continental market access (MA). Estimates suggest that crossing the median border in Africa is as costly as 1043km of road distance, and the time commitment is equivalent to 3.2 days of travel time. These frictions induce an aggregate MA loss between 26% and 78%. Locations across the border from major markets such as Nigeria suffer MA losses of up to 80%.

Given a detailed network and economic geography, optimal road investments still depend on the geographic scale and magnitude of the investment and the planner’s objective (MA or welfare maximization with/without inequality aversion). Structural parameters such as the trade elasticity/elasticity of substitution and returns to scale in infrastructure also noticeably impact optimal spatial allocations, as do heterogeneous road construction costs and border frictions. The paper navigates this complexity via a graduated approach, starting with small-scale PE investments and gradually progressing towards GE simulations where social planners spend large amounts. This requires different network representations and methods.

To increase the scale of investments to entire roads or collections of segments, I build a detailed representation of the transport network connecting 447 important cities and 52 ports along the fastest routes. My network graph has 1379 nodes and 2344 edges (links), summing to 315K km. I apply a network finding algorithm and find 481 potential new links summing to 104K km. Building them all would raise RE by 7.1%, from 0.68 to 0.72, and MA (denominated in road distance) by 5.5%. The most valuable links increasing RE/MA by more than 0.1% are new trans-Saharan roads through Mauritania, Mali, and Chad and reopening the closed Morocco-Algeria border. Some new roads in Central and Southern Africa would also significantly increase RE/MA. When border frictions are activated, the MA gain drops from 5.5% to 3.9%, with particularly high losses on remote and border-crossing links. The median link yields a 50% lower MA gain under frictions.

According to the World Bank ROCKS database, the median 2-lane highway in Africa costs $ 611K 2015 USD per km. Using a heterogeneous road costs prediction model considering road length, terrain ruggedness, and population density, I estimate the cost of the existing transport network at $ 186B USD’15, and the proposed extension of $ 104K km at $ 56B, yielding an average MA return of $ 1.4/km per $ invested in new links, which drops to _∼_ $ 0.7/km/ $ if frictions are added.

Examining road quality measured by travel speed, I find that upgrading all links to _≥_ 100km/h yields a time-denominated MA gain of 42.1%. New links yield a 14.1% MA gain if built at 100km/h on top of the existing (slow) network. Upgrading the entire network to fast links and also building the proposed ones could yield a 50% MA gain. There is significant heterogeneity in the value of upgrading links, with trans-Saharan links yielding the greatest MA gains. Applying border frictions lets the total MA gain from upgrading all links drop to 27%. Again, particularly peripheral and transnational links lose value. I estimate the cost of upgrading the entire network at $ 106B, comprising $ 67.8B full upgrades, $ 36.5B mixed works, and $ 12.3B asphalt resurfacing. Upgrading all roads yields average gains of $ 6/min per $ spent, which drops to $ 2.4/min/ $ under frictions.

To raise average returns, I consider investment packages targeting high marginal gain links (both upgrades and new roads). I propose three packages with links at marginal gains _>_ 1, 2, or 4 $ /min/ $ . They cost $ 60.9B/ $ 36.6B/ $ 17B, yield frictionless MA gains of 38.1%/30.6%/20%, 44
and 26.8%/24%/19.5% under frictions. These packages are globally macroeconomically feasible under frictions if they can raise Africa’s aggregate growth rate (4.1%) by 0.33%/0.2%/0.09%. For the _>_ $ 1 ( $ 60.9B) package, this would require that growth in MA translates into economic growth at a rate of _≥_ 81:1. This is plausible in light of empirical results suggesting economic returns to MA gains as high as 3:1, but there are several factors that could raise the cost of these packages considerably and lower the macroeconomic returns, especially if they are financed through debt.

Calibrating a general equilibrium framework by Fajgelbaum & Schaal (2020) permitting global optimization over the space of networks, I also derive optimal investments in spatial equilibrium, letting a welfare-maximizing social planner spend a fixed infrastructure budget. My calibration accounts for trade in multiple goods and through international ports. I consider cases with/without inequality aversion, cross-border frictions, and increasing returns to infrastructure. I also simulate with different values for the elasticity of substitution ( _σ_ ). Planners with a baseline budget of $ 35B optimally connect large cities and ports with each other and surrounding smaller cities, focusing on populated/productive areas. An inequality-averse planner invests slightly more in the poorer Central African and the Horn of Africa regions. About 30% of the investments are new/proposed links, built particularly around major cities to limit congestion. With increasing returns to infrastructure (low congestion forces), the investments become less concentrated around large cities, and the planner upgrades more roads (building few new ones). This increases welfare gains. With inequality aversion, the gains again become more equitable at minor aggregate losses, and the planner upgrades more roads in Central Africa. Overall, due to concave utility of consumption, remote cities gain more welfare despite receiving less infrastructure.

To focus spending on important roads, I additionally consider a trans-African planner that optimally invests in continental transport roads connecting 47 major (port-)cities. The planner, endowed with $ 15B, also mostly improves regional connectivity under standard assumptions of _σ_ = 3 _._ 8. Much is invested along the West African coastline, but also in better connecting Dakar to Bamako, Khartoum to Egypt, Kampala to Mogadishu via Mombasa, and Kinshasa to Mbuji Mayi. An inequality-averse planner fully connects Kinshasa to Bukavu/Rwanda and further to Mombasa, Mogadishu, and Dar es Salaam. Thus, the planner upgrades a corridor crossing populated Central Africa. To include truly pan-African investments with roads across sparsely populated Northern Central Africa (e.g., Goma to Bangui) and the Sahara within an optimally spent $ 15B budget would require an unrealistically low elasticity _σ_ = 1 _._ 5 coupled with inequality aversion and increasing returns. Further, activating border frictions reduces equilibrium investments along these links again, indicating that such investments should be coupled with trade facilitation measures.

In conclusion, the paper characterizes optimality for road network investments at different spatial scales and economic objectives from a pan-African perspective. It thus provides broad guidance for policymakers interested in improving Africa’s road infrastructure. An overarching finding is that trans-African roads are neither MA nor welfare maximizing compared to benefits from improved local connectivity in populated areas or between nearby cities and coastal ports. This is especially true under border frictions, which reduce the value of trans-African links. Yet, these links are essential for trans-African trade under AfCFTA to pick up. A general policy recommendation is thus to prioritize regional connectivity improvements, reduce border frictions, and invest in trans-African roads along specific high-yield links/corridors. The graphical results provided in the paper and appendix should be helpful in prioritizing continental and regional investments, with detailed data and results available on GitHub. The multi-level approach combining routing engines, PE, and GE analysis can also be adapted to examine specific regions/investments— such as, recently, for the CEMAC region as part of a World Bank regional infrastructure assessment program. Towards this end, the Julia library and reproducibility package are utile.

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