International Trade and Agglomeration in Asymmetric World: Core-Periphery Approach

Research Motivation Our Results/Contribution International Trade and Agglomeration in Asymmetric World: Core-Periphery Approach Alexander V. Sidorov Sobolev Institute of Mathematics, Novosibirsk, Russia St.Petersberg, September 8-9, 2011 Alexander V. Sidorov Asymmetry, Trade and Agglomeration

Preliminary Remarks Spatial economy aspects can t be explained by Perfect Competitive Market Model. Pointed out by R.Mundell (1957), formally proved by D. Starrett (1978) in Starrett s impossibility theorem: Consider an economy with a finite number of locations. If space is homogeneous, transport is costly, and preferences are locally nonsatiated, then there exists no competitive equilibrium involving the transport of goods between locations. Monopolistic-competition model of Dixit and Stiglitz (1977), accepted by P.R.Krugman (1979). The final form of Dixit-Stiglitz-Krugman model is Core-Periphery Model (CP-Model) (1991)

Model Two-sectoral economy: M-sector and A-sector. Two types of labor: industrial (mobile in long run) and agricultural (immobile) A-sector (Perfect Competition, PC): CRS, Homogeneous Good and Zero Transportation Costs M-sector (Monopolistic Competition, MC): Dixit-Stiglitz-type production, Variety of Goods, Iceberg-type Transportation Costs Two regions: Home and Foreign, consumer s preferences are identical for both countries and both types of workers. ( N ) 1 Utility function U = C µ M C 1 µ A, C M = c(s) ρ ρ ds, 0 < µ < 1, ρ = σ 1, σ > 1 elasticity of substitution, mass σ of firms N 0

Short Run and Long Run Equilibria Short Run Equilibrium (International Trade) Industrial labor supposed to be immobile in short run; Perfect Competitive equilibrium in A-sector; Monopolistic-competitive equilibrium in M-sector; Consumption equilibrium over all markets. Long Run Equilibrium (Agglomeration) Industrial labor flows into the region with higher Welfare In Long Run Equilibrium labor shares should be persistent Short Run Equilibrium conditions (except labor immobility)

Notions and Definitions (Exogenous) 0 < µ < 1 expenditure share of M-sector goods: µ = 0 purely PC-economy, µ = 1 purely MC-economy 0 < ρ = σ 1 < 1 CES-function parameter, 1 ρ = 1 σ σ consumer s taste for variety M-sector: f > 0 fixed costs, m marginal costs of labor, τ > 1 Iceberg-type transportation costs: to ship the unit of good we should produce τ > 1 units, as if τ 1 units melt in transit of one unit of the good, ϕ = τ 1 σ (0,1) trade freeness A-sector: fixed costs f a = 0, marginal costs of agricultural labor m a = 1 (normalization), transportation is costless τ a = 1 0 < θ < 1 Home share of immobile (agricultural) labor, 1 θ is Foreign share, W.L.o.G. assume θ 1/2, Symmetry Assumption: θ = 1/2 In Short Run only: 0 λ 1 Home share of industrial labor (immobile in short run), 1 λ is Foreign, λ = 0 and λ = 1 are total agglomeration cases

Notions and Definitions (Endogenous) In Long Run: 0 λ 1 equilibrium Home share of mobile (industrial) labor Equilibrium prices: p HH Home-produced and Home-consumed, p HF Home-produced and Foreign-consumed, p FH, p FF are defined analogously Equilibrium agricultural wage: in fact, w a = 1 for both countries, due to perfect competition and costless transportation Equilibrium industrial wages: w H Home wage, w F Foreign wage Equilibrium firm s masses N H and N F under Free Entry Assumption

Struggling Forces Traditional point of view: agglomeration, or centripetal, forces ( market access effect and cost of living effect ) struggle against dispersion, or centrifugal ones ( market crowding effect ) Two proving grounds with clear outcome: Perfect competition: resultant force is centrifugal Monopolistic competition: resultant force is centripetal So in fact: monopolistic competition struggles against perfect one! In CP model natural correlation of forces measure is expenditure share µ µ = 1 pure monopolistic competition, µ = 0 pure perfect competition

In the World of Symmetry (θ = 1/2) Short Run Equilibrium: Existence and Uniqueness analytically proved by P.Mossay (2006) Long Run Equilibria: F.Robert-Nicoud(2005) Tomahawk diagram (taken from Baldwin et al. (2003)) ϕ = τ 1 σ (0,1) trade freeness, s n = λ the Home share of industrial labor

Troubles with Asymmetry (θ 1/2) Baldwin et al. (2003), Economic Geography and Public Policy:...intense intractability of the CP model means that numerical simulation... is the only way forward. Numerical simulations: Broken Tomahawk for θ < 1/2, taken from Baldwin et al. (2003). Sustain points and break point can be in various relative positions. Berliant and Kung (2009):...conclusions drawn from the use of this bifurcation to generate a core-periphery pattern are not robust. Generically, this class of bifurcations is a myth, an urban legend.

Guys, You re Wrong! Theorem Asymmetric CP model is not intractable! Numerical simulation is useful but not the only way forward. Theorem Though bifurcation is a myth, the most of conclusions are robust under moderate asymmetry. Changes are considerable only under sufficiently large asymmetry. Theorem Asymmetry even simplifies equilibrium structure.

Short Run Equilibrium Moreover p HH = w H m σ σ 1, p HF = w H τ m σ, σ 1 p FH = w F m τ σ, p FF = w F m σ σ 1 σ 1 N H = λl f σ, N (1 λ)l F = f σ (1 µ)(w H λl + w F (1 λ)l + 1 L a ) = 1 L a w H = µ 1 µ L a wh w F ( L λ w H w F + (1 λ) ), w F = w H wf w H

Stumbling Block It is sufficient to find equilibrium relative wage w H w F, as a root of equation: ( 1 ((1 θ) + µθ)(1 ϕ 2 ) ) ( ) σ 1 wh ϕ w F ( ) σ wh (1 ((1 µ)θ + µ)(1 ϕ 2 )) ϕ Is it intractable? Not at all! w F ( ) 2σ 1 wh w F = λ 1 λ Existence and Uniqueness (generalization of P.Mossay (2006)): There exists the unique positive root of this equation: ϕ 1 σ < w H w F < ϕ 1 σ.

Implicit Function Technic Relative wage is implicit function of all exogeneous parameters w H = x(θ,λ,ϕ, µ,σ) defined by equation F (x;θ,λ,ϕ, µσ) = 0. w F Comparative statics of relative wage w.r.t. π {θ,λ,ϕ, µ,σ}: ( ) F x (θ,λ,ϕ, µ,σ) = π π ( F x ) w H (θ,λ,ϕ, µσ) = µ 1 µ L a x(θ,λ,ϕ, µ,σ) L(λ x(θ,λ,ϕ, µ,σ) + (1 λ)) and so on.

Comparative Statics of Relative Wage Relative wage w H w F increases w.r.t. agricultural labor share θ is monotone w.r.t. industrial labor share λ. Type of monotonicity depends on parameter s relation increases when ϕ > ( 1 ) α2 2 µ > 1+µ α 2 decreases when ϕ < 1 µ 1 ) α2 2 µ < α 2 ( 1+µ 1 µ 1 (1 ϕ 2 )α 2 ϕ 1 (1 ϕ 2 )α 2 +ϕ 1 (1 ϕ 2 )α 2 ϕ 1 (1 ϕ 2 )α 2 +ϕ where α = 2θ 1 = θ (1 θ) relative measure of asymmetry in agricultural labor (NB: θ 1/2!)

Price Index and Agricultural Welfare Consumer s price indices (PI) for both regions ( NH NF ) 1 P H = phh 1 σ (j)dj + pfh 1 σ (j)dj 1 σ 0 0 ( NH P F = 0 NF ) 1 phf 1 σ (j)dj + pff 1 σ (j)dj 1 σ 0 Moreover, agricultural consumer welfares VH a = w H a (P H ) µ = 1 (P H ) µ, VF a = w F a (P F ) µ = 1 (P F ) µ, because in equilibrium w H a = w F a = 1

Comparative Statics of Price Indices Price indices Home CPI P H increases w.r.t agricultural labor share θ, while foreign P F decreases, thus relative price index (RPI) P H P F increases w.r.t. θ RPI P H P F Agricultural welfare decreases w.r.t. industrial labor share λ Home agricultural welfare V a H decreases w.r.t agricultural labor share θ, while foreign V a F increases, thus relative welfare V a H V a F decreases w.r.t. θ Relative agricultural welfare V a H labor share λ V a F increases w.r.t. industrial

Industrial Welfare Welfare (equilibrium utility) for industrial workers V H = V F = w F (P F ) µ, relative welfare V H V F = w H w F ( ) µ PF P H w H (P H ) µ, V H and V H /V F increase w.r.t. agricultural labor share θ Let µ ρ (Black Hole), then relative welfare V H w.r.t. industrial labor share λ V F increases Let µ < ρ, then relative welfare V H increases w.r.t. industrial V F labor share λ for ϕ ϕ B = ρ µ 1 ) α2 ρ + µ 2, where α 2 ( 1+µ 1 µ α = 2θ 1 Let µ < ρ and ϕ < ϕ B, then derivative V H (0) > 0 and λ V F changes sign not more than twice for λ (0,1)

Patterns of Relative Welfare Change θ = 0.55, µ = 0.2, ρ = 0.25

Long Run Equilibria Ad hoc dynamics: or the same λ = λ [V H (λ) (λ V H (λ) + (1 λ)v F (λ))] λ = λ (1 λ) [V H (λ) V F (λ)] Agglomerated equilibria: λ 0 = 1 (Core) or λ 0 = 0 (Periphery) Interior equilibria: V H (λ 0 ) = V F (λ 0 ), 0 < λ 0 < 1. Main questions: Stability of agglomerated equilibria Number and stability of interior equilibria

Geographic Desciption of Long Run Equilibria There is no need of any differential equation.

It Is Sufficient to Know: V H (0) < 1 ϕ < ϕ0 S V, V H (1) > 1 ϕ > ϕ1 S below or F V F above sea level, sustain points ϕ0 S, ϕs 1 are the smalllest roots of equation θ(1 µ)ϕ µ ρ ρ (1 θ)(1 µ)ϕ µ ρ ρ + (1 θ(1 µ))ϕ µ+ρ ρ = 1 + (1 (1 θ)(1 µ))ϕ µ+ρ ρ = 1 positions of Height and Cavity (It is sufficient to solve some quadratic equation) break point ϕ B = ρ µ ρ + µ ( 1+µ 1 µ 1 (2θ 1)2 ) 2 (2θ 1) 2 and one more critical value turn point { } ϕ T (1 θ)(θρ (1 + θρ)µ) = max 0, θ((1 θ)ρ + (1 + θρ)µ)

Top View: Black Hole Black Hole Condition: µ ρ Contour plot (map) of V H V F (ϕ,λ) under µ = 0.5, ρ = 0.25 θ = 0.6 Legend Sea : V H < V F Shore : V H > V F Coastline : V H = V F

Old True Tomahawk

Hooked Coastline: ϕ S 1 > ϕt Narrow Isthmus: ϕ S 0 0.277 < ϕu 0.29 µ = 0.25, ρ = 0.75, θ = 0.5025 Wide Isthmus: ϕ S 0 0.3 > ϕu 0.2535 µ = 0.25, ρ = 0.75, θ = 0.525

U-Turn Point ϕ U ϕ U 0.29, λ U 0.7 ϕ U 0.2535, λ U 0.9

Regular Coastline ϕ S 1 ϕt Regular northern coastline: µ = 0.25, ρ = 0.75, θ = 0.560885, ϕ T = ϕ S 1 0.216766 More regular: µ = 0.25, ρ = 0.75, θ = 0.6, ϕ T 0.238 > ϕ S 1 0.184

Northern Frontier

Big (Continental) Asymmetry: θ ρ+µ 2(1 µ)ρ µ = 0.25, ρ = 0.75, θ = 0.89, ϕ S 0 = 1

Catastrophic and Step-by-Step Agglomeration/Deindustrialization Catastrophe : ϕ S 1 > ϕt Step-by-step: ϕ S 1 ϕt

That s All, Folks! Thank You for Attention! Any questions?