# **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.

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