the dynamic heckscher-ohlin model: adi ti a l ia ...the dynamic heckscher-ohlin model: adi ti a l ia...
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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.