Dornbusch, Fischer, Samuelson (DFS) Model
Dornbusch, Fischer, Samuelson (DFS) ModelSeyed Ali MadanizadehSharif U. of Tech.April 2014Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 20141 / 27
IntroductionRicardian Trade TheoryDornbusch, Fischer, Samuelson AER (1977)A convenient parameterization for multi-goods modelExtension to transportation costsSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 20142 / 27
ModelTwo countries H, FSame preferencesMulti goods: Commodity space j 2 [0, 1]Factor of production: Labor LH , LFCRS productionLabor productivity in producing good j : zH (j ) , zF (j )A (j )zH (j )zF ( j )Order goods with decreasing A (j ); i.e. A0 (j ) 0 (See picture) Acontinious and strictly increasing.Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 20143 / 27
ModelCompetitive equilibriumCostless trade: no transportation costs, tari sWage rates: wH , wFBuy j from vendor in H ifpH ( j ) or ifω wHzH (j )wHwFwF pF ( j )zF ( j )zH (j ) A (j )zF ( j )H has comparative advantage in goods where A (j )Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) Modelω.(See picture)April 20144 / 27
ModelFor a given ω, de ne the cuto point j φ (ω ) AGood j where j1(ω )j ar eproduced in H the rest in F .(See picture)j 2 [0, φ (ω )] are produced in Hj 2 [φ (ω ) , 1] are produced in FSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 20145 / 27
DemandHH’s problem:maxQ (j )Z 1log Q (j ) dj0Budget constraintZ 1p (j ) Q (j ) dj M wL0Demandsp (j ) Q (j ) MSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 20146 / 27
Pricesp (j ) p (j ) wHif j 2 [0, φ (ω )]zH (j )wFif j 2 [φ (ω ) , 1]zF (j )Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 20147 / 27
EquilibriumLabor market equilibrium in H :LH 0Z j (ω ) BB MH0MF CC BC djp (j ) A@ p (j ) {z } {z } Q H (j ) Z φ(ω )0 Z φ(ω )01 Q F (j )MHM F djwHwHLFLH djωωLH φ (ω ) (ωLH LF )Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 20148 / 27
SolutionFixed point problem (See picture)ω 1φ ( ω ) LFφ ( ω ) LHSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 20149 / 27
Alternative solution 1Fraction of world income spent on a’s goodsZ φ(ω )p (j ) Q (j )M0dj Z φ(ω )0dj φ (ω )Solve for equilibrium relative wagewH LH) φ (ω ) * world Income φ (ω ) (wH LH wF LF )ω 1φ ( ω ) LFφ ( ω ) LHSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201410 / 27
Alternative solution 2Equilibrium: Income ExpenditureIncome and expenditure in each country n is wn LnThus H’s sales at home are just j wH LH while its export revenues arej wF LF .Full employment in H thus requires thatwH LH j . (wH LH wF LF )Solve for equilibrium relative wage )ω φ ( ω ) LF1 φ ( ω ) LHSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201411 / 27
An exampleLet’s takeA (j ) THTF1θ1θj1jExplanation!Result:ω TH /LHTF /LF11 θSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201412 / 27
Transportation costsdij is the iceberg-cost of sending a good to from j to i.F buys good j from H ifwH dFHzH (j )or ifωSo there is a j 0 such that ω wFzF (j )zH (j ) 1zF (j ) dFHz H (j 0 ) 1zF (j 0 ) d FHSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201413 / 27
Transportation costsH buys good j from H ifwHzH (j )or ifωSo there is a j 00 such that ω wF dHFzF (j )zH (j )dHFzF (j )zH (j 00 )d .(SeezF (j 00 ) HFpicture)Goods j 2 (j 0 , j 00 ) are produced for domestic consumption only.Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201414 / 27
Transportation costs: EquilibriumIn equilibrium: Income ExpenditurewH LH j 0 wF LF j 00 wH LHorωLH A1(ωdFH ) A1(w /dHF ) ωLHSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201415 / 27
Transportation costs: EquilibriumIn the example above we getj 00 TH wH θΦHwhere ΦH TH wH θ TF (dHF wF ) θandTH (dFH wH )j0 ΦFwhere ΦF TH (dFH wH )θθ TF wF θSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201416 / 27
Welfare Gains:Price indexPH expZ 1log p (j ) dj0 expZ j 000log zH (j ) dj log wHZ 1j 00log zF (j ) djlog dHF wFUnder AutarkyPHA expZ 10log wHAlog zH (j ) djIt can be shown that:wH /PH wHA /PHATH wH θΦH!1/θ TF1 THSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModeldHF wFwHθ!1/θApril 201417 / 27
Welfare Gains:If dHF dFH 1 )wH /PH wHA /PHA1 TFTH1/(1 θ )LFLHθ/(1 θ )!1/θIf dHF dFH d and TH TF and LH LF ) wH wF )wH /PH 1 dwHA /PHAθ1/θSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201418 / 27
Extension (Eaton Kortum 2002 Framework)To extend, rst we need to extend the framework to a newcommodity space.Think of each country i’s e ciency at making any good j as therealization of a random variable Zi drawn independently from aprobability distribution Fi (z ). Or by the pair (x1 , x2 ) of laborrequirements.2 .Describe joint distribution by measure µ on Borel sets of R Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201419 / 27
Extension (Eaton Kortum 2002 Framework)Goods produced in H are2: ωx1 x2(x1 , x2 ) 2 R µ (BH (ω )) Fraction of spending on H’s goodsBH ( ω ) Equilibrium: Income ExpenditurewH LH µ (BH (ω )) (wH LH wF LF )or µ (BH (ω )) (ωLH LF )µ (BH (ω )) LFω 1 µ (BH (ω )) LHωLHSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201420 / 27
A convenient parameterizationBased on exponential distribution on R , with cdfΦ (x ) Pr (Xx) 1eλxand densityφ (x ) λeλxWrite X exp (λ)Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201421 / 27
A convenient parameterization"Hazard rate" is constant for this distributionφ (x ) λ1 Φ (x )The mean and standar deviation are both1λIf X exp (λ) and y x θ then y has Frechet Distribution (Eatonkortum 2002).Seyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201422 / 27
A convenient parameterization: Factsλk1If X exp (λ) and y kx then y exp2If X and Y are independent, X exp (λ) and Y exp (µ) thenZ min (X , Y ) exp (λ µ)3If X and Y are independent, X exp (λ) and Y exp (µ) thenPr (X Y ) λλ µSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201423 / 27
A convenient parameterizationApply to DFSLet the joint distribution of (xH , xF ) be described by independentexponentials TH , TF .µ (BH (ω )) Pr (ωxHxF )TH /ω TH /ω TFImposing into the equilibrium condtions:ω µ (BH (ω )) LF1 µ (BH (ω )) LHTH /ω LFTF LHSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201424 / 27
Extension (Eaton Kortum 2002 Framework)Suppose Z X1/θ .Then Z has Frechet distribution.Suppose productivities are distributed with this Frechet distributionwith parameter θ and Ti ; i.e.hiFZi (z ) Pr (Ziz ) exp Ti z θSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201425 / 27
Extension (Eaton Kortum 2002 Framework)n2:BH (wH , wF ) (zH , zF ) 2 R wHzHwFzFIt can easily be shown thatµ (BH (wH , wF )) oTH wH θTH wH θ TF wFθ TH ω θTH ω θ THThus:ω µ (BH (ω )) LF1 µ (BH (ω )) LHTH ωTFω θLFLHTH LFTF LH11 θSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201426 / 27
Extension (Eaton Kortum 2002 Framework)Extending to N countries 1, 2, ., NThe probability that country i is the lowest cost supplier to country nµ (Bi (w1 , ., wN ))Pr De ne Φn Nk 1 Tk (wk dnk )wi dnizimink 6 iTi (wi dni )θ Nk 1 Tk (wk dnk )θwk dnkzkθ; therefore:XniTi (wi dni ) XnΦnθSeyed Ali Madanizadeh (Sharif U. of Tech.) Dornbusch, Fischer, Samuelson (DFS) ModelApril 201427 / 27
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 2014 2 / 27
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Dornbusch at around the same time, (e.g., Hans Genberg and Henryk Kierzkowski, 1979). But the elegance and clarity of Dornbusch’s model, and its obvious and immediate policy relevance, puts his paper in a separate class from the other international macroeconomics papers of its time. A. Still a Useful Policy Tool
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7 Annual Book of ASTM Standards, Vol 14.02. 8 Discontinued 1996; see 1995 Annual Book of ASTM Standards, Vol 03.05. 9 Annual Book of ASTM Standards, Vol 03.03. 10 Available from American National Standards Institute, 11 West 42nd St., 13th Floor, New York, NY 10036. 11 Available from General Service Administration, Washington, DC 20405. 12 Available from Standardization Documents Order Desk .
