The Dynamic Heckscher-Ohlin Model:

A Di ti A l iA Diagrammatic Analysis

日本国際経済学会関西支部研究会2012年3月17日関西学院大学

Eric W. BondDepartment of Economics, Vanderbilt University

Kazumichi IwasaKIER, Kyoto University

Kazuo NishimuraKazuo NishimuraKIER, Kyoto University

Plan of my talkPlan of my talk

Settings of dynamic H O models Settings of dynamic H-O models

Motivation of our studies

Results in H-O models including our recent studies

Purpose of this paper

Findings Findings

Settings of our model

Diagrammatic explanations for results in literature and our models

Conclusions

Dynamic versions of the H-O modelDynamic versions of the H O model

Each country has access to the same technology for producing two Each country has access to the same technology for producing two

goods using a fixed factor (labor) and a reproducible factor (capital)

under conditions of perfect competition and constant returns to scale.

Factors of prod ction are ass med to be mobile bet een sectors Factors of production are assumed to be mobile between sectors

within a country, but immobile internationally, and there are no markets

for international borrowing and lending.

Countries (Home and Foreign) have identical preferences with

homotheticity and a constant intertemporal elasticity of substitution

(CIES). E E E E

Motivation of our previous studiesMotivation of our previous studies

Houthakker and Taylor (1970) find that the share of income spent on foody ( ) p

declines and the share spent on services rises with household income.

Hunter (1991) estimated a linear expenditure system for 11 product

categories across 34 countries, and found that departures from homotheticity

have a significant impact on trade volumes.

Ogaki and Atkeson (1997) find evidence of differences in the IES with the

level of household wealth.

Jensen and Miller (2008) provide evidence that two staple commodities, rice

and wheat, are Giffen goods in China.

M ti ti f i t diMotivation of our previous studies (cont.)(cont.)

How do departures from the assumption of homothetic

preferences and CIES affect the pattern of trade and

the dynamics around the steady states of dynamic H Othe dynamics around the steady states of dynamic H-O

models?

Results in H-O models with homothetic preferencesResults in H O models with homothetic preferences

(i) There is a continuum of steady state equilibriaTK

2T ATK K K (i) There is a continuum of steady state equilibria

under free trade, each of which is a saddle point

characterized by incomplete specialization in

TTK K

AKT TK K production, and is constant.

(ii) The static H-O theorem holds: the country that

0K

TK is capital abundant in the steady state exports

the capital intensive good.

O0K AK TK

(iii) The initially capital abundant country remains

capital abundant, and will export the capital

intensive good on the path to the steady state

TKcf. Atkeson and Kehoe (2000) discuss thepossibility that countries that start thedevelopment process later end up with a

intensive good on the path to the steady state.

(The dynamic H-O theorem holds: The future

trade patterns are determined by the initial p p plower capital labor ratio and level ofincome than countries that develop earlier.

relative factor endowments.)

R l i i di i h l dResults in our previous studies with normal goods

If labor productivities and discount

f h i 2T ATK K K

TK

factors are the same across countries,

then the main results of the benchmark

2T ATK K K TTK K

H-O model will hold as long as goods

are normal in consumption.AK

TK

The primary difference introduced in this

case is that the world capital stock in the 0K

steady state will depend on the

distribution of income across countries.

O0K AK TK TK

Results in our previous studies with normal goodsesu ts ou p ev ous stud es w t o a goods(cont.)

In autarky, Home is the capital abundant (scarce) country

The case of ( : labor productivity in Foreign)L L L

y p ( ) y

in the steady state: ,A A A AK K K K

L L L L

iff labor-intensive good 1 is a necessity (luxury).

L L L L

This can yield the violation of the static H-O theorem.

The dynamic H-O theorem will hold if preferences are

homothetic with CIES T TK K K K homothetic with CIES. 0 0 K K K KL L L L

Results in our previous studiesResults in our previous studieswith an inferior good

In the case where the laborTK

intensive good is inferior,AHK

TTK K

there may be multiple steady states

in autarky and free-trade steady

HK

AK

1E

states where the static H-O

AMK

AK2E

theorem is violated and/or the

saddle-point stability does not hold. O

ALK

ALK A

MK AHK

3ETKLK MK HK K

Purpose of this paperPurpose of this paper

To show that main results in dynamic H-O models (with

non-homothetic preferences) can be derived from

diagrams that represent the basic functions in static

d l hmodels such as

(i) the Rybczynski line Production side(i) the Rybczynski line,

(ii) an income expansion path,

Production side

Demand side

(iii) an excess demand function.

FindingsFindings

We can define their steady state versions and show thatWe can define their steady state versions and show that

(i) the “steady state” Rybczynski line and (ii) the income expansion

path evaluated at the “steady state prices” yield (iii) the “steady

state” excess demand function that specifies the country's excess

demand as a function of its capital stock.

Using the excess demand functions for each of the countries, weg

can derive the locus of home and foreign capital stocks that are

consistent with a steady state equilibrium with free trade.consistent with a steady state equilibrium with free trade.

Also, we can see the stability of steady states and the steady state

t d tt l f th i htrade pattern only from their shapes.

The production sideThe production side

LetLet

( ) : labor endowment in Home (Foreign),L L

( , ) : unit cost function in s

( ) : captital stock in Home (Foreign),

ector .i

K K

a w r i

Under incomplete specialization,

the competitive profit conditions require that

( )a w r p1

2

( , ) ,

( , ) 1,

a w r p

a w r

factor prices, ( ) andand we obtai ( ).n w p r p

The Stolper-Samuelson theorem

Totally differentiating the competitive profit conditions yield

1 1

2 2

( , ) ( , ) ,

( , ) ( , ) 0,w r

w r

a w r dw a w r dr dp

a w r dw a w r dr

where and are equal to liw ira a abor and capital input coefficients

in sector , respectively, and we assume they are all positive.i

r1( , )a w r p

Let good 1 be labor intensive:a a

p

2 ( , ) 1a w r ( )r p

1 21 2 2 1

1 2

( )

.0r rw r w r

w w

a a a a a aa a

apw p r

O

2 ( , )

( )

21 1

( )Then, ( )

rw r

apw p ra aw p w

1

a

O w( )w p2and ( ) 0.war p

The Rybczynski theoremFactor market equilibrium requires that

and KL a Y a Y a Y a Y

1 1 2 2 1 1 2 2

1 2

and ,

( ) (

)

w w r rK

w p L r p K p

L a Y a Y a Y a Y

Y Y

2Y

1 1 2 2a Y a Y L

K

2 2( )

which yield

r wa L a KY p K L

1 1 2 2w wa Y a Y L

2 ( , , )Y p K L a Y a Y K

1

2 21

(

( , , )

( ) ( ), , ,)

r wY p K L

w p L r p KY p K L

O 1Y1( , , )Y p K L

1 1 2 2r ra Y a Y K

12

1( , , ) r wa L a KY p K L

2 ( , , ) ( )Y p K L w rp L

1 2

( ) [ ( ) ( ) ].

( )Then 0 and 1

p K p w p L r p K

Y Y K r pr Kr

2

Then, 0 and 1.( ) ( )

rK K Y w pw L r pr K

The consumption sidep

1 20max ( , ) exp( ) ,u C C t dt

th di t tsubject to

( ) ( ) iC C

the discount rate

the rate of depreciation on capital

:

::

1 2 0

1 2 1 2

( ) ( ) , given.

w p L r p K pC C K K K

pY Y pC C K K

on capital

: the shadow value of capita

:

l

1 2 1 2

1 2 1 2Let ( , ) [ ( ) ( ) ].u C C w p L r p K pC C K

The necessary conditions for optimality are

1 1 2 2 1( , ) , ( ,u C C p u C 2 ) , [ ( )], and lim ( ) ( ) exp( ) 0.t

C r p K t t t

the marginal utility of the shadow value of capital

the consumable capital

Monotonic relation between K and rMonotonic relation between K and r

1

1

( )

YK p r p

C

Generally, there is a negative

relation between capitalr

relation between capital

accumulation and the rental on

capital: the rental rate

decreases when capital stock

( )r r p increases.

This yields the uniqueness of

O A

This yields the uniqueness ofan autarkic steady state and itssaddle-point stability O KAK0Ksaddle point stability.

Non-monotonic relation between K and rNon monotonic relation between K and r

Suppose that labor-intensive good 1

pp g

is inferior.

Then the more capital countries

1

1

( )

YK p r p

C

Then, the more capital countries

accumulate, the less labor-intensive

good is demanded and hence the

rgood is demanded, and hence the

more capital is needed for producing

goods.goods.

So, there is a possibility of a non-

t i l ti b t th ( )r r p

monotonic relation between the

capital stock and the return on

capital and the model will exhibit Ocapital, and the model will exhibit

rich dynamic properties.O KA

LK AHKA

MK

The steady state Rybczynski line

( ) ( )

At any steady state in Autarky,

r p K w p L CY 2 2,Y C1 1

2 2

( ) ( ) ,

hold.

r p K w p L C

Y C K

Y

2 2, K

1 1 2 2

So,

,w wa Y a Y L 1 2 2( ) ( )

( )p C C w p k p L

r p

1 2 ( ) ( )

( ( ) ( )) 1 2

pY Y w p L r p K

a a w p r p i

1 2 ( )pC C w p L K ( ( ), ( )), 1, 2

yield

iw iwa a w p r p i

2 ( )Y K K

O1

1 22 2

1 ( ) ,( ) ( )

w

w w

a w pC C La r p a r p

1( )Y K 2 1,r ra L a L

1 1,Y CO

The steady state1pC 2 ( ) ( ) .C w p L r p K

The steady stateRybczynski line

Lemma 1

Let be the steady state capital stockK

Lemma 1

Let be the steady state capital stock.

Then, the outputs of two goods at the steady state,

K

1 2( ( ), ( )), are derived from the intersectionY K Y K

between the steady state Rybczynski line,

( )p

r p

1 2 2( ) ( ) ,

d th t d t t t i t

C C w p k p L

1 2

and the steady state resource constraint,

( ) ,pC C w p L K

1 2 1 2as ( ( ), ( )) ( , ).Y K Y K C C K

The autarkic steady stateThe autarkic steady state 1 2Notice that ( ), ( ) denotesC K C K

2CTh i i th ith

K the consumption bundles at a steady

state with capital stock ,KThe income expansion path with p

1 1

state with capital stock ,

and that ( ) ( ) is an excess

demand for good 1

K

C K Y K

2AC

demand for good 1.

The steady state Rybczynski line

The budget constraint

2 ( )C K

2 ( )Y K K

2

The steady state resource constraint

1CO

The budget constraint

for households with

capital stock andK1( )C K 1( )Y K

2 ( )

1AC

The steady state resource constraint

capital stock and

investment

K

K

P i i 1Proposition 1

An intersection between the steady state Rybczynski line and

the income expansion path with the steady state price of good 1the income expansion path with the steady state price of good 1

corresponds to an autarkic steady state:

1 ( ) ( )

AA C w p LK

r p

2 1 2and ( , ). A A Au C C

Therefore, it uniquely exists as long as labor intensive good 1

is normal and preferences exhibit neither a satiation level noris normal and preferences exhibit neither a satiation level nor

a minimum subsistence level.

The excess demand for good 1The excess demand for good 1

2C

Homothetic1Z 1 1 1( ) ( )Z C K Y K

Homotheticpreferences

AK KO K1 2 ( )pC C w p L K

1CO1( )C K 1( )Y K

Lemma 2Lemma 2

1Z

1 1 1( ) ( )Z C K Y K With normality in consumption,

AK KOthe steady state in autarky is a

saddle point.p

Intuition: Lemma 2 can be interpreted as indicating that if an increase in the

capital stock above the autarkic steady state creates an excess demand for thep y

labor intensive good, then there will be an increase in the price and a decrease

in the rental on capital in the economy.in the rental on capital in the economy.

The foreign countryThe foreign countryZ H Z L L

The home and foreign countries

Assumption:1 1 1( ) ( )Z C K Y K

1 1 1[ ( ) ( )]H Z H C K Y K

1 1,Z H Z L L

have identical utility functions, ,

identical technologies, , 1, 2,i

u

a i O 2K2K

K1K

1K

AK

AKg , , , ,

, and .

i

A steady state equilibrium with trade is a pair ( , ) such that

( ) ( ) 0 i h ( ) ( ) d ( ) ( )

K K

Z K H Z K K k L k L K k L k L

1 1 1 2 1 2( ) ( ) 0 with ( ) , ( ) and ( ) , ( ) ,

where is the number of households in the foreign country an d

Z K H Z K K k p L k p L K k p L k p L

H Z

1 1(.) (.)Z

iff .L L

Lemma 3

With lit i ti ll thWith normality in consumption, all the

free trade steady states are saddlefree trade steady states are saddle

points.points.

(The intuition for this result is the same as

for Lemma 2.)

Steady state values of and

Letting ( ) be one of the steady stateT TK K Letting ( , ) be one of the steady state

free trade pairs, the values of and at

K K

the steady state are given by

2 1 2 ( ( ), ( ))and T T Tu C K C K

2 1 2 and ( ( ), ( ) .)T T Tu C K C K

The locus of ( , ) with 1 and T TK K H L L

1Z 1 1 1( ) ( )Z C K Y K

2C

1 2 2( ) ( )pC C w p L k p L 1( )k p L 2( )k p L

AK1 2 2

KO

TK

2( )k p L

AK

1 2 ( )pC C w p L K

1CO

1( )k p L1( )k p L

1 2 1( ) ( )pC C w p L k p L AKOO

1( )k p L 2( )k p L TK

R k 1Remark 1

For given technologies, preferences, and a labor endowment in each country,

we can draw, for each country,we can draw, for each country,

(i) the steady state Rybczynski line,

(ii) the income expansion path with ,

and (iii) the stea

p

dy state resource constraints at the highest and the lowest

capital stocks consistent with incomplete specialization.

They yield the steady state excess demand function for each country fromThey yield the steady state excess demand function for each country, from

which we can precisely derive the locus of ( , ).T TK K

Proposition 2 (Heckscher-Ohlin theorem)Proposition 2 (Heckscher Ohlin theorem)

1 1,Z H Z

TK

Let goods be normal and L = L*.

Th th i iti ll it l1 1 1( ) ( )Z C K Y K

1 1 1[ ( ) ( )]H Z H C K Y K

TKThen, the initially capital

abundant country remainsAK KO

capital abundant along the

dynamic general equilibrium

TK TTK K y g q

path to the steady state, and the

capital abundant country exportsAKcapital abundant country exports

the capital intensive good at theTK

0K

steady state.AKOO TKTK0K 0K

An example with inferior goodsAn example with inferior goods

2C The income expansionpath with p

A multiplicity of steady states in autarky

is possible when labor intensive good 11 2 2( ) ( )pC C w p L k p L

2

is inferior at some range of income and

, the labor input coefficient in capitalwa2 , p p

intensive sector 2, is sufficiently

w

small.

11 2

2 2

The steady state Rybczyn 1 ( )( ) (

ski )

line: w

w w

a w pC C La r p a r p

1CO1( )k p L

2 2( ) ( )w wp p

1 2 1( ) ( )pC C w p L k p L

The non-monotonic excess demandThe non monotonic excess demand

2C The income expansionpath with p 1Z

1 1 1( ) ( )Z C K Y K

1 2 2( ) ( )pC C w p L k p L

OK

AK1( )k p L

2( )k p L

ALK A

MK

O HK 2( )k p L

1CO1( )k p L

1 2 1( ) ( )pC C w p L k p L

Proposition 3 (Stability condition)

If the steady state demand function in each country isy y

upward sloping at the value of capital stock in a free trade

steady state, then the steady state is a saddle point.

If it is downward and the discount factor in each country is

the same, then the steady state is unstable.the same, then the steady state is unstable.

The locus of ( , ) with 1 and T TK K H L L

T

C( ) ( )Z C K Y K

TK

2( )k p LE

K1( )k p LALK A

MK

1 1 1( ) ( )Z C K Y KAHK

E

E

b

D

AHK

1( )p

2( )k p LL M

ALK

AMK

A B

B

bD

A

1( )k p L

L

C

b

S ddl i t ith th t ti TKO1( )k p L 2( )k p LA

HKALK A

MKSaddle points with the static

H-O theorem being violated

Remark 2

T

Remark 2

Under the symmetry on

C

TK

2( )k p LE

Under the symmetry on

countries’ fundamentals except

AHK

E

E

b

D

for their initial capital stocks,

the capital abundant country

ALK

AMK

A B

B

bD

remains capital abundant along

the trajectory to the steady state.A

1( )k p L

L

C

bj y y

However, it is possible that theTKO

1( )k p L 2( )k p LAHKA

LK AMKtrade pattern varies along the

path due to inferiority inpath due to inferiority in

consumption.

Concluding remarksConcluding remarks

We have shown that main results in dynamic H-O models (with non-

homothetic preferences) can be derived and/or examined from some

diagrams which represent the basic functions in static models such as the

Rybczynski line, an income expansion path, and an excess demandy y , p p ,

function.

i h l i f d l b d i hFor given technologies, preferences, and a labor endowment in each

country, we have derived the diagrams and shown that they can clarify not

only the existence and the multiplicity of steady states in autarky and

under free trade, but also their stabilities and the static and the dynamic

Heckscher-Ohlin theorems.