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Heckscher-Ohlin Model Seyed Ali Madanizadeh Sharif U. of Tech. April 2014 Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 1 / 21
Heckscher-Ohlin Model
Seyed Ali Madanizadeh
Sharif U. of Tech.
April 2014
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 1 / 21
Outline
Intution
Model Setup
Solution
Closed EcConomyIntegrated EconomySeparation for free trade case
Factor Price Equalization
Stolper-Samuelson Theorem
Rybczynski Theorem
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 2 / 21
Intuition
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 3 / 21
Intuition
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 4 / 21
Setup: 2*2*2 Model (Heckscher-Ohlin Model)
2 counties A,B
2 Factors K , L
Endowments: KA, LA,KB , LB
2 Goods X ,Y
Same CRS production production across countries:
X = F (K , L)
Y = G (K , L)
Assumption 1: F and G are strictly increasing, strictly concave, andcontinuously differentiable, and they have CRS.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 5 / 21
Setup: 2*2*2 Model
Preferences are UJ (X ,Y ) for J = A,B
Assumption 2: UJ (.), J = A,B is strictly increasing, strictlyquasi-concave, and continuously differentiable.
Homothetic preferences are not necessary but it simplifies the solution.Differentiability is convenient so that prices (tangent lines) are unique.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 6 / 21
Social Planner’s problem
HH’s Problemmaxx ,y
U (X ,Y )
s.t. X = F (kx , lx )
Y = F (ky , ly )
K = kx + kyL = lx + ly
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 7 / 21
Solution
HH’s Problemmaxx ,y
U (X ,Y )
s.t. pX + qY = I = rK + wL
Firm x’s problem
minkx ,lx
pF (kx , lx )− (rkx + wlx )
Firm y’s problem
minky ,ly
qG (ky , ly )− (rky + wly )
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 8 / 21
Solution
Market clearing condition:
X = F (kx , lx )
Y = F (ky , ly )
K = kx + kyL = lx + ly
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 9 / 21
Solution
Assume
F (k, l) = kαl1−α
G (k, l) = kβl1−β
β > α
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 10 / 21
Solution
Cost minimization to produce of 1 unit of good x :
minC = rk + wl
s.t. 1 = kαl1−α
FOC
(1− α)(kl
)αλ = w
α
(kl
)−(1−α)λ = r
1− αα
kl=wr
κ̃x(wr
)≡ kxlx=
α
1− αwr
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 11 / 21
Solution
Factor demand :
1 = l(kl
)α= l
(α
1− αwr
)α⇒
lx =(
α
1− αwr
)−αkx =
(α
1− αwr
)1−α⇒
p = mCx = C (1) = rkx + wlx
=
((α
1− α
)1−α+
(α
1− α
)−α)w1−αr α
=1
(1− α)1−α ααw1−αr α
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 12 / 21
Solution
Cost minimization to produce of 1 unit of good y ⇒
κ̃y(wr
)≡ kyly=
β
1− βwr
and
ly =(
β
1− βwr
)−βky =
(β
1− βwr
)1−βq = mCy =
1
(1− β)1−β ββw1−βr β
Relative pricepq=(1− β)1−β ββ
(1− α)1−α αα(wr
)β−αSeyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 13 / 21
Solution
Market Clearing
lx(wr
)X + ly
(wr
)Y = L
kx(wr
)X + ky
(wr
)Y = K
⇒kylx
=K − XkxL− Xlx
κ̃y =K − κ̃xLxL− Lx
β
1− βwr
=K − α1−α
wr Lx
L− Lx
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 14 / 21
Solution
⇒
Lx = φ(wr
)=
β1−β
wr L−K(
β1−β −
α1−α
)wr
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 15 / 21
Solution
X =Lxlx=
(α
1− αwr
)α β1−β
wr L−K(
β1−β −
α1−α
)wr
X = X̃(wr
)=
(α
1− α
)α β1−β
wr L−K(
β1−β −
α1−α
) (wr
)1−αY = Ỹ
(wr
)= ...
For Each wr we get (X ,Y )⇒PPF and the slope ifpq =
(1−β)1−β ββ
(1−α)1−ααα(wr
)β−αGiven p and q = 1⇒ w and r are functions of p. so All the X ,Y , ..are functions of p.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 16 / 21
Solution
At optimum in a close deconomy
pq=UxUy
Assume U (X ,Y ) = γ logX + (1− γ) logY ⇒
p =pq=
γX1−γY
=γ
1− γYX
So we solve for p.
Done!
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 17 / 21
Free trade
κ̃y =kyly=KA − κ̃xLxLA − Lx
Lx =κ̃yLA −KA
κ̃y − κ̃x
Kx =KA − κ̃xLA
κ̃y − κ̃x
if κ̃x < KA
LA < κ̃y
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 18 / 21
Factor Price Equalization
Factor prices are the same across countries, in case of diversificationwhich is κ̃x < K
A
LA ,K B
LB < κ̃y
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 19 / 21
Stolper-Samuelson Theorem
Consider a Small Open Economy (SOE) that takes relativecommodity prices as exogenous.
Suppose it is initially diversified, producing both goods.
TheoremStolper-Samuelson Theorem: An increase in the relative price of acommodity:increases the relative price of the factor that is used intensively inproducing that commodity.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 20 / 21
Rybczynski Theorem
Consider a Small Open Economy (SOE) that takes relativecommodity prices as exogenous.
Suppose it is initially diversified, producing both goods.
TheoremRybcszynski Theorem: An increase in the supply of a factor:increases production of the commodity that is intensive in that factor,and decreases production of the other commodity.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Heckscher-Ohlin Model April 2014 21 / 21
