Optimal Transport Networks in Spatial Equilibrium Pablo D. Fajgelbaum Edouard Schaal UCLA/NBER, CREI/CEPR World Bank Space and Productivity Conference, 09/2017
Introduction How should transport infrastructure investments be allocated in a network? Link-by-link ◮ Direct effect: value of trade/travel time along a link ◮ Indirect effect: reallocations, investment,.. Trade-offs across links ◮ Shape of network will matter
This Paper Develop a framework to study optimal transport networks in general equilibrium Neoclassical trade model (with factor mobility) on a graph Sub-Problems ◮ how to allocate production and consumption? ◮ how to ship goods through the network? (“Optimal Flows”) ◮ how to build infrastructure? (“Optimal Network”) Apply to road networks in different European countries ◮ how large are losses from misallocation of current networks and gains from optimal expansion? ◮ how do these effects vary across countries? ◮ what are the regional effects?
Brief Background Trade costs in quantitative analyses of international trade and economic geography ◮ Eaton and Kortum (2002), Allen and Arkolakis (2014), Redding (2016),... Assessments of actual changes in transport infrastructure ◮ Donaldson (2010), Duranton et al. (2014), Faber (2014),... Standard approach is to assess implications of a particular (not necessarily “best”) change in transport costs Here: trade costs are endogenous through optimal investments in the transport network ◮ Related to optimal flow problem on a network (e.g., Chapter 8 of Galichon, 2016)
Framework 1 Multiple locations and factors of production ◮ Many traded goods and one non-traded good ◮ Neoclassical production technologies ◮ Workers may be immobile or perfectly immobile ⋆ Homothetic preferences of traded and non-traded commodities ◮ Special cases: Ricardian model, Heckscher-Ohlin, Armington, Rosen-Roback... 2 The locations are arranged on a graph ◮ Each location has a set of “neighbors” (directly connected) ⋆ “Neighbors” may be geographically distant ◮ Shipments Q n jk can flow through neighbors only Per-unit cost τ n 3 jk of shipping from j to a neighbor k ◮ Decreasing returns to transport: τ n jk increases with quantity shipped ◮ Positive returns to infrastructure: τ n jk decreases with infrastructure I jk Building infrastructure I jk takes up δ I 4 jk I jk units of a scarce resource (“asphalt”) ◮ δ I jk may vary across links, due to ruggedness, distance... ◮ The transport network is defined by { I jk }
Framework 1 Multiple locations and factors of production ◮ Many traded goods and one non-traded good ◮ Neoclassical production technologies ◮ Workers may be immobile or perfectly immobile ⋆ Homothetic preferences of traded and non-traded commodities ◮ Special cases: Ricardian model, Heckscher-Ohlin, Armington, Rosen-Roback... 2 The locations are arranged on a graph ◮ Each location has a set of “neighbors” (directly connected) ⋆ “Neighbors” may be geographically distant ◮ Shipments Q n jk can flow through neighbors only Per-unit cost τ n 3 jk of shipping from j to a neighbor k ◮ Decreasing returns to transport: τ n jk increases with quantity shipped ◮ Positive returns to infrastructure: τ n jk decreases with infrastructure I jk Building infrastructure I jk takes up δ I 4 jk I jk units of a scarce resource (“asphalt”) ◮ δ I jk may vary across links, due to ruggedness, distance... ◮ The transport network is defined by { I jk }
Framework 1 Multiple locations and factors of production ◮ Many traded goods and one non-traded good ◮ Neoclassical production technologies ◮ Workers may be immobile or perfectly immobile ⋆ Homothetic preferences of traded and non-traded commodities ◮ Special cases: Ricardian model, Heckscher-Ohlin, Armington, Rosen-Roback... 2 The locations are arranged on a graph ◮ Each location has a set of “neighbors” (directly connected) ⋆ “Neighbors” may be geographically distant ◮ Shipments Q n jk can flow through neighbors only Per-unit cost τ n 3 jk of shipping from j to a neighbor k ◮ Decreasing returns to transport: τ n jk increases with quantity shipped ◮ Positive returns to infrastructure: τ n jk decreases with infrastructure I jk Building infrastructure I jk takes up δ I 4 jk I jk units of a scarce resource (“asphalt”) ◮ δ I jk may vary across links, due to ruggedness, distance... ◮ The transport network is defined by { I jk }
Framework 1 Multiple locations and factors of production ◮ Many traded goods and one non-traded good ◮ Neoclassical production technologies ◮ Workers may be immobile or perfectly immobile ⋆ Homothetic preferences of traded and non-traded commodities ◮ Special cases: Ricardian model, Heckscher-Ohlin, Armington, Rosen-Roback... 2 The locations are arranged on a graph ◮ Each location has a set of “neighbors” (directly connected) ⋆ “Neighbors” may be geographically distant ◮ Shipments Q n jk can flow through neighbors only Per-unit cost τ n 3 jk of shipping from j to a neighbor k ◮ Decreasing returns to transport: τ n jk increases with quantity shipped ◮ Positive returns to infrastructure: τ n jk decreases with infrastructure I jk Building infrastructure I jk takes up δ I 4 jk I jk units of a scarce resource (“asphalt”) ◮ δ I jk may vary across links, due to ruggedness, distance... ◮ The transport network is defined by { I jk }
Example: Spain 50 km x 50 km square network, 8 neighbors per interior node The problem of designing the network determines how much to invest in each link
Optimal Network Problem Assume that, given the transport network, the market efficiently allocates resources ◮ Requires tolls if the decreasing returns to transport are not internalized ◮ Free entry to trading activities Planning problem : choose { I jk } to maximize aggregate welfare subject to: detail ◮ Location and consumption decisions by workers given � � I jk ◮ Production and trade decisions of firms given � � I jk ◮ Goods and factors market clearing constraints ◮ Availability of inputs to build the network
Key Efficiency Conditions on Every Link No-arbitrage condition on flows (“efficient road use”): ∂τ n P n jk k ≤ 1 + τ n Q n jk , = if Q n jk + jk > 0 P n ∂ Q n j jk Optimal investment: � � ∂τ n � µδ I P n j Q n jk ≥ − , = if I jk > 0 jk jk ∂ I jk ���� n Building Cost � �� � Gain from Infrastructure Full problem is globally convex under strong enough congestion in transport
Key Efficiency Conditions on Every Link No-arbitrage condition on flows (“efficient road use”): ∂τ n P n jk k ≤ 1 + τ n Q n jk , = if Q n jk + jk > 0 P n ∂ Q n j jk Optimal investment: � � ∂τ n � µδ I P n j Q n jk ≥ − , = if I jk > 0 jk jk ∂ I jk ���� n Building Cost � �� � Gain from Infrastructure Full problem is globally convex under strong enough congestion in transport
Key Efficiency Conditions on Every Link No-arbitrage condition on flows (“efficient road use”): ∂τ n P n jk k ≤ 1 + τ n Q n jk , = if Q n jk + jk > 0 P n ∂ Q n j jk Optimal investment: � � ∂τ n � µδ I P n j Q n jk ≥ − , = if I jk > 0 jk jk ∂ I jk ���� n Building Cost � �� � Gain from Infrastructure Full problem is globally convex under strong enough congestion in transport
Example: Uniform Geography 20 randomly placed cities across uniform geography jk = δ 0 Distance δ 1 Building Cost: δ I jk
Example: Role of Geography Convex case � δ 2 δ CrossingRiver jk AlongRiver jk � jk = δ 0 Distance δ 1 Building Cost: δ I 1 + | ∆Elevation | jk δ 3 4 jk nonconvex
Application to European Countries In 25 European countries we observe: ◮ Road networks (EuroGeographics) ◮ Value Added (G-Econ 4.0) ◮ Population (GPW) Parametrization ◮ Each locations in the model = population centroid of 0.5 x 0.5 degree ( ∼ 50 km) cell ◮ Observed infrastructure I obs assigned to reproduce actual network map jk ◮ Each location produces one traded and one non-traded good ⋆ Productivity in traded sector and endowment of non-traded good calibrated to match value added and population � � β Q n ◮ Transport technology: τ n jk = δ τ jk , benchmark γ = β I γ jk jk ◮ Building costs δ I jk as function of distance and ruggedness from Collier et al. (2016), based on WB’s ROCKS.
Losses from Misallocation Simulate counterfactual optimal network for each country ◮ How much real income is lost in each country due to misplacement of existing roads? 20 RS 15 RS % Gain 10 LV LV GE GE IE FI 5 FI IE MD SK AT AT LT MK LT MD SK PT ES FR CHLI ES HU HU PT CZ CZ SI FR CY IT BE NL DK CHLI ND MK IT CY SI DE DE BE ND NL DK LU LU 0 7 8 9 10 11 Log Income Per Capita Fixed Labor Mobile Labor Linear regression slope (robust SE): Mobile Labor: -2.523 (1.269); Fixed Labor: -2.035 (.712) Average: 4.6% (fixed labor); 4.8% (mobile labor).
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