Dornbusch, Fischer, Samuelson (DFS) Model

Transcription

1 Dornbusch, Fischer, Samuelson (DFS) Model Seyed Ali Madanizadeh Sharif U. of Tech. April 2014 Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

2 Introduction Ricardian Trade Theory Dornbusch, Fischer, Samuelson AER (1977) A convenient parameterization for multi-goods model Extension to transportation costs Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

3 Model Two countries H, F Same preferences Multi goods: Commodity space j [0, 1] Factor of production: Labor L H, L F CRS production Labor productivity in producing good j : z H (j), z F (j) A (j) z H (j) z F (j) Order goods with decreasing A (j); i.e. A (j) < 0 (See picture) A continious and strictly increasing. Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

4 Model Competitive equilibrium Costless trade: no transportation costs, tariffs Wage rates: w H, w F Buy j from vendor in H if or if p H (j) = ω = w H w F w H z H (j) w F z F (j) = p F (j) z H (j) z F (j) = A (j) H has comparative advantage in goods where A (j) ω.(see picture) Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

5 Model For a given ω, define the cutoff point j = φ (ω) = A 1 (ω) Good j where j j ar eproduced in H the rest in F.(See picture) j [0, φ (ω)] are produced in H j [φ (ω), 1] are produced in F Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

6 Demand HH s problem: Budget constraint Demands max log Q (j) dj Q (j) 0 p (j) Q (j) dj = M = wl p (j) Q (j) = M Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

7 Prices p (j) = w H z H (j) p (j) = w F z F (j) if j [0, φ (ω)] if j [φ (ω), 1] Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

8 Equilibrium Labor market equilibrium in H : L H = = = j (ω) 0 M H p (j) }{{} + M F p (j) }{{} =Q H (j) =Q F (j) φ(ω) ( MH + M ) F dj 0 w H w H φ(ω) 0 ( L H + L F ω ) dj ωl H = φ (ω) (ωl H + L F ) dj Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

9 Solution Fixed point problem (See picture) ω = φ (ω) L F 1 φ (ω) L H Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

10 Alternative solution 1 Fraction of world income spent on a s goods φ(ω) 0 Solve for equilibrium relative wage p (j) Q (j) φ(ω) dj = dj = φ (ω) M 0 w H L H = φ (ω) * world Income = φ (ω) (w H L H + w F L F ) ω = φ (ω) L F 1 φ (ω) L H Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

11 Alternative solution 2 Equilibrium: Income = Expenditure Income and expenditure in each country n is w n L n Thus H s sales at home are just j w H L H while its export revenues are j w F L F. Full employment in H thus requires that w H L H = j. (w H L H + w F L F ) Solve for equilibrium relative wage ω = φ (ω) L F 1 φ (ω) L H Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

12 An example Let s take Explanation! Result: A (j) = ( TH T F ( TH /L ω = H T F /L F ) 1 θ ( j ) 1 θ 1 j ) 1 1+θ Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

13 Transportation costs d ij is the iceberg-cost of sending a good to from j to i. F buys good j from H if or if w H d FH z H (j) ω z H (j) z F (j) So there is a j such that ω = z H (j ) z F (j ) w F z F (j) 1 d FH 1 d FH Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

14 Transportation costs H buys good j from H if or if w H z H (j) w F d HF z F (j) ω z H (j) z F (j) d HF So there is a j such that ω = z H (j ) z F (j ) d HF.(See picture) Goods j (j, j ) are produced for domestic consumption only. Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

15 Transportation costs: Equilibrium In equilibrium: Income = Expenditure w H L H = j w F L F + j w H L H or ωl H = A 1 (ωd FH ) + A 1 (w/d HF ) ωl H Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

16 Transportation costs: Equilibrium In the example above we get j = T H wh θ Φ H where Φ H = T H wh θ + T F (d HF w F ) θ and j = T H (d FH w H ) θ Φ F where Φ F = T H (d FH w H ) θ + T F w θ F Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

17 Welfare Gains: Price index { 1 P H = exp = exp Under Autarky 0 { j 0 } log p (j) dj 1 } log w H log z H (j) dj + log d HF w F log z F (j) dj j { 1 } PH A = exp log wh A log z H (j) dj 0 It can be shown that: ( ) w H /P H TH w θ 1/θ ( H wh A = = 1 + T F /PA Φ H H T H ( ) ) dhf w θ 1/θ F w H Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

18 Welfare Gains: If d HF = d FH = 1 w H /P H wh A = /PA H ( 1 + ( TF ) 1/(1+θ) ( ) ) θ/(1+θ) 1/θ LF T H L H If d HF = d FH = d and T H = T F and L H = L F w H = w F w H /P H wh A/PA H = (1 + d θ) 1/θ Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

19 Extension (Eaton Kortum 2002 Framework) To extend, first we need to extend the framework to a new commodity space. Think of each country i s effi ciency at making any good j as the realization of a random variable Z i drawn independently from a probability distribution F i (z). Or by the pair (x 1, x 2 ) of labor requirements. Describe joint distribution by measure µ on Borel sets of R Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

20 Extension (Eaton Kortum 2002 Framework) Goods produced in H are B H (ω) = { (x 1, x 2 ) R 2 ++ : ωx 1 x 2 } µ (B H (ω)) = Fraction of spending on H s goods Equilibrium: Income = Expenditure or w H L H = µ (B H (ω)) (w H L H + w F L F ) ωl H = µ (B H (ω)) (ωl H + L F ) ω = µ (B H (ω)) L F 1 µ (B H (ω)) L H Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

21 A convenient parameterization Based on exponential distribution on R +, with cdf Φ (x) = Pr (X x) = 1 e λx and density Write X exp (λ) φ (x) = λe λx Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

22 A convenient parameterization "Hazard rate" is constant for this distribution φ (x) 1 Φ (x) = λ The mean and standar deviation are both 1 λ If X exp (λ) and y = x θ then y has Frechet Distribution (Eaton kortum 2002). Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

23 A convenient parameterization: Facts 1 If X exp (λ) and y = kx then y exp ( ) λ 2 If X and Y are independent, X exp (λ) and Y exp (µ) then Z = min (X, Y ) exp (λ + µ) 3 If X and Y are independent, X exp (λ) and Y exp (µ) then Pr (X < Y ) = k λ λ + µ Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

24 A convenient parameterization Apply to DFS Let the joint distribution of (x H, x F ) be described by independent exponentials T H, T F. µ (B H (ω)) = Pr (ωx H x F ) = T H /ω T H /ω + T F Imposing into the equilibrium condtions: ω = µ (B H (ω)) L F 1 µ (B H (ω)) L H = T H /ω T F L F L H Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

25 Extension (Eaton Kortum 2002 Framework) Suppose Z = X 1/θ. Then Z has Frechet distribution. Suppose productivities are distributed with this Frechet distribution with parameter θ and T i ; i.e. F Zi (z) = Pr (Z i z) = exp [ T i z θ] Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

26 Extension (Eaton Kortum 2002 Framework) { B H (w H, w F ) = (z H, z F ) R++ 2 : w H z H It can easily be shown that w F z F } Thus: µ (B H (w H, w F )) = ω = T H w θ H T H w θ H + T F w θ F µ (B H (ω)) L F 1 µ (B H (ω)) L H = T H ω θ T F L F L H T = H ω θ T H ω θ + T H ω = ( ) 1 TH L 1+θ F T F L H Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27

27 Extension (Eaton Kortum 2002 Framework) Extending to N countries 1, 2,.., N The probability that country i is the lowest cost supplier to country n [ { }] wi d ni wk d µ (B i (w 1,.., w N )) Pr nk min k =i = z i T i (w i d ni ) θ N k=1 T k (w k d nk ) θ Define Φ n = N k=1 T k (w k d nk ) θ ; therefore: z k X ni X n = T i (w i d ni ) θ Φ n Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Model April / 27