Optimal Transport Networks in Spatial Equilibrium [ Preliminary ]

Transcription

1 Optimal Transport Networks in Spatial Equilibrium [ Preliminary ] Pablo D. Fajgelbaum Edouard Schaal UCLA and NBER, CREI October 2016

2 Motivation Trade costs are ubiquitous in international trade and spatial economics Crucial to rationalize differences across regions in Prices Trade flows Real incomes Central questions involve counterfactuals with respect to trade costs, e.g.: Gains from trade (autarky counterfactual, partial trade liberalization) Effects of transport infrastructure Quality and shape of transport networks are important components of trade costs ( IDB (2003), World Bank (2001, 2009) ) Virtually all theoretical and empirical analyses treat the transport network as exogenous Many questions involve endogeneizing the transport network

3 This Project Develop a framework to study globally optimal transport networks in spatial general equilibrium 1 Embed an optimal transport problem into a neoclassical environment (with labor mobility) with arbitrary many regions on a graph 2 Solve for the globally optimal transport network subject to a network-building technology Potential Applications What would be the aggregate and regional effects of the optimal network, especially in developing countries? How does the optimal network diffuse local shocks? How do transport networks and regional inequalities evolve along the development process? How to build instruments for the spatial distribution of investments in infrastructure? Political economy issues: competing planners?

4 Challenges Optimal network problem is typically 1 highly dimensional (space of all networks); and 2 non-convex (complementarities between network investments)

5 Key Features 1 Dealing with high dimension Follow a social-planner approach Rather than optimizing over the network in the competitive equilibrium, solve one optimization over both the network and the allocation But welfare theorems hold (neoclassical model) Convexify the problem through congestion in transport and continuous infrastructure investments (not 0/1) FOC s: optimal investment in link i j is function of shadow values in i and j only Sidesteps searching over the (very large) space of all networks Dimension of the optimization reduced to dimension of the allocation (# of goods/factors) 2 Dealing with non-convexity Convexify the objective function through congestion in transport Problem is convex if congestion is stronger than the returns to network investments sufficiency of first-order conditions Guarantees convergence of efficient gradient-descent based algorithms In the absence of this condition There may be multiple local maxima, but Karush-Kuhn-Tucker conditions are still necessary We implement these cases combining duality/foc approach with simulated annealing These properties hold regardless of the generality of the (neoclassical) model.

6 Literature Background Impact of new transport technologies or expansion of transport networks Feyrer (2009), Donaldson (2012), Pascali (2014), Faber (2015), Storeygard (2015), Donaldson and Hornbeck (2016), Quantitative frameworks allowing counterfactuals w.r.t trade costs in international trade Eaton and Kortum (2002), Anderson and van Wincoop (2003) Recently, extended to spatial setups Allen and Arkolakis (2014a), Redding (2016), Caliendo et al. (2014), Fajgelbaum et al. (2016), Ramondo et al. (2016),... Infrastructure investments in gravity models with re-optimization of least-cost routes Counterfactuals with respect to transport infrastructure Allen and Arkolakis (2014a), Alder (2016), Redding (2016), Ahlfeldt, Redding and Sturm (2016), Sotelo (2016) Optimal local changes to existing transport infrastructure Allen and Arkolakis (2014b), Allen and Arkolakis (2016) Optimal infrastructure investment with competing planners on the line Felbermayr and Tarasov (2015) Here: global optimization over transport networks in general neoclassical environment Includes: factor endowments, Ricardian, Armington,... and standard urban setups, e.g., Rosen-Roback In such environment, computing least-cost routes is not sufficient Similar to optimal-transport problems studied by Monge (1781) and Kantorovich (1942), and more recently surveyed by Villani (2003) and Galichon (2016) OT literature does not embed optimal-transport in general equilibrium and does not optimize over network Here, optimal-transport and general-equilibrium interrelated

7 Plan for Today 1 Set up the general model Physical environment and planner s problem Market allocation and equivalence with planner s allocation 2 Computational Aspects 3 Illustrations Comparative statics with one good Multiple goods, fixed vs. mobile labor Role of geographical features Introduction of new transport technologies

8 Preferences J = {1,..., J} locations N traded goods (Cj n) aggregated into C j 1 non-traded (H j ) in fixed supply (can make it variable) L j workers located in j (fixed or mobile) Homothetic utility in j, where U (C j, H j ) C j = C T j ({ C n j }) Cj n Cj T is the quantity of the traded good n consumed in j is homogeneous of degree 1 and concave A convenient (not necessary) functional form: C T j = ( N n=1 ζ n j ) σ ( C n) σ 1 σ 1 σ j with σ > 1.

9 Production Neoclassical environment M primary factors in fixed supply in region j, stacked in V j Labor (fixed or mobile) Output of sector n in location j is: Special cases Y n j Endowment economy: F j ( ) = 1 Armington N = J (as many sectors as regions) = zj n ( F j L n j, V j n ) z n j = z j for n = j, and z n j = 0 otherwise (each good produced in one region only) ( ) Ricardian model: F j L n j, = L n j Specific-factors, Heckscher-Ohlin... Rosen-Roback

10 Underlying Geography Locations are arranged on an undirected graph (I, E), e.g.: J = {1,..., J} is the set of nodes, E the set of edges (unordered pairs of J ) Each location j has a set N (j) of neighbors (directly connected) Trade can only take place between connected locations But neighbors do not have to be geographically contiguous Fully connected case is nested: N (j) = J for all j J

11 Transport Network and Trade Costs Because of dimensionality, optimizing over networks is usually intractable We approach the problem by convexifying the network Per-unit cost of shipping Q n jk units of commodity n from j to k N (j): ( ) τ Q n jk, κ jk τ Congestion in transport: Q > 0 κ jk [0, ] is the road capacity (e.g., number of lanes) τ κ < 0, τ (Q, 0) = (no road), τ (Q, ) = 0 (free transport) Nominated in units of the good being shipped (iceberg) Can encompass other cases, e.g.: nominated in final goods, or other factors The transport network is defined by the set of all capacities {κ jk } j,k N (k) A convenient (not necessary) parametrization: τ (Q, κ) = 1 κ Qβ β 0 measures congestion in transport

12 Balanced Flows Constraint We must impose balance of flows of good n in every location j: C n j + k N(j) ( Q n jk + τ ( Qjk, n ) κ jk Q n ) jk } {{ } Consumption + Exports to Neighbors Y n j + i N(j) Q n ij }{{} Production + Imports from Neighbors Qjk n is the quantity of the good n shipped from j to k N (j) Let Pj n be the multiplier of his constraint Potential in optimal-transport literature It will be the price of the commodity n in location j in the decentralized equilibrium Optimal-transport and general-equilibrium problems will be interrelated

13 Technology to Build The Transport Network Technology to build capacity κ jk along link j k: κ jk = G jk ( Ijk ) Building along some links may be more difficult than along others The input I jk is freely mobile and available in fixed supply Can be generalized to using labor or other factors This assumption allows for separation between the optimal flow and network design problem A convenient (not necessary) parametrization: ( ) I γ G jk (I ) = δ jk δ jk is a measure of how difficult is to build (e.g., due to geographic accidents) γ measures the returns to network building (IRS γ > 1) Network-building constraint: I jk K j k N (j)

14 Planner s Problem without Labor Mobility Definition The planner s problem without labor mobility is subject to W = max I jk max C j n,ln j, V j n max Q n jk ω j L j U (c j, h j ) (i) availability of traded and non-traded commodities, ({ }) c j L j C T j C n j for all j, h j L j H j for all j; (ii) the balanced-flows constraint, C n j + ( ) ) ( ) (Q n jk + τ Q n jk, G jk (I jk ) Q n jk z n j F L n j, V n j + Q n ij for all j, n; k N (j) i N (j) (iii) the network-building constraint, (iv) local labor-market clearing, j k N (j) j I jk K; L n j L j for all j n and local factor market clearing for the{ remaining } factors; and (v) non-negativity constraints on flows Q n jk and network investment {I jk }

15 Planner s Problem with Labor Mobility Definition The planner s problem with labor mobility is subject to (i)-(v) above; (vi) free labor mobility, (vii) aggregate labor-market clearing, max I jk max C j n,ln j, V j n,l j max u Q jk n L j u = L j U (c j, h j ) for all j; L j L; and j

16 Sub-Problems in the Planner s Problem Given weights {ω j }, the planner solves three related problems: W = max max max ω I jk C j n,y j n Q jk n j L j u j j 1 Optimal transport: Choose the gross flows Q n jk through the network Related to optimal transport problems surveyed by Villani (2003), Galichon (2016) Duality results ensure existence of multipliers and allows for tractable resolution Key differences with that literature { } Here, Y n j, C n j are endogenous (OT problem takes them as given) Here, we must define the optimal route (OT problem typically studies coupling of sources to destinations, but see Galichon (2016) ch.8) In gravity models, optimal transport simplifies to least-cost-route, solved independently from general-equilibrium Key assumption in that literature: each commodity is produced in only one location or technologies (including transport) are linear { } Here, many sources can serve each market, interacting with endogenous Y n j, C n j { } 2 Neoclassical allocation: Choose C n j, Y n j (and {L j } if labor is mobile) Standard small-open economy allocation of consumption and output given goods prices (multipliers) subject to the PPF 3 Optimal network: Choose the network investments {I jk } j,k N (j) Some heuristical approaches in operational research, but few general results

17 Convexity Proposition Given the network investments { I jk }, the optimal transport-cum-neoclassical allocation problem is a convex optimization problem if Qτ (Q, ) is convex. The full planner s problem with fixed labor (resp. mobile labor) is a convex (resp. quasiconvex) optimization problem if Qτ ( Q, G jk (I ) ) is convex in Q and I for all j and k N (j). Key implications Sufficiency of first-order conditions Guarantees convergence of efficient gradient-descent based algorithms What if this condition does not hold? A constraint qualification ensures that Karush-Kuhn-Tucker conditions are still necessary, but no longer sufficient Local vs. global optimality

18 Sufficient Condition in the Log-Linear Parametrization When the shipping congestion is given by τ (Q, κ) = 1 κ Qβ (1) and network-building is given by κ jk = ( Ijk δ jk ) γ (2) then convexity is ensured by β γ Intuition: increase in marginal shipping costs resulting from higher traffic along a link must offset the reduction in marginal shipping cost resulting from capacity investments along that link

19 Properties of the Optimum 1 Optimal-flows problem (given {κ jk }) Flows between connected locations are function of the differential in multipliers 1 Q n jk = max κ jk Pn β k β + 1 P j n 1, 0 Pj n is the Lagrange multiplier of flow constraint of good n in j (price of n in j in decentralized equilibrium) { } 2 Network-design problem (given Q n jk ) Road capacity increases with gross flows: ( γ κ jk = P n j µδ jk n ) γ ( Q n ) β+1 1+γ jk µ is the multiplier of the network-building constraint ( price of asphalt ) Capacity decreases with difficulty to build δ jk κ jk = 0 is an outcome if only if prices are identical in j and k Only a function of prices at j and k! Reduces dimensionality Globally optimal network emerges by combining the optimization conditions in every link Similar properties hold without imposing the specific functional forms (1) and (2)

20 Decentralization given the Network Decentralization of the optimal transport and neoclassical allocation problems given the network Some historical examples of transport networks built by private companies, e.g., railways companies in 19th century, but those examples remain few In the decentralization, the network is taken as given, constructed by a benevolent government Shipping companies trading between locations minimize shipping costs

21 Decentralization given the Network General neoclassical economy J regions, N sectors, M + 1 factors (including labor) Competitive equilibrium Profits made by an individual company shipping good n from j to k are: [ ( ) ] π n jk = max p n q jk n k 1 t n jk p n j (1 + τ jk ) q n jk, t jk : sales tax to correct for congestion externality Free entry: π n jk = 0 Tax revenue equals rents from shipping sector Expenditure per capita in region j: where e j = W j + b j Π Π = p H j H j + r l j V l j + t n jk pn k qn jk j j l j k n }{{}}{{}}{{} Non-Traded Good All other factors Rents from Shipping Sector

22 Welfare Theorems Proposition If the sales tax on shipments of product n from j to k is then: 1 tjk n = 1 + τjk n ( ) ), 1 + (ε jk Qjk n, κ jk + 1 τjk n (i) If labor is immobile, the competitive allocation is efficient under specific weights ω j and the planner s allocation can be implemented by the market with specific transfers b j. (ii) If labor is perfectly mobile, the competitive allocation is efficient if and only if b j = 1 L (equal ownership). In either case, the price of good n in location j, pj n, equals the multiplier on the balanced-flows constraint in the planner s allocation, Pj n.

23 Computation: Convex Case The convex case β γ is well-behaved but still high-dimensional. At worst: N J 2 shipping quantities Qjk n, J 2 variables κ jk (fully connected graph) {{ } } (N + 1) J aggregate consumption levels Cj n j, n ( j { } ) N J + 1 Lagrange multipliers λ n j, j,n N L employment levels L n j with labor mobility FOCs: intractable nonlinear system of many variables Convexity however ensures that efficient gradient-descent methods converge Powerful free/open-source convex solvers based on interior-point methods such as IPOPT, CVXOPT, GAMS... can handle very large systems Exploit sparsity of underlying geography to speed up calculations

24 Computation: Convex Case Primal approach { C n j }, sup {inf } { } Q jk n,{κ jk} λ n j,µ L ({ Cj n } {, Q n } jk, {κjk }, { Pj n } ), µ Convergence is guaranteed, may compute gradient/hessian by hand But slow for systems with many locations/products Dual approach {inf } λ n j {,µ C n j }, sup { } Q jk n,{κ jk} L ({ Cj n } {, Q n } jk, {κjk }, { Pj n } ), µ Preferred approach, often used { in } optimal { } transport problems Use FOCs and substitute for Cj n, Qjk n, { } κ jk, then minimize over Lagrange multipliers The dual problem is always a convex minimization ({ } ) problem, guaranteed to converge Optimization over N J + 1 variables λ n j, µ without constraints In practice, takes a fraction of a second and 7 iterations for hundreds of locations

25 Computation: Nonconvex Case A qualitative jump occurs when degree of increasing returns γ exceed congestion β Proposition Computations suggest the network becomes extremely sparse as the planner prefers to concentrate flows on wide highways In the one-traded good case and γ > β, the optimal transport network is a disjoint union of trees. A tree is a connected graph with no cycles (loops) Cycles cannot be optimal because it is always strictly better to delete one edge to concentrate flows on the other

26 Computation: Nonconvex Case The case with γ > β is a nonconvex optimization problem Many local maxima ( lock in?) Some global resolution methods, but computationally intensive Our approach: Solve the optimal flow/allocation given the network using duality (convex subproblem) ( ) γ Q 1+β γ γ+1 jk Iterate on the network FOCs: κ jk = µ δ jk Always converges in practice, but towards a local optimum We thus combine this duality/foc approach with simulated annealing Simulated annealing is an easy-to-implement heuristical approach Often used for traveling salesman problem Improves results, but no guarantee to converge to global maximum In the future: branch-and-bound methods provide arbitrarily close approximation to the global optimum

27 Illustrations 1 Simple example: one good, regular geometry 1 Comparative statics over K 2 Optimal network vs. rescaling 3 Lock-in 4 Random cities across space 2 Multiple goods 1 Fixed labor: convex vs. nonconvex 2 Mobile labor 3 Ricardian example 3 Geographical features and new transport technologies

28 Simple Example Parameters α = 0.5 γ = 1 (CRS in network-building) β = 1 (some congestion) Fundamentals 13x13 square network Homogeneous ( ) Li, H i, δ ij = (1, 1, 1) Productivity Z i = 0.1 everywhere but the center Z i = 1 at the center

29 Optimal Network: Simple Example K=1

30 Optimal Network: Simple Example K=100

31 Optimal Network: Simple Example K=100000

32 Optimal Network: Simple Example Optimal spatial distribution of prices and consumption as K grows 1.15 Price (rel. to center) 0.15 Stdev log prices 1.1 K=10 K=100 K= dist. to center log 10 K 1 C i (rel. to center) 0.15 Stdev log cons K=10 K=100 K= dist. to center log 10 K

33 Optimal Network: Simple Example Aggregate welfare as K grows: optimal vs. uniform improvement Welfare (cons. equivalent) Optimal Rescaled log 10 K

34 Inefficient Network Lock-in Optimal network vs. building on existing network Network initially centered on city in SW and emergence of a new city in NE

35 Inefficient Network Lock-in Welfare gains of moving from lock-in to optimal network 1.04 Gain from moving to the optimal network Welfare (cons. equivalent) Productivity of new city

36 Random Cities Across Space, Convex Case 20 random cities, central node more productive Convex case β = 1 = γ

37 Random Cities Across Space, Non-Convex Case 20 random cities, central node more productive Nonconvex case γ = 2 > β = 1

38 Random Cities Across Space, Non-Convex Case 20 random cities, central node more productive Using annealing to refine the solution

39 Illustrations 1 Simple example: one good, regular geometry 1 Comparative statics over K 2 Optimal network vs. rescaling 3 Lock-in 4 Random cities across space 2 Multiple goods 1 Fixed labor: convex vs. nonconvex 2 Mobile labor 3 Ricardian example 3 Geographical features and new transport technologies

40 Multiple Goods and Fixed Labor: Convex Case 11 goods, CES aggregator with σ = 2, convex case (β = γ = 1) 1 agricultural good may be produced everywhere 10 industrial goods, each produced in one location only Upper left panel: circle size = population Other panels: circle size = production

41 Multiple Goods and Fixed Labor: Nonconvex Case 11 goods, CES aggregator with σ = 2, nonconvex case (β = 1 < γ = 2) 1 agricultural good may be produced everywhere 10 industrial goods, each produced in one location only Upper left panel: circle size = population Other panels: circle size = production

42 Multiple Goods and Labor Mobility: Convex Case 11 goods, CES aggregator with σ = 2, convex case (β = γ = 1) 1 agricultural good may be produced everywhere 10 industrial goods, each produced in one random location Upper left panel: circle size = population Other panels: circle size = production

43 Multiple Goods and Labor Mobility: Nonconvex Case 11 goods, CES aggregator with σ = 2, nonconvex case (β = 1 < γ = 2) 1 agricultural good may be produced everywhere 10 industrial goods, each produced in one random location Upper left panel: circle size = population Other panels: circle size = production

44 Ricardian Example: Convex Case, Fixed Labor 2 sectors, productivity peaks in center for sector 1, flat in 2nd sector convex case (β = γ = 1)

45 Ricardian Example: Convex Case, Mobile Labor 2 sectors, productivity peaks in center for sector 1, flat in 2nd sector convex case (β = γ = 1)

46 Illustrations 1 Simple example: one good, regular geometry 1 Comparative statics over K 2 Optimal network vs. rescaling 3 Lock-in 4 Random cities across space 2 Multiple goods 1 Fixed labor: convex vs. nonconvex 2 Mobile labor 3 Ricardian example 3 Geographical features and new transport technologies

47 Role of Geographical Features Take into account geographical features like mountains, rivers, etc. Recall the network-building technology: i j N (i) 1 γ δ ij κij K δ ij captures the costs of building a route between i and j

48 Role of Geographical Features 20 random cities across uniform geography, β γ Building costs: δij = δ0 + δ1 Euclidean Distanceij

49 Role of Geographical Features Adding a (Gaussian) mountain Building costs: δ ij = δ 0 + δ 1 Euclidean Distance ij

50 Role of Geographical Features Adding a river and allowing for bridges Building costs: δ ij = δ 0 + δ 1 Euclidean Distance ij + δ 2 CrossingRiver ij

51 Role of Geographical Features Allowing for water transport Building costs: δ ij = δ 0 + δ 1 Euclidean Distance ij + δ 2 CrossingRiver ij δ 3 AlongRiver ij

52 Role of Geographical Features Non-convex case (after annealing) Building costs: δ ij = δ 0 + δ 1 Euclidean Distance ij + δ 2 CrossingRiver ij δ 3 AlongRiver ij

53 New Transport Technology and City Sizes Initial geography with strong dependence on water transport Traveling along the river is cheap, ground transport expensive Optimal network leads to a large city at the river crossing

54 New Transport Technology and City Sizes Improvement in ground transport technology Traveling over land increases (e.g., railways) Optimal network leads to shrinking of the city at the river crossing

55 Conclusion and Road Ahead We develop and implement a new framework to compute globally optimal transportation networks in spatial neoclassical models (with labor mobility) In progress Strong duality results from optimal transport theory beyond convex case We have many applications in mind What would be the aggregate and regional effects of the optimal network, especially in developing countries? How does the optimal network diffuse local shocks? How do transport networks and regional inequalities evolve along the development process? How to build instruments for the spatial distribution of investments in infrastructure? Political economy issues: competing planners? How to introduce other transport technologies like air transport?