Factor Price
Equalization
in
a Dynamic
Economy
Joseph
E.
Stiglitz
Co
wles Foundation, Yale
University
Two of the most
important propositions of the modern
theory of inter-
national trade are
extensions of the Heckscher-Ohlin analysis
of compara-
tive
advantage: Free
international trade completely equalizes
factor
prices (and thus ensures
world Pareto optimality), and
the removal of
protective barriers
decreases the return of the scarce factor
in terms of
both commodities and
increases that
of
the abundant
factor.' Both
of
these propositions
were proved under seemingly general
conditions,
although
in
the context of
a static model.
But whether these
results still hold in a dynamic
economy has, almost
without exception, gone
uninvestigated in the literature.2
in
a dynamic
model, moreover, there is
the additional question of whether
international
trade,
even
if
it equalized
rental rates on machines, would
equalize
rates
of
interest,
in
particular if
the two countries had different
rates
of time
preference. Recently,
Samuelson (1965) showed that if
there is non-
specialization (and the
other assumptions of the Samuelson-Heckscher-
Ohlin model
are
satisfied), then interest rates as well as rental
rates are
equalized.
It
has long been
recognized, however, that if factor supplies
are variable,
specialization is much more
likely to occur. Indeed, one
of
the principal
reasons Ohlin
(1933) did
not argue for complete factor price
equalization
The research
described in this
paper was
carried out under grants
from
the
National
Science
Foundation and the
Ford
Foundation and was
completed while at
the
Institute for
Development
Studies, University
College, Nairobi,
under a grant
from
the Rockefeller
Foundation. I
wish to thank D.
Cass, R.
Cooper, and the referee
for
helpful
discussions and
comments.
'
See Stolper
and Samuelson
(1941) and
Samuelson (1948,
1949, 1953-54).
The
crucial
assumptions are: (a)
endowment ratios
are not
separated by one or
more
factor intensity
reversals, and
(b) both
countries remain
incompletely specialized.
(In addition,
of course, the
two countries
must have identical
constant returns
to
scale
production functions.)
For a thorough
discussion of
these
assumptions,
see
Johnson (1958).
2
See
Inada
(1968). Other
articles investigating
dynamic
international
trade
models
include those of
Bardhan (1965,
1966), Kenen
(1965), and Oniki
and Uzawa
(1965).
456
FACTOR PRICE EQUALIZATION 457
but only
for a
"tendency"
for
equalization
was his
concern about
elastic
factor
supplies: "There
can
be little doubt that
as
a rule
supply reactions
tend to offset the
price-equalizing
tendencies
of
trade"
(p. 124).
He
argued
that,
as the
return
to
the abundant
factor
increases,
its
supply
(relative
to the
supply
of
the other
factor)
increases.3
Thus,
in
the
long
run,
free
international trade
tends to increase the
disparity
in
factor
supplies between the two
countries,
and
if
this
increase
is
sufficient
will
result
in
specialization
and
nonequalization
of
factor
prices.
Indeed,
Ohlin
went so
far as to
suggest that
"trade
means
specialization"
(p.
125).
The
purpose
of this
paper
is to
investigate
in
detail
the
long-run
supply
responses of
capital and their
implications
for the
classical
propositions.
We
focus on
the behavior of a
two-country
model
in
which
the
long-run
rate of
interest
in
each
country
is
fixed,
for
example,
by the
pure
rate
of
time
preference
in
the case of
"rational"
savings
behavior,
or
by
the
savings
behavior of
capitalists
if
workers save
nothing.
In
this
case,
unless
the
two
countries have identical
long-run interest
rates, at least one of
the
two
countries
must specialize.
For
in
long-run
equilibrium,
nonspecializa-
tion
at
a
common
price ratio
implies the same rate
of
interest as well as
the same
wage-rentals ratio. The world
price
ratio
may be
that of
one or the
other
country, or
something
in
between
if
both
specialize.
In
the un-
specialized
country, factor
prices are
unchanged, but the
relative
scale of
the
two
industries will
vary as a
result of trade. In the
specialized
country,
the
relative
price
of
imports
is
below the
autarky
(no-trade) price.
If
imports of
the specialized
country are
capital
goods, the
capital-labor
ratio in
the
remaining
consumption-goods
industry
must
rise to
restore
the old
rate of
interest,
thereby
raising the
wage-rentals
ratio as well
as
wages
in
terms of
both
goods. Whether
this
makes
factor
price
differentials
between
the two
countries
larger
or
smaller
depends
on
whether
the
consumption-goods
exporter's
wage-rental
ratio is
larger or
smaller
than
the
capital-goods
exporter's
wage-rental
ratio,
and this
in
turn
depends
on
whether
the
consumption-goods
sector
is more or
less
capital-intensive
than
the
capital-goods sector.
Thus,
free
trade,
in
the case of
the
normal
capital
intensity
hypothesis,
increases
factor price
differentials
and (in
the
specialized
country) increases
the
return
of
the
relatively scarce
factor
(labor) in
terms
of
both
commodities, as
well as
relative to
the
returns to
capital. Since
the
capital-labor
ratio in
each
sector in
the
unspecialized
country
is
unchanged,
whether its
aggregate
capital-labor
ratio
goes up
or
down
depends
on
whether
it
exports the
capital-intensive or
the labor-
intensive
commodity.
On
the
other
hand,
whether the
(specialized)
consumption-goods
exporter
increases or
decreases his
capital-labor
ratio
depends
not
only
on
whether
consumption
is
capital- or
labor-intensive
but
also
on
the
extent
to
which
the price of
consumption
after
trade is
greater than
that
before
trade, for
the
higher
price of
consumption
goods
raises
profits
(the return to
capital).
If
the
consumption-goods
sector is
I
See
also Walsh
(1956),
Vanek
(1959),
and
Caves
(1960).
458
JOURNAL
OF POLITICAL
ECONOMY
capital-intensive, the Ohlin
proposition
that
trade
increases factor
supply
differentials is valid,
but
if
it
is
labor-intensive, it may not
be.
On
the
other
hand,
if
only
one of
the
two
countries
specializes
and
it
exports capital
goods, since the interest rates
are
unchanged,
the
capital-
labor ratio in the
capital goods
industry (the only
remaining one) is
unchanged
and
the
wage-rentals ratio
is
unchanged. Thus factor
price
ratio
differentials after trade
are
identical with those before trade.
But
while in the
unspecialized country
both factors have the
same
factor
prices
in
terms of
both commodities,
in
the specialized
capital-goods
exporting country, both
factors are
better off in terms
of
consumption
goods,
since
its relative
price has
fallen (as compared with the
autarky
situation). Since
in
each
industry the
free-trade equilibrium
capital-labor
ratio
is identical with
the pre-trade
capital-labor ratio, the
country
with
the
lower rate of
interest and
the higher
aggregate capital-labor ratio
(and
which
will
export
the
capital-intensive commodity)
will increase its
capital-labor ratio,
and conversely for
the high interest
rate economy,
so
factor
supply
differentials are always
increased in this case.
After setting up
the basic model in
Section 1, and
analyzing the pre-trade
equilibrium
in
Section
11, we investigate
the long-run
free-trade equilibrium
in
Section
Ill.
In
Section IV, we show
that, because the
high interest rate
(high
time
preference) country is
willing to trade future
consumption
for
present consumption
with the low
interest rate country,
it always has
a
lower
long-run
consumption with
free trade than
pre-trade, and
con-
versely
for
the low
interest rate
economy. In Section
V, the patterns
of
specialization
are
investigated.
Section VI considers the
effects
of
trade
policy
on
the
long-run equilibrium; it
is shown that a
tariff (export sub-
sidy) may lead to a
higher
consumption per capita in
both countries.
The
reason
for this is
simple: Assume, for
instance, that the
exporter of
con-
sumption goods is
specialized, the
importer
unspecialized. The exporter,
by
imposing,
for
example, an export
subsidy, raises the
domestic price
of
consumption goods
above the foreign.
Since the rate of
interest is fixed,
this
leads to
an
increase
in
the
capital-labor ratio, and to
an increase in
coin-
sumption per
capita. The increased
demand for capital
goods leads the
capital goods
exporter to shift
resources to that sector,
which leads, if it is
the
capital-intensive
sector, to an
increase in its
aggregate capital-labor
ratio
and per capita
consumption.
Section
VII
considers some
extensions of the analysis.
In the Appendix,
the
nature of
the
dynamic path is
analyzed.
I. The
Basic
Model
A.
Technology'
As
usual, we
assume that the
production functions have
constant returns
to
scale,
are
identical
in both countries,
and satisfy the
usual assumptions of
FACTOR PRICE EQUALIZATION 459
concavity and
differentiability:
C =
C(KC, Lc)
=
LcC(Kc/LC,
1)
=Lcc(k)
(
Z =
Z(KZ, LU) LZ(KZ/LZ, 1)
=Lzz(k,),
where C is the
output of consumption goods,
Z
that
of
capital goods,
Kc
and
L,
the capital
and labor used
in the
consumption-goods sector,
and
kc
their
ratio;
similarly
for
capital goods. One
of
the
sectors
is
assumed
to
be more
capital-intensive than the
other;
that
is,
at
every
factor
price
ratio,4 the
capital-labor ratio
in
one sector is greater than that
in the other.
The aggregate
capital-labor ratio we shall denote by k
=
K/L.
B. Prices of
Factors and Commodities
The economy is
assumed to be competitive, so
each factor is paid its
marginal product.
Thus
r
=
max
[pc'(kj,)
z'(kz)];
w =
max
{p[c
-
kcc'(kc)],
z
-
kz7'(kz)},
(2)
where
r
is the
rental rate on machines,
wv
the wage
rate, and is the price
of
consumption goods
in
terms of capital, which,
if
there is
no
specialization,
is5
z'(k,)/c'(k,).
If
specialization occurs, then the
price will be
determined
to make
the value
of
exports equal to
the
value
of
imports.
C. Labor
Supply
Labor is assumed
to be growing at the same,
constant rate, n,
in both
countries,
so
that
the ratio
of
their labor supplies is
fixed,
LB/L`
=
a.
The
assumption
of
identical growth rates is, of
course, necessary for
the
existence of a
balanced growth path
in
which one
country
is not infinitesi-
mal relative to the
other.6
D.
Savings
In
most of the
analysis of this paper, we will
consider two alternative
hypotheses about
savings. First, rational
savings: We assume that the
aggregative
savings
behavior of
our economy
may be described by the
4
The
assumption
of no factor
intensity
reversals
is, of
course, crucial
in the
proof
of the
factor
price
equalization
theorem (see
Johnson
1958).
To ensure
the
existence
of
an
equilibrium,
we shall
also assume that
both
production
functions
satisfy the
Inada
conditions.
I
As
usual, if
there is
no
specialization,
the
wage-rentals
ratio
uniquely determines
the
capital
intensity in
each
sector and
relative
prices. As
the
wage-rentals
ratio
increases, the
price of
the good
which is
intensive
in labor
increases
relative
to the
price of
the other
good.
6
That is, if
one of
the
countries is
growing
more slowly
than
the other,
it
will
eventually
become
infinitesimal
in
relation to the
other.
This
is
subsumed
in
the
special
case of a
= 0
or
infinity.
460
JOURNAL
OF POLITICAL ECONOMY
behavior of
a
representative consumer
who
chooses
his
savings
rate
at
each
moment of time so
as to maximize
his
intertemporal
utility.7
For
most of
the analysis,
we employ the
conventional
assumption
that
the
utility
function is of the form8
W=
{
U[c(t)]e<
)ndt,
8
>
n,
(3)
where
6
is the
pure rate
of time preference.9
Utility
maximization
requires
that the
marginal
rate of
substitution
equal
the
marginal
rate
of
transfor-
mation,
or
dIn
U'(c)
+ ,
= r
(4)
Thus,
in
steady state,
r=6.
(5)
Although
the
technologies
of
the
two
economies are
identical,
there
is
no reason
to assume
that they have
identical
tastes; neither
8
nor U(c)
need
be the same for the
two.
Second,
Marxian
savings:
In
the Marxian
model,
a
constant
fraction
of
profits,
s,
is
saved, and none of
wages
(see Uzawa
1961).
Then
=
sr- ,
(6)
or,
in
steady state,
r
-
-
(7)
If we
set
6
-
n/s,
the
long-run behavior
is
identical for the two
models.
It should
be
emphasized, however,
that although
the long-run
implica-
tions of
these two
savings assumptions are
identical, the short-run
dy-
namics are
quite
different (see
Appendix).
In
the
following
discussion, we
shall denote
with a
superscript
A
the
country
with
the
higher
rate of time
preference
(the lower
savings rate),
and
by
a
superscript
B
the other
country.10
7 For
other
studies
employing this
approach
in other
contexts,
see Uzawa
(1968a,
1 968b).
8
Later,
we
consider an
alternative
form of the
utility
function.
We
weight
utility of per
capita
consumption by
the size of
the
population;
alternatively,
we
may think
of
6
-
n as the
"net rate"
at which
future
utility
is
discounted.
From
the
individual's
viewpoint,
the rate of
return
on capital
equals
rentals plus
expected capital
gains. In
long-run
equilibrium,
however, capital
gains
are
zero,
so we still
obtain
equation
(5).
10
Throughout
the
discussion, we
assume
3
:A+
:
(
SB). If
A
=
6B(sA
SB),
pre-trade
prices are
identical
in the
two
countries; even
when the
possibility
of
trade
opens up,
no trade will in
fact
take
place.