international trade lecture 3: the heckscher-ohlin...

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

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

Identical Technologies

CRS and Perfect Competition

Identical Homogeneous Demand

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

country h is said to be V1 abundant, f is V2 abundant.

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

country h is said to be V1 abundant, f is V2 abundant.

Heckscher-Ohlin Model: DataFactor Intensities

Heckscher-Ohlin Model: DataRelative Factor Endowments

Heckscher-Ohlin Model: DataWorld Factor Endowments

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.

Heckscher-Ohlin Model: A Special Case

Figure 8.1

Figure 8.2

Heckscher-Ohlin Model: A Special Case

Figure 8.1 Figure 8.2

Heckscher-Ohlin Theorem: a Formal ApproachZero Profit ConditionConsider a single country,[

c1(w1, w2)c2(w1, w2)

[a11 a12a21 a22

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.

c1(w1, w2)c2(w1, w2)

[a11 a12a21 a22

c1 = a11w1 + a12w2 = p1

c2 = a21w1 + a22w2 = p2

c1(w1, w2)c2(w1, w2)

[a11 a12a21 a22

c1 = a11w1 + a12w2 = p1

c2 = a21w1 + a22w2 = p2

Heckscher-Ohlin Theorem: a Formal ApproachZero Profit Condition[

dc1dc2

[a11 a12a21 a22

] [dw1

[dp1dp2

By Cramer’s Rule,[a22/D −a12/D−a21/D a11/D

] [dp1dp2

Since X1 is V1 intensive, so that

a11a12

>a21a22

a11a22 > a12a21 a11a22 − a12a21 ≡ D > 0 (7)

Let p2 = 1 and dp2 = 0,[dw1

]dp2=0

< 0 (8)

dc1dc2

[a11 a12a21 a22

] [dw1

[dp1dp2

] [dp1dp2

a11a12

>a21a22

a11a22 > a12a21 a11a22 − a12a21 ≡ D > 0 (7)

]dp2=0

< 0 (8)

dc1dc2

[a11 a12a21 a22

] [dw1

[dp1dp2

] [dp1dp2

a11a12

>a21a22

a11a22 > a12a21 a11a22 − a12a21 ≡ D > 0 (7)

]dp2=0

< 0 (8)

dc1dc2

[a11 a12a21 a22

] [dw1

[dp1dp2

] [dp1dp2

a11a12

>a21a22

a11a22 > a12a21 a11a22 − a12a21 ≡ D > 0 (7)

]dp2=0

< 0 (8)

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)

da11dp1

< 0da12dp1

> 0da21dp1

< 0da22dp1

> 0 (9)

da11dp1

< 0da12dp1

> 0da21dp1

< 0da22dp1

> 0 (9)

Heckscher-Ohlin Theorem: a Formal ApproachFactor Market Clearing Condition

[a11 a21a12 a22

Apply Eq. (7), [a22/D −a21/D−a12/D a11/D

] [V1V2

Divide the first equation by the second in (11),

a22 − a21 V2V1−a12 + a11

[a11 a21a12 a22

] [V1V2

a22 − a21 V2V1−a12 + a11

[a11 a21a12 a22

] [V1V2

a22 − a21 V2V1−a12 + a11

da11dp1

< 0da12dp1

> 0da21dp1

< 0da22dp1

> 0 (9)

a22 − a21 V2V1−a12 + a11

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.

da11dp1

< 0da12dp1

> 0da21dp1

< 0da22dp1

> 0 (9)

a22 − a21 V2V1−a12 + a11

da11dp1

< 0da12dp1

> 0da21dp1

< 0da22dp1

> 0 (9)

a22 − a21 V2V1−a12 + a11

da11dp1

< 0da12dp1

> 0da21dp1

< 0da22dp1

> 0 (9)

a22 − a21 V2V1−a12 + a11

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.

Heckscher-Ohlin Model: A Formal ApproachRecall that country h is V1 abundant and country f is V2 abundant.

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.

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.

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.

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.

Factor-Price-Equalization Theorem (FPE)Unit Value Isoquant and Unit Value Isocost

For both countries,

a22a21

>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

Factor-Price-Equalization Theorem (FPE)World Edgewood Box

Figure 8.6

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.

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.

Rybczynski Theorem: Graphical Presentation

Figure 8.7

Figure 8.8

Rybczynski Theorem: Graphical Presentation

Rybczynski Theorem: Formal Proof[a11 a21a12 a22

Total derivative of (10):

a11dX1 + a21dX2 = dV1 =

[V11X1

]dX1 +

[V21X2

a12dX1 + a22dX2 = dV2 =

[V12X1

]dX1 +

[V22X2

]dX2 (13)

Dividing the total factor endowments V1 and V2,[V11V1

[V21V1

X2=dV1V1

(14)[V12V2

[V22V2

X2=dV2V2

a11dX1 + a21dX2 = dV1 =

[V11X1

]dX1 +

[V21X2

a12dX1 + a22dX2 = dV2 =

[V12X1

]dX1 +

[V22X2

]dX2 (13)

[V21V1

X2=dV1V1

(14)[V12V2

[V22V2

X2=dV2V2

a11dX1 + a21dX2 = dV1 =

[V11X1

]dX1 +

[V21X2

a12dX1 + a22dX2 = dV2 =

[V12X1

]dX1 +

[V22X2

]dX2 (13)

[V21V1

X2=dV1V1

(14)[V12V2

[V22V2

X2=dV2V2

Rybczynski Theorem: Formal Proof

The share of factor j used in good i – λij

Proportional change in a variable – ˆ[λ11 λ21λ12 λ22

Invert the equation system,[λ22/Dλ −λ21/Dλ

−λ12/Dλ λ11/Dλ

] [V1V2

Dλ ≡ λ11λ22−λ12λ21 > 0 whereλ22

λ11λ22 − λ12λ21> 1 given 0 < λij < 1

Rybczynski Theorem: Formal Proof

The share of factor j used in good i – λij

Proportional change in a variable – ˆ[λ11 λ21λ12 λ22

Invert the equation system,[λ22/Dλ −λ21/Dλ

−λ12/Dλ λ11/Dλ

] [V1V2

Dλ ≡ λ11λ22−λ12λ21 > 0 whereλ22

λ11λ22 − λ12λ21> 1 given 0 < λij < 1

Rybczynski Theorem: Formal ProofThe magnitudes and signs of the mapping in (17) are as follows.[

> 1 < 0< 0 > 1

] [V1V2

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

> 1 < 0< 0 > 1

] [V1V2

X1 > V1 > V2 = 0 > X2 X2 > V2 > V1 = 0 > X1 (19)

> 1 < 0< 0 > 1

] [V1V2

X1 > V1 > V2 = 0 > X2 X2 > V2 > V1 = 0 > X1 (19)

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.

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

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

Stolper-Samuelson Theorem: Graphical Presentation

Figure 8.9

Figure 8.10

Stolper-Samuelson Theorem: Graphical Presentation

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.

Stolper-Samuelson Theorem: A Short ProofValue of marginal product conditions for competitive equilibrium

w1 = p1MP11 = p2MP21 w1/p1 =MP11 w1/p2 =MP21

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

w1/p1 ↑ w1/p2 ↑w2/p1 ↓ w2/p2 ↓

w1 > p1 > p2 = 0 > w2

w1/p1 ↑ w1/p2 ↑w2/p1 ↓ w2/p2 ↓

w1 > p1 > p2 = 0 > w2

w1/p1 ↑ w1/p2 ↑w2/p1 ↓ w2/p2 ↓

w1 > p1 > p2 = 0 > w2

Stolper-Samuelson Theorem: A Formal ProofStart from equation (5) from zero profit condition,[

a11 a12a21 a22

] [dw1

[dp1dp2

Rewrite the system of equation into[V11X1

]dw1 +

[V12X1

]dw2 = dp1[

]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

Stolper-Samuelson Theorem: A Formal ProofStart from equation (5) from zero profit condition,[

a11 a12a21 a22

] [dw1

[dp1dp2

Rewrite the system of equation into[V11X1

]dw1 +

[V12X1

]dw2 = dp1[

]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

Stolper-Samuelson Theorem: A Formal Proof

Invert the equation system,[θ22/Dθ −θ12/Dθ

−θ21/Dθ θ11/Dθ

] [p1p2

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

The S-S Theorem is given formally by the magnification relationships:

w1 > p1 > p2 = 0 > w2 w2 > p2 > p1 = 0 > w1 (24)

Stolper-Samuelson Theorem: A Formal Proof

Invert the equation system,[θ22/Dθ −θ12/Dθ

−θ21/Dθ θ11/Dθ

] [p1p2

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

The S-S Theorem is given formally by the magnification relationships:

w1 > p1 > p2 = 0 > w2 w2 > p2 > p1 = 0 > w1 (24)

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).

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=

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).

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