On the dynamics of the Heckscher-Ohlin theory

Transcription

1 On the dynamics of the Heckscher-Ohlin theory Lorenzo Caliendo The University of Chicago 2010

2 Introduction "...free commodity trade will, under certain speci ed conditions, inevitably lead to complete factor-price equalisation.." Samuelson 1949 The factor price equalization (FPE) hypothesis states that if countries have access to the same technology, then free trade must lead to complete FPE. Although general as it seams this hypothesis, it was considered in a static setup in which countries open up to trade in a given moment of time. However, will this result hold if we allow countries to accumulate factors? How International trade in goods a ects factor accumulation and prices?

3 Introduction The factor proportions theory is one of the most in uential theories of international trade. When factors are capital and labor: Heckscher-Ohlin theory. Static, Factor Price Equalization (FPE), small open economy, constant savings. This paper extends the 2x2x2 Heckscher-Ohlin model to a dynamic setup. Endogenous savings - two large economies.

4 Findings Initial conditions matter. Specialization is the most likely outcome. While a small country can grow without the retarding force of a terms of trade deterioration, this is not the case for a large country. No FPE across trading partners. Davis and Weinstein (2001, 2003) Schott (2003), Debare and Demiroglu (2003) and Cuñat (2000). Patern of specialization. Imbs and Wacziarg (2003). Methodologically, the model is stable and several aplications can be considered.. more on this later.

5 Literature Oniki - Uzawa (1965), Bardhan (1965a, 1965b, 1966) Stiglitz (1970), Deardor (1973) Eaton (1987), Mainford (1998) Chen (1992), Baxter (1992) Ventura J. (1997), Cuñat and Ma ezzoli (2004) Atkenson Kehoe (2000), Bajona and Kehoe (2006a) (2006b)

6 Environment 2 Countries i = fn, Sg 2 Tradable goods yj i Factors of production Kj i and L i j. 1 Final good Y i - Not tradable x i j the demand of intermediate inputs 1 and 2. Identical Cobb Douglas Technologies j = f1, 2g 2 Immobile c i and i i the consumption and investment of the household. Prices: p j the price of the intermediate inputs and r i and w i the rental price of capital and the wage rate. We normalize the price of the nal good to 1. Trade is balanced date by date j All variables per capita, intensive form. Assume that k N (0) > k S (0) p j xj i = p j yj i j

7 Neoclassical Growth Model Household- rm in each country The problem they solve is the following: Z max exp ( 0 ρt) log c i (t) dt subject to: c i (t) + i i (t) = G k i ; p k i (t) = i i (t) δk i (t) k i (0) given

8 2x2x2 Heckscher-Ohlin G k i ; p solution to the static sub problem: subject to: j G k i ; p = max x i 1 γ x i 2 1 γ (1) p j xj i = p j yj i (2) j yj i = kj i θj l i 1 j θj j = f1, 2g (3) k i = kj i (4) j 1 = lj i (5) j k i j 0, l i j 0, y i j 0 (6) Note that we are allowing for corner solutions in which one or both of the countries specialize. market

9 World prices The price of the intermediate goods p = fp 1, p 2 g are determined in the world commodity market and are such that it clears: xj i i k i ; p = yj i k i ; p j = f1, 2g (7) i where xj i k i ; p and yj i k i ; p are the optimal conditional demand and supply of intermediate inputs in each country.

10 Initial Conditions Regions of Specialization K 10 L S 0 S 9 8 Case 4 Case Case 3 Case 1 K N 5 K S FPE Set 1 0 N L N L

11 Initial Conditions K 10 L S 0 S 9 8 Case 4 Case Case 3 Case 1 K N 5 4 K S 3 2 FPE Set N L N L State Space I

12 Initial Conditions K 10 9 Case 4 L S Case 2 0 S 8 7 Case 3 Case 1 K N K S 3 2 FPE Set N L N L State Space II

13 Dynamic problem Current value Hamiltonian in each country. H i = log c i (t) + q i (t) G k i (t) ; p (t) c i (t) δk i (t) q i (t) is the current value co-state variable. The necessary rst order conditions are: G k i c i 1 = q i (8) q i = G k i k i ; p (ρ + δ) q i (9) k i = G k i ; p c i δk i (10) lim e ρt k i q i = 0 (11) t! is the partial derivative of total production with respect to k i, the marginal product of capital in country i

14 Solution The following di erential equations and the commodity market equilibrium condition (7) describe the equilibrium dynamics of this model: ċ N = c N G k N k N ; p (ρ + δ) ċ S = c S G k S k S ; p (ρ + δ) k N = G k N ; p c N δk N k S = G k S ; p c S δk S xj i i k i ; p = yj i k i ; p j = f1, 2g i k i (0) given, lim t! e ρt k i q i = 0

15 Characterize the dynamics in each region. Inside the FPE set. Regions A (A ) and B (B ) Outside the FPE set. Regions C (C ), D (D ), E, F

16 25 Phase Diagram Dynamic H O Model State Space I 20 k(t) South Region B 5 Region A Region C Region D k(t) North Box I Region A Region B Region C - D Phase z(t)

17 15 Phase Diagram Dynamic H O Model State Space II 10 k(t) South Region B' 5 Region A' Region E Region F Region C'D' k(t) North Box II

18 Inside the FPE set. Growth rate of consumption c i (t) = c i (R (t) (0) e (ρ+δ))t Relative consumptions stay constant over time. c S (0) c N (0) = ϖ (12) c S (t) c N (t) = cs (0) e (R (t) c N (0) e (R (t) (ρ+δ))t (ρ+δ))t = ϖ

19 Reduce the system Using (12) we can reduce the system to a set of three di erential equations, given by: ċ N = c N G k N k N ; p (ρ + δ) (13) k N = G k N ; p δk N c N (14) k S = G k S ; p δk S ϖc N (15) Terms of Trade - Factor Prices

20 Steady state r = ρ + δ Ψ w = (ρ + δ) ρ + δ p1 p2 = Θ 1 k γ (θ2 θ 1 ) where r is the rental price of capital, w the wage rate, Ψ = Γ (γ) Γ (θ 1 ) γ Γ (θ 2 ) 1 γ and Γ (x) = x x (1 x) 1 x.

21 Steady state The steady state world aggregate stock of capital and aggregate consumption are given by: k 2 c 2 = = γ 1 γ ρ + δ (1 1 γ Ψ ρ + δ 1 1 γ γ) Ψ ρ + δ 1 1 γ (16) (17) where c is the world per capita consumption: c = c N + c S.

22 Note that the steady state levels for the country aggregates are a function of the initial wealth distribution. Steady state Proposition 1 There exists a unique steady state level for the world aggregate variables. The steady state level for the country aggregates is a function of the initial wealth distribution, hence there are and in nite number of such steady states. c N = c S = ϖ c (18) ϖ 1 + ϖ c (19) k N = 1 1 ρ 1 + ϖ c w k S = 1 ϖ ρ 1 + ϖ c w (20) (21)

23 Solution inside Linearized system. 3 real eigenvalues - 1 negative and two positive. Linearized system - eigenvalues Saddle path stable. Eigenvalues are not a function of the initial conditions. Do we stay inside the FPE set? State Space

24 Look at the boundary De ne z S (t) = k S (t) /k (t). Suppose that at t = 0 we start at the boundary, then z S θ2 1 γ 1 (0) = 1 θ 2 γ 2 If ż S (t) > 0 countries move strictly inside the cone. Lemma (1) Suppose that countries factor supplies belong to the lower bound of the cone of diversi cation. If k (0) < k then ż S (t) > 0 and if k (0) > k then ż S (t) < 0.

25 This implies that if we are in region A then the economies stay in region A, however if we are in region B, the economies might leave the FPE set. In particular, if they start in the boundary of the set, the ray dividing region B from region C, they move to region C. State Space

26 Phase Diagram Inside the Cone Stable Arms Parameters Θ1 = 0.75 Θ2 = 0.25 γ = 0.5 ρ = 0.05 δ = Region A Steady States 3.5 k(t) South 3 kn(0), ks(0) inside the cone Region D Lower Bound Cone kn(0), ks(0) at the boundary of the cone k(t) North

27 Phase Diagram Inside the FPE set Stable Arms State Space II Parameters Θ1 = 0.65 Θ2 = 0.25 γ = 0.7 ρ = 0.05 δ = Region A' Steady States k(t) South kn(0), ks(0) inside kn(0), ks(0) at the boundary Lower Bound FPE set Region E Region F Region C'D' k(t) North

28 Slope of the saddle path (inside) Compare this slope with the slope of the cone of diversi cation. Lemma (2) The slope of the saddle path is larger than the slope of the lower boundary of the cone. lim t! k S (t) k N (t) = ψ0 k k N = α (1 ϖ) ϖ λ S α (1 ϖ) λ ψ 0 k k N (1 γ) θ 2 > S (1 θ 2 ) γ2 (1 γ) θ 2

29 Phase Diagram Inside the FPE set Stable Arms State Space I 8 Parameters kn(0), ks(0) 7.5 Θ1 = 0.75 inside Region B Θ2 = γ = 0.5 ρ = δ = Region B k(t) South Steady States Region A Region C Lower Bound k(t) North o p

30 Phase Diagram Inside the FPE set Stable Arms State Space II Parameters Θ1 = 0.65 Θ2 = 0.25 γ = 0.7 ρ = 0.05 δ = Region B' kn(0), ks(0) inside Region B' k(t) South Steady States Region A' Lower Bound Region E Region F Region C',D' k(t) North p o

31 Outside the FPE set. The laws of motion governing transition for both countries are: ċ N = G K N k N ; p (ρ + δ) c N ċ S = G k S k S ; p (ρ + δ) c S k N = G k N k N ; p δ k N + G L N k N ; p k S = G k S k S ; p δ k S + G L S k S ; p c S G K N 6= G K S and G L N 6= G L S. The aggregate production function in each country G k i ; p will inherit the properties of the production function(s) active in each sector. We can have corner solutions (i.e. y1 S = 0, y 2 N = 0) c N

32 Specialized steady states S-specialized steady state k N = k S = γ (1 θ2 ) + γ (θ 1 θ 2 ) (1 γ) (1 θ 2 ) 1 θ2 Ψ 1 γ ρ + δ 1 θ 2 Ψ ρ + δ 1 1 γ N-specialized steady state k N = k S = 1 θ1 Ψ 1 γ 1 θ 1 ρ + δ γ (1 θ1 ) (1 γ) (θ 1 θ 2 ) (1 γ) (1 θ 1 ) Ψ ρ + δ 1 1 γ

33 Slope of the saddle path (outside) Let us label the roots in ascending order as ψ 1, ψ 2, ψ 3 and ψ 4 where the rst two are the stable ones. Two paths that can lead us to the steady state. Pike is the one associated to the less negative root (ψ 2 ). Backroad with the most negative root (ψ 1 ). Convergence through the pike is slower than through the backroad and the path of adjustment of capital stocks are toward the pike since as t! the eigenvalue that dominates the system is ψ 2. Di erential equations

34 Slope of the saddle path (outside) We can solve for the slope of the pike as a function of the eigenvalue and it is given by the following expression: S (ψ 2 ) = (ψ 2 )2 f ψ 2 + a ψ 2 g b where a, b, f and g are constants. Proposition 7 The slope of the pike is lower than the slope of the lower bound of FPE set.

35 Phase Diagram Region C D Dynamic Heckscher Ohlin Model Parameters Θ1 = 0.75 Θ2 = 0.25 γ = 0.5 ρ = 0.05 δ = Region A Steady States Lower boundary of the cone Pike 2.8 k(t) South Pike Region C Region D 2.55 Lower boundary of the cone Backroad k(t) North prices, allocations Region D, prices, allocations Region C

36 4.5 Phase Diagram Ouside the FPE set Region E F C',D State Space II Region A' Region B' 4 k(t) South Lower boundary Backroad Region E Pike 2.5 Region F 2 Region C',D' k(t) North

37 Lemma (4) Trajectories with initial conditions belonging to the region B below p leave the cone in nite time and converge to the specialized steady state through the Pike from region C.

38 Phase Diagram Regions B C Dynamic Heckscher Ohlin Model Parameters Θ1 = 0.75 Θ2 = 0.25 γ = 0.5 ρ = 0.05 δ = Region B o m 2.88 p Lower boundary of the cone Pike k(t) South Region C 2.82 "Specialized" Steady State Backroad k(t) North Figure 5

39 25 Phase Diagram Dynamic H O Model Pike N S degree line k(t) South 15 Backroad N S 10 Specialized steady state N S Autarky steady state Pike S N 5 Specialized steady state S N Backroad S N k(t) North

40 10 Phase Diagram State Space II degree line 7 Pike N S k(t) South Specialized steady state Autarky steady state 3 Specialized steady state 2 Pike S N k(t) North

41 Conclusion Contribution: solution to a two large country model and showing that it is tractable and stable We nd that while a small country can grow without the retarding force of a terms of trade deterioration, a large capital intensive rich country can experience terms of trade deteriorations as a consequence of trading with a large, poor, labor intensive trading partner. We also show that the model can help to explain why countries experience non monotonic changes in their pattern of specialization as they grow.

42 Conclusion The model can also help to explain why large capital intensive rich countries can optimally experience periods of booms in consumption. We nd that neither starting to trade inside the cone nor starting to trade outside the cone can guarantee FPE between countries in nite time. One of the stylized growth facts observed in the data is the lack of convergence across countries. The model is consistent with the data. Extensions, applications.

43 FPE Equilibrium Factor prices Terms of Trade: K (t) γ 1 r (t) = γθ 2 (22) L K (t) γ w (t) = (1 γ) Θ 2 (23) L p 1 (t) p 2 (t) = Θ 1 K (t) L (θ2 θ 1 ) (24) Back

44 Back k S (t) = v 11 e ψ 1 t + v 12 e ψ 2 t k N (t) = v 11 e ψ 1 t + v 12 e ψ 2 t Recall that, ψ 1 < 0, ψ 2 < 0 and jψ 1 j > jψ 2 j Slope of the saddle path: k S (t) k N (t) = v 11e ψ 1t + v 12 e ψ 2t v 11 e ψ 1 t + v 12 e ψ 2 t

45 Back k S (t) = v 11 e ψ 1 t + v 12 e ψ 2 t k N (t) = v 11 e ψ 1 t + v 12 e ψ 2 t Recall that, ψ 1 < 0, ψ 2 < 0 and jψ 1 j > jψ 2 j Slope of the saddle path: k S (t) k N (t) = v 11e ψ 1t + v 12 e ψ 2t v 11 e ψ 1 t + v 12 e ψ 2 t lim t! k S (t) k N (t) = v 11e ψ 1 t + v 12 e ψ 2 t v 11 e ψ 1 t + v 12 e ψ 2 t = v 12e ψ 2 t v 12 e ψ 2 t = v 12 v 12

46 Inside the Cone. ċ N = c N γθ (k/2) γ 1 (ρ + δ) (25) k N = γθ (k/2) γ 1 k N + (1 γ) Θ (k/2) γ δk N c N (26) k S = γθ (k/2) γ 1 k S + (1 γ) Θ (k/2) γ δk S ϖc N (27) The linearized system is (where the variables with "~" are deviations with respect to the steady state) 0 d c N dt d k N dt d k S dt 1 0 C B A 0 dr dk N c N j 1 drk N ϖ j + dw j + ρ dk N dk N drk S j + dw j dk N dk N dr c N j dk S drk N j + dw j dk S dk S drk S dk S j + dw dk S j + ρ 1 0 C c N k N k S 1 A

47 Back Cubic: λ 3 2ρλ 2 + ā (1 + ϖ) + ρ 2 λ āρ (1 + ϖ) = 0 Roots: λ 0, λ 1, λ 2 λ 0 = λ 1 = ρ r ρ (1 γ) (ρ + δ) r 2 ρ + ρ (1 γ) (ρ + δ) λ 2 = ρ 1 negative eigenvalue λ 0. Saddle path stable. 2 ρ+δ(1 γ ρ+δ(1 γ γ) γ)

48 Households Problem: subject to Z max e ρt log c i (t) dt 0 c i (t) + i i (t) = r i (t) k i (t) + w i (t) k i (t) = i i (t) δk i (t) k i (0) > 0 given

49 Firms Final Goods producer: Technology: max fxj i g Y i p j xj i j Intermediate Good producer: Technology: Y i = x1 i γ x i 1 2 γ where γ 2 (0, 1) max fkj i,l j i g p jyj i r i kj i w i lj i y i j k i j θj l i j 1 θj where θ 1 > θ 2 and θ j 2 (0, 1)

50 Market clearing and Trade Balance Markets Clear c i + i i = Y i lj i = L i, j kj i = k i j xj i = yj i i i Also, trade balance holds, which implies for each i = fn, Sg: j p j xj i = p j yj i (28) j Back

51 Example transition CASE 2 Finite 0.6 Consumption levels C North C South Time 0 x 10 3 Change in Capital Change in C North Change in C South Time 1 Ratio of Consumptions Ratio C South /North Time

52 Example transition CASE 2 Finite Time r north r south Time w north w south

53 Example transition CASE 2 Finite 1 Labor 0.5 L sector 1 North L sector 2 North Time Labor L sector 1 South L sector 2 South Time

54 Example transition CASE 2 Finite 3 Capital 2 1 K sector 1 North K sector 2 North Time 1.5 Capital K sector 1 South K sector 2 South Time back

55 Example transition CASE P capital intensive good / P labor intensive good

56 Example transition CASE r south / w south

57 Example transition CASE Production Labor intensive good South back