Factor Proportions and the Heckscher-Ohlin Theorem
Author(s): R. W. Jones
Source: The Review of Economic Studies, Vol. 24, No. 1 (1956 - 1957), pp. 1-10
Published by: Oxford University Press
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Factor Proportions and the
Heckscher-Ohlin Theorem'
1. Recent contributions to the pure theory of international trade have relied heavily
on the variable-proportions account of trade developed by Eli Heckscher and Bertil Ohlin2,
who linked export-import patterns to factor endowments and methods of production.
Their hypothesis is expressed in summary form in the Heckscher-Ohlin Theorem: A country
exports those commodities produced with relatively large quantities of the country's relatively
abundant factor. Perhaps the most controversial application of this theorem to
appear recently is that based on Leontief's input-output studies of the American economy.
According to Leontief, America is not capital-rich compared with the rest of the world, as
popularly supposed.3 His data suggest that American exports require a higher proportion
of labor to capital than do American imports (or rather, American import-competing
commodities). Leontief's conclusions then follow from his unqualified support of the
Heckscher-Ohlin dictum.
However, there are significant exceptions to the Heckscher-Ohlin theorem.4 The
theorem rests upon both a concept of factor abundance related to pre-trade factor price
ratios and special shapes for the production functions. Should these strict conditions be
modified, the Heckscher-Ohlin theorem in some cases no longer holds, in others becomes
meaningless. The purpose of this article is to clarify the meaning of the HeckscherOhlin
theorem add to demonstrate that there is no a priori basis for accepting the Leontief
conclusions; his data are not inconsistent with the commonly-held notion of American
factor endowments.
2. The two-country, two-factor, two-commodity framework customary in the variable
proportions account of trade is retained in what follows unless explicitly altered. That
is, each commodity in the two countries is sold in a purely competitive market and produced
under constant returns to scale. No transfer costs of any kind exist, so that the prices of
internationally traded commodities are equalised in the two countries. Fixed endowment
quantities of two homogeneous factors of production, capital and labor, are fully employed
in each country. The quality (but not quantity) of each factor in the two countries is
identical, as are the production functions. This assumption is made so that differences in
trading patterns and pre-trade price ratios are ascribable to differences in factor endowments
on the supply side and/or to differences in demand conditions, rather than to differences
in technologv.
1 In preparing this article I have benefited greatly by conversations with Professor Robert Solow and
my colleagues, Louis Lefeber and Michael Moss.
2 Heckscher, Eli: " The Effect of Foreign Trade on the Distribution of Income," Ekonomisk Tid
XXI (1919), pp. 497-512, reprinted as chapter 13 in Readings in the Theory of International Trade (P
Blakiston, 1949), pp. 272-300. Ohlin, Bertil: Interregional and International Trade, (Cambridge, Harvard
University Press, 1933).
3 Leontief, W., "Domestic Production and Foreign Trade ; the American Capital Position Reexamined,"
the Proceedings of the American Philosophical Society, Volume 97, No. 4, September, 1953,
reprinted in Economia Internazionale, Volume VII (February, 1954), pp. 9-38.
' The tendency towards factor-price equalisation, a corollary of the Heckscher-Ohlin theorem, has
been challenged by S. F. James and I. F. Pearce in " The Factor-Price Equalisation Myth," the Review
of Economic Studies, 1951-52, Vol. XIX (2), No. 49. For a statement and discussion of the factor-price
equalisation theorem see Samuelson, P. A. : " International Trade and the Equalisation of Factor Prices,"
Economic Journal, June, 1948, and " International Factor-Price Equalisation Once Again ", June, 1949,
and Lionel MacKenzie, " Equality of Factor Prices in World Trade," Econometrica, July, 1955, 239-257.
1
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REVIEW OF ECONOMIC STUDIES
Within the framework set out by these assumptions the concepts of
abundance and factor intensity each admit of more than one interpretat
of relative factor abundance is examined first; this part of the discussion is
strong Samuelson notion of factor intensity: For any set of factor pric
resource allocation in each country entails that one commodity (X) is always
a greater amount of capital per unit of labor ( px) than the capital/labor ratio
in the other commodity (Y).1
In addition, initially it is assumed that trade does not lead to comple
in either country.
3. Ohlin's definition of relative factor abundance rests on the pre-trade
prices in the two countries. Denoting the countries by the subscripts 1
and labor by C and L respectively, country 1 is relatively capital abund
sense if before trade is initiated:
2
By definition capital is relatively cheaper in the capital-rich country before trade. As
suggested by Samuelson in his factor-price equalisation articles, if the strong factor intensity
property holds, the assumptions underlying the Ohlin theory imply a unique relationship
between factor price ratios and commodity price ratios. This relationship is identical for
the two countries. Equality of commodity prices with trade must involve an equalisation
~7y
(1) _ _... .._.__
2 :
(1) (2)
FIGURE I
of the returns to the same factor in each country. Furthermore, any
ratio is associated uniquely with a ratio of commodity prices, as illus
The fact that capital is relatively cheaper in country 1 before trade impl
intensive commodity (X) is relatively less expensive in that country. Thu
capital-rich country must export the capital intensive commodity. With
of factor abundance and the strong assumptions of section 2 the Hecksch
although true, is trivial. The converse of the theorem also holds: if
1 Samuelson, P. A. : footnote on p. 165 of the 1948 article, op. cit. and p. 182 of the
2
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FACTOR PROPORTIONS AND THE HECKSCHER-OHLIN THEOREM 3
the capital intensive commodity, capital must have been the relatively cheap factor of
production in that country before trade.
However, this seems not to be the notion of factor abundance employed by Leontief
and most others who discuss this question. Applications of the Ohlin dictum hint at a
relationship between physical factor endowments and patterns of trade, rather than the
latter and the pre-trade constellation of factor prices. Indeed, Ohlin, himself, makes this
point succintly:
. . . the real problem is to demonstrate what lies behind such inequality in prices, or,
more precisely, to show in what way differences in equipment come to be expressed in
differences in money costs and prices.1
4. An alternative definition of relative factor abundance goes back to physical factor
endowments. According to this definition a country is relatively capital abundant if and
only if it is endowed with a higher proportion of capital to labor than the other country.
Thus country 1 is relatively capital-rich if:
(2) el > C2Li L2
where the bars denote the fixed factor quantities available in each country. This definition
and the Ohlin (1) definition are certainly not equivalent. Pre-trade factor price ratios
are determined both by conditions of supply and demand and are uniquely related to pretrade
commodity price ratios, whereas factor endowment proportions may be outweighed
by dissimilarities in demand in influencing price ratios.a Differences in relative endowments
impart a bias on the supply side, but not sufficient to insure validity of the HeckscherOhlin
theorem if this new definition (2) of factor abundance is used.
The nature of the bias introduced by differences in factor endowments is most clearly
seen in the relationship between the transformation schedules of the two countries. If
both countries were endowed with the same factor proportions, the constant-returnsto-scale
assumption would be sufficient to insure that the transformation schedules of the
two countries, if not identical, would be radial blowups of each other. Differences in
relative factor endowments are reflected by transformation schedules differing in shape.
Suppose that country 1 is relatively capital abundant by the non-Ohlin definition (2).
The transformation schedule of the capital-rich country must be flatter than that of the
other country along any ray from the origin, as illustrated in Figure 2. That is, if the
output of the two commodities is in the same proportion in both countries, the relatively
capital abundant country will be able to expand its production of the capital intensive commodity
at a lower opportunity cost than the other country. In this sense the relatively
capital abundant country has a production bias in favor of the capital intensive commodity.3
That this production bias may not be sufficient to insure that before trade capital and
the capital intensive commodity are relatively cheaper in country 1 than in country 2 is
seen in Figure 2.' The pre-trade price ratio, Px/Py, is greater in country 1 at b than in
Ohlin, op. cit., p. 13. Italics mine.
'In an original version of this article I made use of the production box diagrams of each country to
prove this relationship between the slopes of the two transformation schedules. Since then, T. Rybczynski,
in his " Factor Endowment and Relative Commodity Prices," Economica (New Series), Volume XXII,
No. 88, Nov., 1955, pp. 336-341, has demonstrated the equivalent proposition that should the supply of
one factor (capital) in a country be augmented, the transformation schedule becomes flatter.
'For example, cf. James & Pearce, op. cit., p. 113.
'Two sets of community indifference curves are drawn in Figure 2. Essentially this amounts to assuming
that each country behaves like a single rational consumer in a competitive market. For a demonstration
of the graphical procedure, cf. Leontief, W.: " The Use of Indifference Curves in the Analysis
of Foreign Trade," the Quarterly Journal of Economics, Volume XLVII (May, 1933), pp. 493-503, and reprinted
as Chapter 10 in Readings in the Theory of International Trade.
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4 REVIEW OF ECONOMIC STUDIES
! (2)
I~ a
x
FIGURE 2
country 2 at a, despite the production bias. Thus with trade the capital-rich country
concentrates on the production of the labor intensive commodity in order to export ei
of this commodity (Y) in exchange for imports (ig) of the capital intensive commodity.
In this case differences in tastes outweigh differences in factor endowments and the
Heckscher-Ohlin theorem (using definition (2) of factor abundance) does not hold. Conversely,
without specifying demand conditions it is invalid to infer factor endowment
proportions from the pattern Qf trade.'
5. The previous comments may be used to examine the meaning of the term, " comparative
advantage." In a Ricardian, one-factor, constant-returns world, technology
alone determines which commodity is exported from each country. Furthermore, each
country is said to possess a comparative advantage in producing that commodity. Thus
the concept of comparative advantage satisfied the following two criteria: (i) comparative
advantage is determined by production conditions alone, and (ii) each country exports that
commodity in which it enjoys a comparative advantage.
However, in a two-factor, variable proportions model of trade, the concept of comparative
advantage does not always satisfy these criteria, so some sacrifice in the meaning of the
term must be made. Under the assumptions of Section 2, and employing the Ohlin
definition of factor abundance in terms of pre-trade factor prices, it is possible to define
the capital-rich country to have a comparative advantage in producing the capital intensive
commodity. It is then true, but trivial, to state that each country exports that commodity
in the production of which it enjoys a comparative advantage. However, the first criterion
is not satisfied, for a knowledge of factor endowment proportions alone is not sufficient
with Ohlin's definition, to determine the " comparative advantage " commodity, so define
Demand conditions must be considered as well.
An alternative approach consists in selecting the " comparative advantage " commodity
for each country merely from a knowledge of factor endowments. Thus, using the defin
ition of factor abundance related to endowment proportions rather than to factor prices,
state that the capital abundant country has a comparative advantage in producing the
capital intensive commodity. Unlike the Ohlin definition, the alternative definition satisfies
the first criterion, although demand conditions may invalidate the second.
1 Even the assumption of identical taste patterns in the two countries is not sufficient to assure validity
of the Heckscher-Ohlin theorem, as there is no necessary relationship among the slopes of different indifference
curves along any ray. It is easy to demonstrate, however, that if both countries' taste patterns are
identical and homothetic (implying unitary income elasticities), the Heckscher-Ohlin theorem must hold with
either definition of factor abundance.
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FACTOR PROPORTIONS AND THE HECKSCHER-OHLIN THEOREM 5
The meaning of comparative advantage in production is reflected in the production
bias arising from differences in factor endowments. Whatever the conditions of demand
and patterns of consumption, with free trade there is a unique relationship between the
ratios of commodities produced in each country. In the country possessing a greater relative
supply of capital, the ratio of the quantity of the capital intensive commodity produced
to the production of the labor intensive commodity must always be larger than in the other
country. Retaining the assumptions in section 4, the ratio X/Y in country 1 must, in
free trade, always be larger than the X/ Y ratio of production in country 2. In this sense
it is meaningful to state that the capital-rich country has a comparative advantage in producing
the capital intensive commodity.
6. As suggested in the previous sections, the relationship between factor price and
commodity price ratios is unique granted constant returns to scale and the strong definition
of factor intensity. This is clearly demonstrated if the analysis is extended to include
more than two commodities produced in each country. Suppose each country produces
three commodities, X, Y, Z, each subject to constant returns to scale. Furthermore, suppose
that for any given factor price ratios commodity X always employs a higher capital/labor
ratio than Y, which employs a higher capital/labor ratio than Z. That is, let
(3) Px > Py > Pz
Regardless of demand conditions in the two countries, if commodity X is cheaper relative
to Y in country 1 than in country 2, so also must Y be cheaper relative to Z in country 1.
This binding relationship among commodity price ratios may be proved in straightforward
fashion. Since each commodity is produced subject to constant returns
X - Lxf(px)
(4) Y Ly g( py)
Z = Lzh(pz).
Optimal resource allocation requires the ratio of factor prices to be equal to the ratio
of the marginal physical products of labor to capital in each commodity:
PL f( px) - Pf '( px) g( py) - py g'( py) h( pz) - pz h'( pz)
(5) Pc f'( px) g'( py) h'( pz)
Pick any two commodities, e.g., Y and Z. The ratio of the prices of Y and Z in equilibrium
is given by the ratio of the marginal physical products of either factor in Z and Y:
P6 y h'(pz)
(6) Pz g'( py)
This price ratio varies inversely with the capital/labour ratio employed in any commodity.
For:
d(Py/Pz) g' h" d pz/d py - h' g"
dpy (g')2 Differentiating
equation (5), solving for d
d Py g" (h')2 g -h'9 g"
(8) PZ ) hg'
d py (g')2
But equation (5) may be solved for g
g h
(9) gI h' + PY PzThis
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6 REVIEW OF ECONOMIC STUDIES
Substituting (9) into (8) yields
d(PY/Pz) g"_(h_ ) (10)
7= h(g) ( - Pz).dpy ()
Since g" is negative (reflecting a falling marginal physical product of capital) a
commodity Y has been assumed more capital intensive than Z for any common f
price ratio, d(Py/Pz) / dpy must be negative.1
Commodities Y and Z were picked arbitrarily. With any number of commodities,
(1, . t. .>n),
(11) d(P4iP) < 0 if pi > 8.dpi
Therefore an X to Y commodity price ratio in country 1 lower than in country 2 must
reflect a capital to labor ratio employed in any commodity in country I that is higher than
that found in the production of the corresponding commodity in country 2. This latter
relationship, and equation (11), implies a lower Y to Z commodity price ratio in country
1 as well, regardless of demand conditions.2
This strong relationship among commodity price ratios has implications for the concept
of comparative advantage as well. Ordering the commodities with respect to the capital/
labor ratios employed in production is to rank them in order of comparative advantage.
Demand conditions merely determine the dividing line between exports and imports;
it is not possible to break the chain of comparative advantage by exporting, say, the third
and fifth commodities and importing the fourth when they are ranked by factor intensity.3
7. The preceding analysis seems unduly restricted by the strong factor-intensity
criterion suggested by Samuelson. That is, previously technology has assumed X to
require a higher proportion of capital than Y for any set of factor prices common to both
commodities. By using an alternative concept of factor intensity it is possible to include
cases heretofore ruled out these cases not only qualify the meaning of the HeckscherOhlin
theorem but may also invalidate the Leontief procedure.
The alternative notion of factor intensity has meaning only when factor endowments
are specified. Given the initial level of resources, it is possible to define X as capital
intensive if px > py for any equilibrium combination of output.4 Alternately, X is capital
intensive if px is greater than the endowment ratio, OIL. Stated in terms of common factor
price ratios, commodity X is capital intensive if px > py for all common possible equilibrium
factor price ratios once a set of factor endowments is specified.
The relationship between this definition of factor intensity and the Samuelson
definition is clear: to be labelled capital intensive ii la. Samuelson, commodity X must
require a higher capital/labor ratio than Y for any common factor price ratio, whether or
not this ratio can exist in equilibrium once factor endowments are specified. This requirement
is not satisfied by all pairs of constant-returns production functions. On the other
1 I wish to thank Professor Harry Johnson for recommending a more simple form of the original
proof. The relationship between factor intensities and commodity price ratios proved here has also been
demonstrated in slightly different fashion by James Meade, Trade and Welfare (Oxford University Press,
1955), pp. 600-601.
2 Although demand conditions are important in determining the relationshin between pi in one country
and p in the other. Note that the relationships proved here apply as well in the two-commodity case
and serve as verification of the previous remarks.
3 This is similar to the relationships that hold in the one-factor case. Cf. Harberler, G. : The Theory
of International Trade (London ; Wm. Hodge and Co., Ltd., 1936), p. 137.
4 Constant returns to scale in each commodity insures that a greater px tha'n py at one point along the
contract curve is sufficient for px > fy everywhere on the contract curve.
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FACTOR PROPORTIONS AND THE HECKSCHER-OHLIN THEOREM 7
hand, any pair of linear homogeneous production functions may be subjected to the
suggested alternative criterion.'
The significance of this different interpretation is clearly seen if two production functions
that do not satisfy the strong Samuelson condition are examined. By continuity, if
px > py for some set of common factor price ratios and px < py for some other set,
capital to labor ratios employed in the production of the two commodities must be equal
for some factor price ratio.2 In Figure 3 a member of each isoquant family is depicted.3
If the endowment ratio in country 1, C1/L1, is greater than the A-ratio, while the endowment
ratio in country 2, (2112, is less than the X-ratio, X must be the labor intensive commodity
in country 1 and the capital intensive commodity in country 2.4
c x
y
i:<
A
/
/
O L
FIGUR E 3
Should the endowment proportions in both countries lie on the sam
X-ratio, all the previous conclusions of this paper hold, despite the fact that
criterion is not satisfied (except in the special case of complete specialization
in section 9).5
1 Except in the singular case in which the contract curve and the diagonal of the p
identical.
aAssume for simplicity the existence of only one such critical factor price ratio. Th
main case considered by James and Pearce, op. cit. In their excellent demonstration of
may be relatively cheaper in the labor abundant country before trade and that factor
rather than be equalised once trade is initiated, they concentrate on the case in which
price ratios exist (although their terminology and graphical demonstration differ subs
employed in this paper). Both ratios lie between the endowment proportions of the tw
the same commodity is capital intensive in both countries. However, our primary conc
effect of trade on factor prices but rather the relationship between pre-trade factor pr
countries arnd the pattern of trade. Suffice it to say that in the James and Pearce cas
one-critical-factor-price-ratio case, it is possible for the capital abundant country to e
tensive commodity.
a The pioneering examination of these cases is that of Lerner, A.: "Factor Prices and International
Trade," Economica (New Series), Volume XIX, No. 73, February, 1952, pp. 1-15.
4 A heuristic proof, which can be made quite rigorous, is sketched here. As in Figure 3, assume the
X-isoquant to be more concave than the Y-isoquant, reflecting the fact that there is less possibility for factor
substitution in X-production than in Y-production. That is, for any factor price ratio common to both
industries (excluding the price ratio at A), the critical x-capital/labor ratio must more closely correspond
to px than to py. In country 1 the X-ratio is " labor-intensive" relative to factor endowments. Therefore
px must be less than py at any point on the contract curve. In country 2, px > py at all points.
6 In the general case of more than one A-ratio the previous conclusions of this paper hold if factor endowments
for both countries lie between the same two consecutive critical factor ratios, Xr and Xr+1. As
has been pointed out to me by Professor Pearce, the above conditions, although sufficient, are not necessary
for the previous conclusions. For if factor endowments in the two countries are separated by an even
number of X-ratios the same commodity is capital intensive in both countries and the capital abundant
country may export the capital intensive commodity (although not in all cases, cf. n. 2 above). The case
cited in the text is the special case in which only one >-ratio exists ; if both sets of factor endowments lie
on the same side of this >-ratio, its existence creates no new problems and all previous conclusions hold.
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8 REVIEW OF ECONOMIC STUDIES
8. If the factor endowment ratios in each country
as suggested above, px < py in country 1, and px > py in country 2. But the literal
meaning of the Heckscher-Ohlin theorem is, in this case, open to question, as is Leontief's
procedure of comparing capital/labor ratios in exports and import-competing industries.
_~~~~~~~~ -j
FIGURE 4
The contract curves in the two countries are shown in Figure
country 1, Px/Py is an increasing function of px (cf. section 6). In c
decreasing function of px But, as is seen in Figure 4, px and py
greater than either px or py in country 2 ; these sets of ratios are s
A-ratio. The relationship between commodity and factor price ra
country is shown in Figure 5.
Country 1 is relatively capital abundant; not only is C11L1 greater than C212 but,
as Figure 4 demonstrates, px in country 1 must always be higher than in country 2, which
implies (cf. Figure 5) that capital must be relatively cheaper in country 1, regardless of
demand conditions. Commodity X, however, need not be relatively cheaper in country 1
before trade, as shown by points A and B in Figure 5.1 It may be the case that the relatively
capital abundant country exports its labor intensive commodity. America may be capitalrich,
either by the Ohlin definition or in terms of factor endowment proportions, and export
commodities employing relatively less capital than in its import-competing industries.
Closer inspection reveals that, in a formal sense, the Heckscher-Ohlin theorem as
applied to both countries in this case cannot possibly hold. For the exports of each country
are either both capital intensive (relative to the other commodity produced in the
country) or labor intensive. It is impossible for both countries to export that commodity
requiring a relatively high quantity of the factor in which the country is relatively well
endowed.
This is only a formal objection to the Heckscher-Ohlin theorem, however. For no
matter which commodity is exported from the capital-rich country, it must embody a
higher proportion of capital than either commodity produced in the other country.
Although validated in this sense, the Heckscher-Ohlin theorem cannot be put into reverse
in the Leontief manner. In concluding that America must be labor abundant, Leontief
compared the capital/labor ratios in American exports and American import-competing
industries; no comparison was made with factor proportions abroad. If both American
'Note also that in Figure 5 an equality of commodity prices with free trade could never result in factor
price equalisation.
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FACTOR PROPORTIONS AND THE HECKSCHER-OHLIN THEOREM 9
P.
ry
A (1 _) ~ ~ ~; /
D (2) _
S I
-- -- | ~ ~ ~~~I I
X- ratio
te ~~~~~~I I
PCII
I I
possiblet
teotor pile.
ratios in
sountry 2
porsibleg
faoter pise.
ratios In
soItry Ig
(2) ratio(
possible ratios possible xrlttio
in oountry 2 in oeuntr:y 1
FIGURS S
exports and import-competing products are produced with more capital intensive methods
than abroad, Leontief's paradoxical conclusion is no longer valid. In this case, and there
is no a priori reason to believe this case is more unlikely than any other, Leontief 's method
tells us nothing about American factor endowments relative to the rest of the world.
9. The case of complete specialization offers the best support to the Heckscher-Ohlin
theorem and Ohlin's factor-pri'ce equalisation views. Suppose commodity K is produced
by capital intensive methods in both countries,' country 1 is relatively capital abundant
by the definition that compares endowment proportions, and its transformation schedule
is everywhere flatter than country 2's transformation schedule. In this case before trade
(and after trade) capital must be realtively cheaper in country 1 than in country 2, and one
Assume the existence of no miore thai one critical X-ratio.
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10 REVIEW OF ECONOMIC STUDIES
country must specialize completely in its export commodity. The Heckscher-Ohlin
theorem must hold and Ohlin is justified in his supposition that trade brings about a
tendency, necessarily incomplete, towards factor price equalisation.'
10. Thus in the two-commodity model, with identical constant-returns-to-scale
technology in the two countries, the Heckscher-Ohlin theorem may not hold in all cases.
If production functions are of the strict Samuelson type, the Heckscher-Ohlin theorem must
hold if relative factor abundance is tied to pre-trade factor prices. However, it may not
be valid to use the Heckscher-Ohlin theorem to infer relative endowment proportions
from existing trading patterns if demand conditions in the two countries are quite dissimilar.
If the Sameulson restriction is lifted, strict application of the Heckscher-Ohlin theorem
may not be valid if factor endowments differ considerably. And the Leontief procedure
of comparing factor intensities in only one country may be invalid.
As suggested in section six, similar binding relationships exist in a three-or-more
commodity world. However, lifting the two-factor assumption poses problems for
analysis ; unless two or more factors are completely substitutable, the concept of factor
intensity, upon which, in part, the Heckscher-Ohlin theorem rests, loses some of its meaning.2
Finally taking into account differences in technology or quality of the factors of production
as well as endowment proportions makes for sterile analysis, as trading patterns could then
be explained by differences in factor endowments, or in the quality of the factors, or in
technology, or in conditions of demand.
Swarthmore, Pa. R. W. JONES.
I As pointed out by Professor Pearce
follows if factor endowments are sep
2 The case in which there are more fa
price equalisation have been handled b
and Goods in General Equilibrium," th
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