Optimal Transport Networks in Spatial Equilibrium

Transcription

1 Optimal Transport Networks in Spatial Equilibrium Pablo D. Fajgelbaum Edouard Schaal UCLA/NBER, CREI/CEPR World Bank Space and Productivity Conference, 09/2017

2 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

3 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?

4 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)

5 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 can flow through neighbors only 3 Per-unit cost τ n of shipping from j to a neighbor k Decreasing returns to transport: τ n increases with quantity shipped Positive returns to infrastructure: τ n decreases with infrastructure I 4 Building infrastructure I takes up δ I I units of a scarce resource ( asphalt ) δ I may vary across links, due to ruggedness, distance... The transport network is defined by {I }

6 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 can flow through neighbors only 3 Per-unit cost τ n of shipping from j to a neighbor k Decreasing returns to transport: τ n increases with quantity shipped Positive returns to infrastructure: τ n decreases with infrastructure I 4 Building infrastructure I takes up δ I I units of a scarce resource ( asphalt ) δ I may vary across links, due to ruggedness, distance... The transport network is defined by {I }

7 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 can flow through neighbors only 3 Per-unit cost τ n of shipping from j to a neighbor k Decreasing returns to transport: τ n increases with quantity shipped Positive returns to infrastructure: τ n decreases with infrastructure I 4 Building infrastructure I takes up δ I I units of a scarce resource ( asphalt ) δ I may vary across links, due to ruggedness, distance... The transport network is defined by {I }

8 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 can flow through neighbors only 3 Per-unit cost τ n of shipping from j to a neighbor k Decreasing returns to transport: τ n increases with quantity shipped Positive returns to infrastructure: τ n decreases with infrastructure I 4 Building infrastructure I takes up δ I I units of a scarce resource ( asphalt ) δ I may vary across links, due to ruggedness, distance... The transport network is defined by {I }

9 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

10 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 } to maximize aggregate welfare subject to: Location and consumption decisions by workers given { } I Production and trade decisions of firms given { } I Goods and factors market clearing constraints Availability of inputs to build the network detail

11 Key Efficiency Conditions on Every Link No-arbitrage condition on flows ( efficient road use ): Pk n Pj n 1 + τ n + τ n Q n Q n, = if Qn > 0 Optimal investment: µδ I ( P n j }{{} Qn τ n ), = if I > 0 I n Building Cost }{{} Gain from Infrastructure Full problem is globally convex under strong enough congestion in transport

12 Key Efficiency Conditions on Every Link No-arbitrage condition on flows ( efficient road use ): Pk n Pj n 1 + τ n + τ n Q n Q n, = if Qn > 0 Optimal investment: µδ I ( P n j }{{} Qn τ n ), = if I > 0 I n Building Cost }{{} Gain from Infrastructure Full problem is globally convex under strong enough congestion in transport

13 Key Efficiency Conditions on Every Link No-arbitrage condition on flows ( efficient road use ): Pk n Pj n 1 + τ n + τ n Q n Q n, = if Qn > 0 Optimal investment: µδ I ( P n j }{{} Qn τ n ), = if I > 0 I n Building Cost }{{} Gain from Infrastructure Full problem is globally convex under strong enough congestion in transport

14 Example: Uniform Geography 20 randomly placed cities across uniform geography Building Cost: δ I = δ 0Distance δ 1

15 Example: Role of Geography Convex case Building Cost: δ I = δ 0Distance δ 1 ( ) δ2 CrossingRiver 1 + Elevation δ 3 δ AlongRiver 4 nonconvex

16 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 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 Transport technology: τ n = δτ ( Q n I γ, benchmark γ = β Building costs δ I as function of distance and ruggedness from Collier et al. (2016), based on WB s ROCKS.

17 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? % Gain RS RS MD MD GE MK MK LV LV IE FI FI SK IE AT LTSK HU PT PT ES FR CHLI CZ CZ CYSI FR ND ITBE DK NL CY IT CHLI SI ND DE BE DK NL Log Income Per Capita Fixed Labor Mobile Labor Linear regression slope (robust SE): Mobile Labor: (1.269); Fixed Labor: (.712) LU Average: 4.6% (fixed labor); 4.8% (mobile labor).

18 Where should infrastructure be placed? Back of the envelope for how optimal investments should be undertaken Simulate 50% optimal expansion of each network Regress optimal infrastructure growth on characteristics of each cell Investment Population 0.104*** Income per Capita Consumption per Capita 0.179*** Infrastructure *** Differentiated Producer 0.133*** R These observables explain 30% of spatial distribution of optimal investments Rest is geography and pre-existing links

19 Regional Effects (Spain) Optimal Expansion reallocation

20 Which regions grow? Dependent variable: Employment growth in the counterfactual Reallocation Expansion Population Income per Capita Consumption per Capita *** *** Infrastructure *** Infrastructure Growth 0.013* 0.032** Differentiated Producer 0.013** 0.023*** R These regressions pool the outcomes across all locations in the convex case.

21 Conclusion We developed a framework to study optimal transport networks 1 Neoclassical model (with labor mobility) on a graph 2 Optimal Transport Flows with congestion 3 Optimal Network design Forces not (yet) included: Indirect effects through further investments (e.g., in building structures) Other factors (land, labor) in the infrastructure construction cost Investments in trade hubs Optimal network investments around second best (e.g., distortions) Agglomeration and spillovers in production Dynamics Potential applications for future work Optimal urban network International trade facilitation Developing countries Political economy and competing planners Instruments for location of infrastructure

22 Planner s Problem Definition The planner s problem with labor mobility is W = max I subject to (i) availability of traded and non-traded goods, (ii) the balanced-flows constraint, C n j + Q n + τ n ( k N (j) (iii) free labor mobility, ( Q n, I (iv) the network building constraint, with bounds I I I ; and max max u C j,l j,... Q n c j L j C T j (C j ) and h j L j H j for all j; ) ) Q n ) = F n j (L n j,... + Q n ij for all j, n; (Multiplier: P n j ) i N (j) L j u L j U (c j, h j ) for all j; j k N (j) (v) factor market clearing and non-negativity constraints. δ I I = K back

23 Example: Role of Geography Non-convex case Building Cost: δ I = δ 0Distance δ 1 ( ) δ2 CrossingRiver 1 + Elevation δ 3 δ AlongRiver 4 back

24 Example: Spain (a) Underlying Graph (b) Actual Road Network (c) Measured Infrastructure I obs back

25 Regional Effects (Spain) Optimal Reallocation back