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International TradeLecture 3: The Heckscher-Ohlin Model
Yiqing Xie
School of EconomicsFudan University
Oct. 11, 2013
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 1 / 33
Outline
Heckscher-Ohlin Model: an Intuitive Approach
Heckscher-Ohlin Theorem: a Formal Approach
Factor-Price-Equalization Theorem
Rybczynski Theorem
Stolper-Samuelson Theorem
Summary
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 2 / 33
Heckscher-Ohlin Model: Model Assumptions
Two Goods: X1 and X2
Two Factors: V1 and V2, Vij is industry i’s use of factor j
X1 = F1(V11, V12) X2 = F2(V21, V22)
V 1 = V11 + V21 V 2 = V12 + V22 (1)
Two countries: h and f
V11V12
>V21V22
andV h1
V h2
>V f1
V f2
(2)
Identical Technologies
CRS and Perfect Competition
Identical Homogeneous Demand
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 3 / 33
Heckscher-Ohlin ModelFactor Intensities and Factor Abundance
Factor Intensities – Characteristics of technologies
Definition of factor intensities:
If at a given factor-price ratio w1/w2,
optimal factor input ratios are V11V12
> V21V22
X1 is said to be V1 intensive and X2 is V2 intensive.
Factor Abundance – Characteristics of countries
Let V kj give country k’s endowment of factor j.
Then if V h1
V h2>
V f1
V f2
country h is said to be V1 abundant, f is V2 abundant.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 4 / 33
Heckscher-Ohlin ModelFactor Intensities and Factor Abundance
Factor Intensities – Characteristics of technologies
Definition of factor intensities:
If at a given factor-price ratio w1/w2,
optimal factor input ratios are V11V12
> V21V22
X1 is said to be V1 intensive and X2 is V2 intensive.
Factor Abundance – Characteristics of countries
Let V kj give country k’s endowment of factor j.
Then if V h1
V h2>
V f1
V f2
country h is said to be V1 abundant, f is V2 abundant.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 4 / 33
Heckscher-Ohlin Model: DataFactor Intensities
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 5 / 33
Heckscher-Ohlin Model: DataRelative Factor Endowments
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 6 / 33
Heckscher-Ohlin Model: DataWorld Factor Endowments
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 7 / 33
Heckscher-Ohlin Model: TheoremEach country will export the good using intensively its abundant factor.
Step 1: Comparative advantage is indirect.
Differences in relative endowments between countries+
Differences in relative factor intensities between goods=
Comparative advantage
Step 2: Autarky prices reflect comparative advantage.
Each country has a relatively low price for the good usingintensively its abundant factor.
Step 3: Free trade prices must lie between the two autarky prices.
In free trade, each country exports the good using intensively itsabundant factor.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 8 / 33
Heckscher-Ohlin Model: TheoremEach country will export the good using intensively its abundant factor.
Step 1: Comparative advantage is indirect.
Differences in relative endowments between countries+
Differences in relative factor intensities between goods=
Comparative advantage
Step 2: Autarky prices reflect comparative advantage.
Each country has a relatively low price for the good usingintensively its abundant factor.
Step 3: Free trade prices must lie between the two autarky prices.
In free trade, each country exports the good using intensively itsabundant factor.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 8 / 33
Heckscher-Ohlin Model: TheoremEach country will export the good using intensively its abundant factor.
Step 1: Comparative advantage is indirect.
Differences in relative endowments between countries+
Differences in relative factor intensities between goods=
Comparative advantage
Step 2: Autarky prices reflect comparative advantage.
Each country has a relatively low price for the good usingintensively its abundant factor.
Step 3: Free trade prices must lie between the two autarky prices.
In free trade, each country exports the good using intensively itsabundant factor.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 8 / 33
Heckscher-Ohlin Model: A Special Case
Figure 8.1
Figure 8.2
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 9 / 33
Heckscher-Ohlin Model: A Special Case
Figure 8.1 Figure 8.2
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 9 / 33
Heckscher-Ohlin Theorem: a Formal ApproachZero Profit ConditionConsider a single country,[
c1(w1, w2)c2(w1, w2)
]=
[a11 a12a21 a22
] [w1
w2
]=
[p1p2
](3)
ci: the production cost pi: the good price wj : the factor price
aij : the optimal amount of factor j used in industry i
c1 = a11w1 + a12w2 = p1
c2 = a21w1 + a22w2 = p2
dc1 = a11dw1 + a12dw2 + [w1da11 + w2da12] = dp1 (4)dc2 = a21dw1 + a22dw2 + [w1da21 + w2da22] = dp2
The term in brackets is ZERO:
aij is optimally chosen, small changes in these values have no effect incost.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 10 / 33
Heckscher-Ohlin Theorem: a Formal ApproachZero Profit ConditionConsider a single country,[
c1(w1, w2)c2(w1, w2)
]=
[a11 a12a21 a22
] [w1
w2
]=
[p1p2
](3)
ci: the production cost pi: the good price wj : the factor price
aij : the optimal amount of factor j used in industry i
c1 = a11w1 + a12w2 = p1
c2 = a21w1 + a22w2 = p2
dc1 = a11dw1 + a12dw2 + [w1da11 + w2da12] = dp1 (4)dc2 = a21dw1 + a22dw2 + [w1da21 + w2da22] = dp2
The term in brackets is ZERO:
aij is optimally chosen, small changes in these values have no effect incost.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 10 / 33
Heckscher-Ohlin Theorem: a Formal ApproachZero Profit ConditionConsider a single country,[
c1(w1, w2)c2(w1, w2)
]=
[a11 a12a21 a22
] [w1
w2
]=
[p1p2
](3)
ci: the production cost pi: the good price wj : the factor price
aij : the optimal amount of factor j used in industry i
c1 = a11w1 + a12w2 = p1
c2 = a21w1 + a22w2 = p2
dc1 = a11dw1 + a12dw2 + [w1da11 + w2da12] = dp1 (4)dc2 = a21dw1 + a22dw2 + [w1da21 + w2da22] = dp2
The term in brackets is ZERO:
aij is optimally chosen, small changes in these values have no effect incost.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 10 / 33
Heckscher-Ohlin Theorem: a Formal ApproachZero Profit Condition[
dc1dc2
]=
[a11 a12a21 a22
] [dw1
dw2
]=
[dp1dp2
](5)
By Cramer’s Rule,[a22/D −a12/D−a21/D a11/D
] [dp1dp2
]=
[dw1
dw2
](6)
Since X1 is V1 intensive, so that
a11a12
>a21a22
a11a22 > a12a21 a11a22 − a12a21 ≡ D > 0 (7)
Let p2 = 1 and dp2 = 0,[dw1
dp1
]dp2=0
> 0
[dw2
dp1
]dp2=0
< 0 (8)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 11 / 33
Heckscher-Ohlin Theorem: a Formal ApproachZero Profit Condition[
dc1dc2
]=
[a11 a12a21 a22
] [dw1
dw2
]=
[dp1dp2
](5)
By Cramer’s Rule,[a22/D −a12/D−a21/D a11/D
] [dp1dp2
]=
[dw1
dw2
](6)
Since X1 is V1 intensive, so that
a11a12
>a21a22
a11a22 > a12a21 a11a22 − a12a21 ≡ D > 0 (7)
Let p2 = 1 and dp2 = 0,[dw1
dp1
]dp2=0
> 0
[dw2
dp1
]dp2=0
< 0 (8)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 11 / 33
Heckscher-Ohlin Theorem: a Formal ApproachZero Profit Condition[
dc1dc2
]=
[a11 a12a21 a22
] [dw1
dw2
]=
[dp1dp2
](5)
By Cramer’s Rule,[a22/D −a12/D−a21/D a11/D
] [dp1dp2
]=
[dw1
dw2
](6)
Since X1 is V1 intensive, so that
a11a12
>a21a22
a11a22 > a12a21 a11a22 − a12a21 ≡ D > 0 (7)
Let p2 = 1 and dp2 = 0,[dw1
dp1
]dp2=0
> 0
[dw2
dp1
]dp2=0
< 0 (8)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 11 / 33
Heckscher-Ohlin Theorem: a Formal ApproachZero Profit Condition[
dc1dc2
]=
[a11 a12a21 a22
] [dw1
dw2
]=
[dp1dp2
](5)
By Cramer’s Rule,[a22/D −a12/D−a21/D a11/D
] [dp1dp2
]=
[dw1
dw2
](6)
Since X1 is V1 intensive, so that
a11a12
>a21a22
a11a22 > a12a21 a11a22 − a12a21 ≡ D > 0 (7)
Let p2 = 1 and dp2 = 0,[dw1
dp1
]dp2=0
> 0
[dw2
dp1
]dp2=0
< 0 (8)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 11 / 33
Heckscher-Ohlin Theorem: a Formal ApproachZero Profit Condition
Increase in p1 ⇒ Increase in w1 and decrease in w2
Increase in w1 and decrease in w2
⇒ Increase in a12, a22 and decrease in a11, a21
da11dp1
< 0da12dp1
> 0da21dp1
< 0da22dp1
> 0 (9)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 12 / 33
Heckscher-Ohlin Theorem: a Formal ApproachZero Profit Condition
Increase in p1 ⇒ Increase in w1 and decrease in w2
Increase in w1 and decrease in w2
⇒ Increase in a12, a22 and decrease in a11, a21
da11dp1
< 0da12dp1
> 0da21dp1
< 0da22dp1
> 0 (9)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 12 / 33
Heckscher-Ohlin Theorem: a Formal ApproachZero Profit Condition
Increase in p1 ⇒ Increase in w1 and decrease in w2
Increase in w1 and decrease in w2
⇒ Increase in a12, a22 and decrease in a11, a21
da11dp1
< 0da12dp1
> 0da21dp1
< 0da22dp1
> 0 (9)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 12 / 33
Heckscher-Ohlin Theorem: a Formal ApproachFactor Market Clearing Condition
[a11 a21a12 a22
] [X1
X2
]=
[V1V2
](10)
Apply Eq. (7), [a22/D −a21/D−a12/D a11/D
] [V1V2
]=
[X1
X2
](11)
Divide the first equation by the second in (11),
X1
X2=
a22 − a21 V2V1−a12 + a11
V2V1
(12)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 13 / 33
Heckscher-Ohlin Theorem: a Formal ApproachFactor Market Clearing Condition
[a11 a21a12 a22
] [X1
X2
]=
[V1V2
](10)
Apply Eq. (7), [a22/D −a21/D−a12/D a11/D
] [V1V2
]=
[X1
X2
](11)
Divide the first equation by the second in (11),
X1
X2=
a22 − a21 V2V1−a12 + a11
V2V1
(12)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 13 / 33
Heckscher-Ohlin Theorem: a Formal ApproachFactor Market Clearing Condition
[a11 a21a12 a22
] [X1
X2
]=
[V1V2
](10)
Apply Eq. (7), [a22/D −a21/D−a12/D a11/D
] [V1V2
]=
[X1
X2
](11)
Divide the first equation by the second in (11),
X1
X2=
a22 − a21 V2V1−a12 + a11
V2V1
(12)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 13 / 33
Heckscher-Ohlin Theorem: a Formal Approach
da11dp1
< 0da12dp1
> 0da21dp1
< 0da22dp1
> 0 (9)
X1
X2=
a22 − a21 V2V1−a12 + a11
V2V1
(12)
The production ratio X1/X2 rises with p1/p2.
The relative supply of good X1 rises with the relative price of X1.
The price ratio (p1/p2) at which a country just begins to produceX1 is higher in the V2 abundant country.
The price ratio (p1/p2) at which a country stops producing X2 ishigher in the V2 abundant country.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 14 / 33
Heckscher-Ohlin Theorem: a Formal Approach
da11dp1
< 0da12dp1
> 0da21dp1
< 0da22dp1
> 0 (9)
X1
X2=
a22 − a21 V2V1−a12 + a11
V2V1
(12)
The production ratio X1/X2 rises with p1/p2.
The relative supply of good X1 rises with the relative price of X1.
The price ratio (p1/p2) at which a country just begins to produceX1 is higher in the V2 abundant country.
The price ratio (p1/p2) at which a country stops producing X2 ishigher in the V2 abundant country.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 14 / 33
Heckscher-Ohlin Theorem: a Formal Approach
da11dp1
< 0da12dp1
> 0da21dp1
< 0da22dp1
> 0 (9)
X1
X2=
a22 − a21 V2V1−a12 + a11
V2V1
(12)
The production ratio X1/X2 rises with p1/p2.
The relative supply of good X1 rises with the relative price of X1.
The price ratio (p1/p2) at which a country just begins to produceX1 is higher in the V2 abundant country.
The price ratio (p1/p2) at which a country stops producing X2 ishigher in the V2 abundant country.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 14 / 33
Heckscher-Ohlin Theorem: a Formal Approach
da11dp1
< 0da12dp1
> 0da21dp1
< 0da22dp1
> 0 (9)
X1
X2=
a22 − a21 V2V1−a12 + a11
V2V1
(12)
The production ratio X1/X2 rises with p1/p2.
The relative supply of good X1 rises with the relative price of X1.
The price ratio (p1/p2) at which a country just begins to produceX1 is higher in the V2 abundant country.
The price ratio (p1/p2) at which a country stops producing X2 ishigher in the V2 abundant country.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 14 / 33
Heckscher-Ohlin Model: A Formal ApproachRecall that country h is V1 abundant and country f is V2 abundant.
Figure 8.3
Figure 8.4
Each country will export the good using intensively its abundant factor.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 15 / 33
Heckscher-Ohlin Model: A Formal ApproachRecall that country h is V1 abundant and country f is V2 abundant.
Figure 8.3 Figure 8.4
Each country will export the good using intensively its abundant factor.Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 15 / 33
Heckscher-Ohlin Model: A Formal ApproachIncome Distribution Effect of Trade
Autarky: the scarcity of one factor make the good using that factorintensively expensive.
Trade makes that good cheaper, leads the country to produce lessof that good and more of the good which does not use that factorintensively.
This is going to lower the demand for the scarce factor, and thiswill drive down its price in equilibrium.
The reverse argument can be made about the abundant factor.
Trade increases the return to the abundant factor, lowers the return tothe scarce factor.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 16 / 33
Heckscher-Ohlin Model: A Formal ApproachIncome Distribution Effect of Trade
Autarky: the scarcity of one factor make the good using that factorintensively expensive.
Trade makes that good cheaper, leads the country to produce lessof that good and more of the good which does not use that factorintensively.
This is going to lower the demand for the scarce factor, and thiswill drive down its price in equilibrium.
The reverse argument can be made about the abundant factor.
Trade increases the return to the abundant factor, lowers the return tothe scarce factor.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 16 / 33
Factor-Price-Equalization Theorem (FPE)
Assumptions:
Both countries have identical CRS technologies.
Trade is completely costless so that goods price are equalized.
Both countries produce both goods in free-trade equilibrium.
The Factor-Price-Equalization Theorem
(A) if trade is costless such that trade equalizes commodity pricesbetween countries and
(B) if countries are not “too different" such that both continue toproduce both goods after trade,
then the price of each factor is equalized across countries.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 17 / 33
Factor-Price-Equalization Theorem (FPE)
Assumptions:
Both countries have identical CRS technologies.
Trade is completely costless so that goods price are equalized.
Both countries produce both goods in free-trade equilibrium.
The Factor-Price-Equalization Theorem
(A) if trade is costless such that trade equalizes commodity pricesbetween countries and
(B) if countries are not “too different" such that both continue toproduce both goods after trade,
then the price of each factor is equalized across countries.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 17 / 33
Factor-Price-Equalization Theorem (FPE)Unit Value Isoquant and Unit Value Isocost
For both countries,
a22a21
>V 2
V 1
>a12a11
Identical technologies,equalized commodity prices⇒ Same unit-value isoquants
Same unit value isoquants⇒ Same isocost line⇒ FPE
Endowment point at E1 or E2
Figure 8.5
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 18 / 33
Factor-Price-Equalization Theorem (FPE)World Edgewood Box
Figure 8.6
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 19 / 33
Rybczynski Theorem
Assumptions and Intuition
Start from FPE, subject to producing both goods:
Hold commodity prices constant⇒Hold factor prices constant⇒Hold optimal aij ’s constant⇒Changes in endowments can be absorbed through changes in thecomposition of output rather than changes in factor prices.
The Rybczynski Theorem
Holding commodity prices constant, an increase in theendowment of factor j leads to a more than proportion increase inthe output of the good using that factor intensively, and to a fall inthe output of the other good.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 20 / 33
Rybczynski Theorem
Assumptions and Intuition
Start from FPE, subject to producing both goods:
Hold commodity prices constant⇒Hold factor prices constant⇒Hold optimal aij ’s constant⇒Changes in endowments can be absorbed through changes in thecomposition of output rather than changes in factor prices.
The Rybczynski Theorem
Holding commodity prices constant, an increase in theendowment of factor j leads to a more than proportion increase inthe output of the good using that factor intensively, and to a fall inthe output of the other good.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 20 / 33
Rybczynski Theorem: Graphical Presentation
Figure 8.7
Figure 8.8
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 21 / 33
Rybczynski Theorem: Graphical Presentation
Figure 8.7 Figure 8.8
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 21 / 33
Rybczynski Theorem: Formal Proof[a11 a21a12 a22
] [X1
X2
]=
[V1V2
](10)
Total derivative of (10):
a11dX1 + a21dX2 = dV1 =
[V11X1
]dX1 +
[V21X2
]dX2
a12dX1 + a22dX2 = dV2 =
[V12X1
]dX1 +
[V22X2
]dX2 (13)
Dividing the total factor endowments V1 and V2,[V11V1
]dX1
X1+
[V21V1
]dX2
X2=dV1V1
(14)[V12V2
]dX1
X1+
[V22V2
]dX2
X2=dV2V2
(15)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 22 / 33
Rybczynski Theorem: Formal Proof[a11 a21a12 a22
] [X1
X2
]=
[V1V2
](10)
Total derivative of (10):
a11dX1 + a21dX2 = dV1 =
[V11X1
]dX1 +
[V21X2
]dX2
a12dX1 + a22dX2 = dV2 =
[V12X1
]dX1 +
[V22X2
]dX2 (13)
Dividing the total factor endowments V1 and V2,[V11V1
]dX1
X1+
[V21V1
]dX2
X2=dV1V1
(14)[V12V2
]dX1
X1+
[V22V2
]dX2
X2=dV2V2
(15)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 22 / 33
Rybczynski Theorem: Formal Proof[a11 a21a12 a22
] [X1
X2
]=
[V1V2
](10)
Total derivative of (10):
a11dX1 + a21dX2 = dV1 =
[V11X1
]dX1 +
[V21X2
]dX2
a12dX1 + a22dX2 = dV2 =
[V12X1
]dX1 +
[V22X2
]dX2 (13)
Dividing the total factor endowments V1 and V2,[V11V1
]dX1
X1+
[V21V1
]dX2
X2=dV1V1
(14)[V12V2
]dX1
X1+
[V22V2
]dX2
X2=dV2V2
(15)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 22 / 33
Rybczynski Theorem: Formal Proof
The share of factor j used in good i – λij
Proportional change in a variable – ˆ[λ11 λ21λ12 λ22
] [X1
X2
]=
[V1V2
](16)
Invert the equation system,[λ22/Dλ −λ21/Dλ
−λ12/Dλ λ11/Dλ
] [V1V2
]=
[X1
X2
](17)
Dλ ≡ λ11λ22−λ12λ21 > 0 whereλ22
λ11λ22 − λ12λ21> 1 given 0 < λij < 1
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 23 / 33
Rybczynski Theorem: Formal Proof
The share of factor j used in good i – λij
Proportional change in a variable – ˆ[λ11 λ21λ12 λ22
] [X1
X2
]=
[V1V2
](16)
Invert the equation system,[λ22/Dλ −λ21/Dλ
−λ12/Dλ λ11/Dλ
] [V1V2
]=
[X1
X2
](17)
Dλ ≡ λ11λ22−λ12λ21 > 0 whereλ22
λ11λ22 − λ12λ21> 1 given 0 < λij < 1
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 23 / 33
Rybczynski Theorem: Formal ProofThe magnitudes and signs of the mapping in (17) are as follows.[
> 1 < 0< 0 > 1
] [V1V2
]=
[X1
X2
](18)
Given V1 > 0 and V2 = 0 or V1 = 0 and V2 > 0,
The Rybczynski theorem (“Magnification" effect):
X1 > V1 > V2 = 0 > X2 X2 > V2 > V1 = 0 > X1 (19)
Small open economyEast and South-East Asia Development
High saving and investment rates (falling birth rates)
⇒ Increase in the relative capital abundance
⇒ Sectoral shifts toward manufacturing
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 24 / 33
Rybczynski Theorem: Formal ProofThe magnitudes and signs of the mapping in (17) are as follows.[
> 1 < 0< 0 > 1
] [V1V2
]=
[X1
X2
](18)
Given V1 > 0 and V2 = 0 or V1 = 0 and V2 > 0,
The Rybczynski theorem (“Magnification" effect):
X1 > V1 > V2 = 0 > X2 X2 > V2 > V1 = 0 > X1 (19)
Small open economyEast and South-East Asia Development
High saving and investment rates (falling birth rates)
⇒ Increase in the relative capital abundance
⇒ Sectoral shifts toward manufacturing
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 24 / 33
Rybczynski Theorem: Formal ProofThe magnitudes and signs of the mapping in (17) are as follows.[
> 1 < 0< 0 > 1
] [V1V2
]=
[X1
X2
](18)
Given V1 > 0 and V2 = 0 or V1 = 0 and V2 > 0,
The Rybczynski theorem (“Magnification" effect):
X1 > V1 > V2 = 0 > X2 X2 > V2 > V1 = 0 > X1 (19)
Small open economyEast and South-East Asia Development
High saving and investment rates (falling birth rates)
⇒ Increase in the relative capital abundance
⇒ Sectoral shifts toward manufacturing
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 24 / 33
Stolper-Samuelson Theorem: Intuition
Note that the opening of trade shift production in each countrytoward the sector which uses intensively the country’s abundantfactor.
The problem is that, at constant factor prices, the expandingsector is going to demand factors in different proportions to thosebeing released by the contracting sector.
Relative to the contracting sector, the expanding sector willdemand “too much" of the abundant factor and “too little" of thescarce factor.
Constant factor prices will NOT lead to an open to trade equilibrium.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 25 / 33
Stolper-Samuelson Theorem: Intuition
At constant factor prices,
X2 releases factors in the proportion a22/a21.
X1 demands factors in the proportion a12/a11.
a22/a21 > a12/a11
Price changes due to the opening of trade
⇒ Changes in outputs
⇒ Excess demand for the abundant factorExcess supply of the scarce factor
⇒ Increased price of the abundant factorDecreased price for the scarce factor
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 26 / 33
Stolper-Samuelson Theorem: Intuition
At constant factor prices,
X2 releases factors in the proportion a22/a21.
X1 demands factors in the proportion a12/a11.
a22/a21 > a12/a11
Price changes due to the opening of trade
⇒ Changes in outputs
⇒ Excess demand for the abundant factorExcess supply of the scarce factor
⇒ Increased price of the abundant factorDecreased price for the scarce factor
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 26 / 33
Stolper-Samuelson Theorem: Graphical Presentation
Figure 8.9
Figure 8.10
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 27 / 33
Stolper-Samuelson Theorem: Graphical Presentation
Figure 8.9 Figure 8.10
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 27 / 33
Stolper-Samuelson Theorem: The Theorem
Stolper-Samuelson Theorem:
Holding factor endowments constant, an increase in the price of one
good leads to a more than proportional increase in the price of the
factor used intensively in producing that good, and to a fall in the price
of the other factor.
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 28 / 33
Stolper-Samuelson Theorem: A Short ProofValue of marginal product conditions for competitive equilibrium
w1 = p1MP11 = p2MP21 w1/p1 =MP11 w1/p2 =MP21
w2 = p1MP12 = p2MP22 w2/p1 =MP12 w2/p2 =MP22
An increase in p = p1/p2 raises w1/w2, and therefore raises theratio of V2/V1 in both industries.
MPi1 raises and MPi2 falls.
w1/p1 ↑ w1/p2 ↑w2/p1 ↓ w2/p2 ↓
Wage of V1 rises relative to both commodity prices.
Wage of V2 falls relative to both commodity prices.
w1 > p1 > p2 = 0 > w2
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 29 / 33
Stolper-Samuelson Theorem: A Short ProofValue of marginal product conditions for competitive equilibrium
w1 = p1MP11 = p2MP21 w1/p1 =MP11 w1/p2 =MP21
w2 = p1MP12 = p2MP22 w2/p1 =MP12 w2/p2 =MP22
An increase in p = p1/p2 raises w1/w2, and therefore raises theratio of V2/V1 in both industries.
MPi1 raises and MPi2 falls.
w1/p1 ↑ w1/p2 ↑w2/p1 ↓ w2/p2 ↓
Wage of V1 rises relative to both commodity prices.
Wage of V2 falls relative to both commodity prices.
w1 > p1 > p2 = 0 > w2
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 29 / 33
Stolper-Samuelson Theorem: A Short ProofValue of marginal product conditions for competitive equilibrium
w1 = p1MP11 = p2MP21 w1/p1 =MP11 w1/p2 =MP21
w2 = p1MP12 = p2MP22 w2/p1 =MP12 w2/p2 =MP22
An increase in p = p1/p2 raises w1/w2, and therefore raises theratio of V2/V1 in both industries.
MPi1 raises and MPi2 falls.
w1/p1 ↑ w1/p2 ↑w2/p1 ↓ w2/p2 ↓
Wage of V1 rises relative to both commodity prices.
Wage of V2 falls relative to both commodity prices.
w1 > p1 > p2 = 0 > w2
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 29 / 33
Stolper-Samuelson Theorem: A Short ProofValue of marginal product conditions for competitive equilibrium
w1 = p1MP11 = p2MP21 w1/p1 =MP11 w1/p2 =MP21
w2 = p1MP12 = p2MP22 w2/p1 =MP12 w2/p2 =MP22
An increase in p = p1/p2 raises w1/w2, and therefore raises theratio of V2/V1 in both industries.
MPi1 raises and MPi2 falls.
w1/p1 ↑ w1/p2 ↑w2/p1 ↓ w2/p2 ↓
Wage of V1 rises relative to both commodity prices.
Wage of V2 falls relative to both commodity prices.
w1 > p1 > p2 = 0 > w2
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 29 / 33
Stolper-Samuelson Theorem: A Formal ProofStart from equation (5) from zero profit condition,[
a11 a12a21 a22
] [dw1
dw2
]=
[dp1dp2
](5)
Rewrite the system of equation into[V11X1
]dw1 +
[V12X1
]dw2 = dp1[
V21X2
]dw1 +
[V22X2
]dw2 = dp2
⇒
[w1V11p1X1
]dw1w1
+[w2V12p1X1
]dw2w2
= dp1p1[
w1V21p2X2
]dw1w1
+[w2V22p2X2
]dw2w2
= dp2p2
(20)The terms in brackets are the shares of each factor’s earnings (j) inthe total revenue of the industry (i), denoted by θij , and 0 < θij < 1.
[θ11 θ12θ21 θ22
] [w1
w2
]=
[p1p2
](21)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 30 / 33
Stolper-Samuelson Theorem: A Formal ProofStart from equation (5) from zero profit condition,[
a11 a12a21 a22
] [dw1
dw2
]=
[dp1dp2
](5)
Rewrite the system of equation into[V11X1
]dw1 +
[V12X1
]dw2 = dp1[
V21X2
]dw1 +
[V22X2
]dw2 = dp2
⇒
[w1V11p1X1
]dw1w1
+[w2V12p1X1
]dw2w2
= dp1p1[
w1V21p2X2
]dw1w1
+[w2V22p2X2
]dw2w2
= dp2p2
(20)The terms in brackets are the shares of each factor’s earnings (j) inthe total revenue of the industry (i), denoted by θij , and 0 < θij < 1.[
θ11 θ12θ21 θ22
] [w1
w2
]=
[p1p2
](21)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 30 / 33
Stolper-Samuelson Theorem: A Formal Proof
Invert the equation system,[θ22/Dθ −θ12/Dθ
−θ21/Dθ θ11/Dθ
] [p1p2
]=
[w1
w2
](22)
where Dθ ≡ θ11θ22 − θ12θ21 > 0 and 0 < θij < 1.
The magnitudes and signs of the mapping in (22) are as follows.[> 1 < 0< 0 > 1
] [p1p2
]=
[w1
w2
](23)
The S-S Theorem is given formally by the magnification relationships:
w1 > p1 > p2 = 0 > w2 w2 > p2 > p1 = 0 > w1 (24)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 31 / 33
Stolper-Samuelson Theorem: A Formal Proof
Invert the equation system,[θ22/Dθ −θ12/Dθ
−θ21/Dθ θ11/Dθ
] [p1p2
]=
[w1
w2
](22)
where Dθ ≡ θ11θ22 − θ12θ21 > 0 and 0 < θij < 1.
The magnitudes and signs of the mapping in (22) are as follows.[> 1 < 0< 0 > 1
] [p1p2
]=
[w1
w2
](23)
The S-S Theorem is given formally by the magnification relationships:
w1 > p1 > p2 = 0 > w2 w2 > p2 > p1 = 0 > w1 (24)
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 31 / 33
Stolper-Samuelson Theorem: Policy Implication
There will be political fights over changes in trade policy.
While free trade results in aggregate gains in income, those gainsare very unevenly distributed. Some factor owners generally lose.
This is in turn the source of considerable political controversy overprotection and liberalization
A country’s scarce factors may lose following trade liberalization.
There is a sense in which American unskilled workers competeagainst workers in the developing world.
However, the policy options are not just free trade versusrestricted trade, but possibly include free trade versus variousmeasures to help adversely affected workers (education, training,relocation assistance).
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 32 / 33
SummaryA country’s comparative advantage, production and trade aredetermined by underlying factor endowments intersected withtechnologies.
Relative factor endowments across countries+
Relative factor intensities across industries=
Comparative advantage
Changing the underlying factor endowment can have very biasedeffects on production and trade (Rybczynski).
Higher savings rates and capital formation in Asia naturally lead toa shift in capital intensive manufacturing toward Asia.
While free trade results in aggregate gains in income, those gainsare very unevenly distributed. Some factor owners generally lose(Stolper-Samuelson).
Yiqing Xie (Fudan University) Int’l Trade - H-O Oct. 11, 2013 33 / 33