STOR The General Validity of the Heckscher-Ohlin Theorem Alan V. Deardorff The American Economic Review, Vol. 72, No. 4 (Sep., 1982), 683-694. Stable URL: http://linksjstor.org/sici?sici=0002-8282%28198209%2972%3A4%3C683%3ATGVOTH%3E2.0.CO%3B2-Q The American Economic Review is currently published by American Economic Association. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http ://w w w .j stor. org/j ournal s/aea. html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to creating and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor.org. http://www.j stor.org/ Tue Aug 16 11:39:28 2005 The General Validity of the Heckscher-Ohlin Theorem By Alan V. Deardorff Despite a number of attempts to extend it, the Heckscher-Ohlin Theorem remains woefully restricted in terms of the generality of the assumptions needed for its proof. In its traditional form as a statement about the commodity composition of trade, the theorem is valid only in the highly abstract environment of the two-factor, two-good, two-country model that has been the mainstay of trade theory for half a century.1 A weaker version of the theorem, involving the factor content of trade rather than its commodity composition, has been proved for any number of factors, goods, and countries. But this generality has so far been bought only at the cost of other and perhaps equally restrictive assumptions.2 The purpose of this paper is to provide statements and proofs of the *University of Michigan. I have benefitted greatly from the comments of Paul DeGrauwe, Wolfgang Mayer, Michael Mussa, Carl Simon, and Robert Stern, as well as others who participated in seminars at the Universities of Michigan, Chicago, and Cincinnati and at the National Bureau of Economic Research. An anonymous referee also made valuable comments. 'With additional goods and factor-price equalization, the theorem runs afoul of the indeterminacy of production and trade noted by Paul Samuelson (1953) and James Melvin (1968). Without factor-price equalization, trade impediments and intermediate production can cause reversals of the pattern of trade, as shown in my 1979 article. If additional factors and countries are also added, it becomes difficult even to state the theorem, let alone to prove it. 2Melvin stated such a theorem for the three-good, two-factor case, and Jaroslav Vanek (1968) proved it for the general case, but only by assuming factor-price equalization. Trent Bertrand (1972) and Jon Harkness (1979) were able to drop the factor-price equalization assumption, but replaced it with restrictions on consumption of either goods or factors by country of origin. It is unclear to me, however, that these restrictions can ever be met in a model that is otherwise plausible. Harkness (1978) has also used the factor-content version of the theorems as the motivation for a regression equation relating commodity trade to factor intensities. This bears some resemblance to my analysis below in Section V. See, however, James Anderson (1981) and Edward Learner and Harry Bowen (1981). who have criticized his result, with a reply by Harkness (1981). Heckscher-Ohlin (H-O) Theorem that are more general than those currently available. The approach taken is similar to that used in my 1980 article for an extension of the theory of comparative advantage in terms of correlations between autarky prices and trade. In Section I, I lay out the assumptions on which the analysis is based. These assumptions provide the outlines of a reasonably general model in which there may be any numbers of goods and factors and in which factor prices may be equalized or not. Trade impediments in the form of tariffs or export taxes are permitted, though transportation costs are not.3 In Section II, I discuss the difficulty of inferring the factor content of trade in a world of unequal factor prices, and I provide several alternative definitions of that concept for use later in the paper. These, together with the assumptions of Section I, are used in Section III to prove a theorem on the value, at autarky prices, of the factor content of trade. In Section IV this theorem is used to prove a first corollary that makes a more direct statement about the pattern of trade. It says that, worldwide, there must be a negative correlation between the factor content of trade by factor and country, and the corresponding autarky factor prices. It is precisely these autarky factor prices that constitute the Ohlin measure of factor scarcity. Therefore, the corollary provides a formal statement of the general idea that countries tend to export the services of those factors which they have in relative abundance, and it may be regarded as a generalization of the factor-content version of the H-0 Theorem. Finally, a generalization of the commodity version of the theorem is proved in Section V. The concept of a covariance is generalized 3 For ease of exposition I do not allow here for transportation costs. These were included in an earlier version of this paper, following the technique in my 1980 article. 683 684 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1982 to relate three variables rather than two. It is then shown that this generalized covariance, or "comvariance," among certain measures of factor abundance, factor intensity, and trade must be positive. This result may be interpreted as the general mathematical expression of the notion that countries tend to export those goods which use intensively their abundant factors. A concluding Section VI discusses further the concept of factor intensity and the particular definition of it that is required for this result. I. Assumptions I consider a world in which / primary factors combine to produce m goods in n countries. Factors cannot be traded but goods, at least potentially, can. A country's production, consumption, and trade are described by a pair, (V,XJ) = (LJ,Cj+TJ), where L7 = (L{,...,Lj) is an /-vector representing country j 's employment of the / factors; \J = (X{,...,X^) is an m-vector of net outputs of goods in country j; CJ = {C{,...,Cm) is an m-vector of final demands for goods in country j; and TJ = {T{,...,T^) is an m-vector of country 7's net exports. Elements of TJ will be negative for goods which are imported. Technology is characterized by a set % of all feasible pairs, (L,X), and is common to all countries. Of this set, I implicitly assume whatever is necessary to permit the existence of the equilibria I will be studying.4 This assumption of internationally identical technologies is consistent with the spirit of the H-O model, which directs attention to factor endowments rather than technologies as the incentive for trade. Demand is characterized for each country, j, by a vector-valued demand function DJ(pJ, YJ), where pJ is an m-vector of domestic goods prices in some numeraire and YJ is income in the same units. Demands are assumed to be homogeneous of degree zero in prices and income, and to satisfy the weak 4 They must be closed, convex, and, for any given ly, bounded from above. axiom of revealed preference:5 Assumption 1: (1) yD(p\Yl)>plD(p\Y2)] -[p2i)(p1,y1)>PW^2)] where superscripts denote any two alternative price and income combinations for the given country, j (country superscript omitted). Demanders, incidentally, are assumed always to spend all of their incomes, so that the extreme left-hand and extreme right-hand products in (1) are equal to Y1 and Y2, respectively. Primary factors of production are available in fixed supply in each country, and are given by vectors LJ. Equilibrium will require that factors be fully employed. Goods may also serve as intermediate inputs to production, and if that is ever their sole use, then particular C( will be zero in equilibrium. An autarky equilibrium for any country consists of vectors of net output, X" = C", of prices (of goods), pa, and of wages (of factors), wa, that are feasible, demanded, and that maximize profits.6 These requirements are captured in the following three assumptions: Assumption 2: (2) (L,Ca)e%; Assumption 3: (3) Ca = D(f,paCa); 5 To assume that this axiom holds for a country as a whole is of course very strong, almost as strong as the usual trade-theoretic assumption of community indifference curves. Though without theoretical justification, the assumption now has some very tentative empirical support in Hal Varian (1980) and Steven Landsburg (1981), who both searched for violations of the strong axiom of revealed preference and failed to find any. 6For Assumption 4, as well as 7 below, to follow from profit maximization, I also assume perfect competition and the absence of increasing returns to scale, externalities, production taxes and other domestic distortions. VOL. 72 NO. 4 DEARDORFF: THE HECKSCHER-OHLIN THEOREM 685 Assumption 4: (4) paCa-wflL^paC-waL for all(L,C)e9C. A trade equilibrium is somewhat more complicated. There is first a vector of world prices, pw, which is not associated with any particular country and is defined relative to some international numeraire. In addition, residents of each country, j, face a vector of domestic prices, p'J, that may differ from pw due to their country's taxes on trade. A trade equilibrium, then, consists of these vectors of world and domestic prices, the latter for each country, plus vectors of goods produced, traded, and demanded by each country, X'J = C'J +T'7, and finally vectors of factor wages, w'7. Assumptions analogous to those made above require that production and trade be feasible, that demands be met, and that profits be maximized: Assumption 5: (5) (L,c'+T)£%; Assumption 6: (6) C = D{tf,p'C); Assumption 7: (7) pr(C'+T')-w'I>prX-w'L; for all (L,X)eOC. In addition it is assumed that trade is balanced at world prices and that differences, if any, between world and domestic prices result, on average, more from taxes on trade than from subsidies: Assumption 8: (8) p^T'^O; Assumption 9: (9) (p'-p")T'^0. This last assumption is a weaker version of what I have called elsewhere (1980) the assumption of natural trade. It limits the use of trade subsidies that might otherwise artificially give rise to any conceivable pattern of trade. It says that the total revenue from taxes on trade, plus rents arising from other forms of trade restriction, must be at least as great as the total paid out in subsidies to trade.7 Thus trade policy constitutes on average more of an impediment to trade than a stimulant. Finally, equilibrium in international trade for the world as a whole requires that world markets clear. Assumption 10: n (10) 2 T"' = 0. j = l It should be noted that the assumptions made here are not intended to be sufficient to determine an equilibrium. I have not, for example, specified the quantitative size of the tariffs that may be implicit in Assumption 9, or even their form as ad valorem or specific. I have only made those assumptions that are needed for the results I intend to prove, and that means that the results will be valid for a variety of more explicit models such as might be used for other purposes. II. The Factor Content of Trade If techniques of production are everywhere the same, then there is no difficulty defining the factor content of trade as just the factors required directly and indirectly to produce the vector of traded goods, using these common techniques. This is the case, for exam- 7To see this, consider each term in the expansion of the inner product in (9). Negative terms indicate either exports (T'>0) for which traders receive less than the world price (p' p7)- In either case, such price differences represent either taxes on trade or the rents that accrue to the holders of scarce import or export licenses, and the total of the negative terms is the total tax revenue plus rent. Positive terms similarly indicate either exporters getting more than the world price or importers paying less, both of which require subsidies to trade. 686 THE A MERICA N ECONOMIC REVIEW SEPTEMBER 1982 pie, if technologies are homogeneous and internationally identical and factor prices are internationally equalized, as assumed by Vanek (1968). But if factor prices may be unequal, as permitted here, then there exists a variety of techniques that might be used to impute factor content to each good, and the factor content of trade may therefore be defined in a corresponding variety of ways. Rather than limit myself throughout to a specific definition of factor content of trade, I will instead state two more assumptions that deal with it, one or both of which will be required in proving results later in the paper. Then I will describe three specific ways that the factor content of trade might be defined, illustrating different concepts of factor content when techniques of production differ internationally. Only one of these definitions, it turns out, is appropriate for the most general result to be proved later in the paper, but all three satisfy the first of the two assumptions below and that is all that is needed for the basic theorem of Section III. In general terms I represent the factor content of a country's trade by an /-vector of factors, S7. Each element of the vector, S£, represents the net export of factor h, or import if negative, that is said to be embodied in country j's total vector of trade in goods. The first assumption made about this vector is that, if these amounts of factors could be traded instead of goods, then this factor trade could substitute for the commodity trade in terms of permitting the same levels of consumption to be achieved. Formally, SJ is assumed to satisfy the following condition Assumption 11: (11) (I>'-s;,C"')e9C. That is, if endowments of factors are reduced by the amounts of factors regarded as exported, and increased by the amounts regarded as imported, it would become possible to produce entirely domestically the same vector of goods that is consumed with trade. With this assumption, since LJ — SJ is a vector of factors capable of producing CtJ, one may also think of Sy as the difference be- tween factor endowments and the factor content of consumption, just as trade in goods is the difference between production and consumption of goods. The second assumption that I will sometimes need about the factor content of trade is that net exports of each factor by the world as a whole, as measured by S7, are nonnegative. That is, what all countries together are regarded as exporting of each factor is at least as great as what they regard themselves, in the aggregate, as importing. Assumption 12: n (12) 2 Si>0;h = \,...,l. 7 = 1 This condition, which would hold with equality if SJ represented actual trade in factors, will be shown in a moment to hold for two specific definitions of the factor content of trade. To construct such specific definitions of factor content I need the additional assumptions of constant returns to scale and the absence of joint production, so that techniques of production can be represented by the amount of inputs used per unit of output of each good. I can then represent various production techniques with (/X n) matrices, G, whose elements ghi denote the direct-plus-indirect requirement of factor h per unit of output of good i. Various such matrices may be used to represent techniques in use in different locations and circumstances. Factor content based on domestic coefficients: Perhaps the simplest specific definition of the factor content of trade would impute factors to goods on the basis of domestic techniques of production. Let Gdj be the direct-plus-indirect factor requirements matrix based on techniques of production actually in use in country j with trade, or, for those goods not produced, the cost-minimizing techniques given y's domestic factor prices. I then define the domestic factor content of trade Sdj as (13) SdJ = GdJTJ. VOL. 72 NO. 4 DEARDORFF: THE HECKSCHER-OHHN THEOREM 687 Note that this definition clearly satisfies Assumption 11, since (5) implies GdJ(CtJ+TJ) - Sdj. Actual factor content of trade: A second specific definition imputes to traded goods those factors actually used in their production wherever that took place. Let G'j be the actual factor requirements matrix for country j's trade. It is constructed by tracing backwards through the complete production history of each good that enters Tj, and adding up the factors actually used in its production, plus those used in producing intermediate inputs to its production, perhaps in a different country, and so on. Once this matrix is constructed, I define the actual factor content of trade, StJ, as simply: (14) S'' = G'J'T'. This definition of factor content also satisfies Assumption 11, since technologies are everywhere identical. Factor content based on actual content of consumption: The factor content of trade need not always be measured as a matrix times the vector of traded goods. To see this, consider repeating the process described above of measuring the factors actually used in producing goods, but apply it to the vectors of goods consumed by countries rather than those traded. Letting g'7 represent this actual factor requirements matrix for country j's consumption bundle, I can then define a third measure of the factor content of trade as follows: (15) Scj = Lj-GCJCJ. This is just the difference between country j's endowment of factors and the factors that may be said to be embodied in its consumption. Like (14), (15) is consistent with (11) due to the identical technology assumption. To see how these three definitions compare to one another, and also to check whether they satisfy Assumption 12, consider the sum across all countries of various factor trade vectors. Taking the last one first, one can be sure that the actual content of worldwide consumption is no greater than world factor endowments, n n (16) 2 gc;c"'< 2 ly, since these are the factors actually used. It follows that (17) 2 s^>o, with equality if factors are fully employed.8 What I have called the actual factor content of trade, S'J, has a similar but even stronger property. Since every exported good of one country is an import of another, and since the actual factor content is the same for both, it must follow that (18) 2 S"' = 0. 7 = 1 Finally, the domestic factor content of trade may be of either sign for the world as a whole, since imports tend to be inefficient to produce domestically and thus "contain" large amounts of factors that enter negatively in SdJ. Thus (19) 2 S^' = 0. y = i < Assumption (12), then, is satisfied by both Sc and S', but may not be satisfied by S^. The use of domestic techniques of production to impute factor content of trade may therefore lead to difficulties in generalizing the H-O Theorem. Why this is true will be explained further in Section VI below. There are of course other ways that might be used to measure the factor content of trade, some of which would undoubtedly satisfy one or both of Assumptions 11 and Inequality would arise even with full employment, however, if transport costs were included in the model, for then Sc would include as exports all factors used up in transporting goods. 688 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1982 12. But not all methods work, as may be seen by considering the method that is probably most common in practice: that is to use the techniques employed by a single country to impute factor content in all countries. The difficulty with this single country method is that it is likely to impute excessive factor content to other countries' exports, making it impossible for them to produce their with-trade consumption bundles if these resources were removed. Thus the single country method of imputing factor content need not satisfy Assumption 11. III. A Theorem on the Factor Content of Trade I can now prove that, for any country, the value at autarky factor wages of the factor content of trade must be negative. THEOREM: If Assumptions 1 through 9 hold, and if S7 is defined so as to satisfy (11), then (20) wa7S7<0. PROOF: I omit the superscript j in what follows. Begin by rearranging (9) and using (8) to get (21) p'T'p'Ca-w'L. Rearrange and use (21): (23) p'C'-p'Ca3s -jfV>0. Thus (24) p'Op'CA Substituting (6) and (3) for C and Ca, (1) then implies (25) paC'>paC". Finally, (11) entitles one to use (4) to get (26) paCa-waL>p"C'-wa(L-S) or (27) waS = (wf,...,w?,wf,...,w?"). Also let Es be a vector of the same length containing the net exports by each country of each factor, arranged in the same order: (30) Es=(sl,...,S},S?,...,Sr). Then the corollary states that these two vectors must be negatively correlated: COROLLARY 1: // the world contains n countries, all satisfying Assumptions 1 through 9, and if the factor content of trade also satisfies Assumptions 11 and 12, then (31) Cor(Wa, Es)<0 where Cor( ) is the simple correlation between the elements of the two vectors that are its arguments. PROOF: The sign of the correlation is the same as that of the corresponding covariance. That, in turn, depends on the inner product of the two vectors minus the product of their means multiplied by their length. Thus the corollary will be proven if it can be established that (32) WflEs-«/ WaEs<0 whereji bar over a vector denotes its mean. Now W" must be positive for the model to be meaningful. And it follows easily from (12) and (30) that (33) Es3»0. Thus the second term in (32), including its sign, is nonpositive. Finally, the inner product, WaEs, is just the sum of the n inner products, yvaJsJ\ that were shown in the theorem to be negative. This proves (32) and thus Corollary 1. This result gives substance to a very general statement of the H-0 Theorem in its factor-content interpretation. Recalling that autarky factor prices inversely reflect the abundance of those factors in the Ohlin sense, equation (31) says that countries will on average tend to be net exporters of their abundant factors and net importers of their scarce factors. Proof of this proposition does not depend on factor-price equalization or on any peculiar characteristic of demand behavior as have other attempts to generalize the theorem. On the contrary, the present model even allows for tariffs and requires of demand only the weak axiom of revealed preference. The result does, however, require that the imputation of factor content to trade be consistent with both Assumptions 11 and 12. The specific definitions in Section II that accomplish this are those that rely on techniques of production actually used, wherever that production takes place. In particular, the factor content of imports is to be measured by the factors which produce them aboard. This is in contrast to what has usually been done in testing the factor proportions theory empirically. Wassily Leontief s (1953) test and others like it have always used domestic input-output tables to evaluate the factor content of imports as well as exports. V. Implications for the Pattern of Commodity Trade The link between the factor content of trade and the commodity composition of trade is seen most easily by looking at an implication of the factor content of trade for a particular factor. Suppose in general that a country's factor content of trade is measured as some matrix of factor requirements, G7, times its vector of 690 THE A MERICAN ECONOMIC REVIEW SEPTEMBER 1982 commodity trade, TJ, (34) Sj = GJ'TJ\ as in either the domestic factor content or actual factor content definitions of Section II. Letting g/, be the elements of GJ, the factor content of trade for a particular factor h becomes m (35) St= 2 4T>. 1 = 1 Now gJhj are direct plus indirect per unit factor requirements, and can be used to construct measures of factor intensity. A particular measure that will be convenient here is the following: (36) »uj = ^hZUp7^ where w is any vector of factor wages. These 0's have the form of "factor shares," since they relate values of factor inputs to values of outputs. However, since neither the wages nor the prices used in equation (36) are necessarily those faced by the producers in choosing g's, the 6's may not be observable as factor shares. Still, they provide a better measure of factor intensities than the g's alone, since they are unit free. In any case, equation (35) can be multiplied through by wf;, and each term in the summation can be both multiplied and divided by world prices to yield the following result: m (37) wh*i= 2 efoipYT/). 1 = 1 From Assumption 8, balanced trade, the mean of the terms in parentheses is zero. It follows that the inner product on the right-hand side of equation (37) also gives the sign of the correlation between the 0's and the values of commodity trade. It therefore implies the following: If a country is a net exporter (importer) of a particular factor content, then there must be a positive (negative) correlation across commodities between the value of its net exports at world prices and the intensity with which commodities use the factor. More intuitively, this means that countries tend to export those goods which are intensive in the factors whose content they export. This connection between commodity and factor-content trade, together with the results of earlier sections of the paper, suggest that it should be possible to construct another corollary that would correlate commodity trade with factor intensity and factor abundance and thus generalize the commodity version of the H-0 Theorem. The difficulty here is the need to correlate three variables symmetrically, and I am not aware of any standard method of doing this. I therefore define a generalization of the concept of the covariance. Let x, y, and z be three vectors all of length N, and let x, y, and z be their respective means, x = ^=xxt/N, etc. I define the comvariancexo among x, y, and z as follows: (38) N com(x,y,z) = 2 {x-x)(y-y)(Zi-z). i-i I will show below that this definition characterizes the kind of relationship that is stated in the H-O Theorem. To measure factor abundance, I first define a vector of world-average autarky prices as follows: (39) **=2Hj=(*h- will be high (positive) for countries with relatively cheap, and hence abundant, factors. For factor intensity, I use the measure 0hij defined above in (36), with wh now explicitly defined in (39), and using the actual-factor-requirements-of-trade matrix g'7 defined in Section II. (41) 0ktj = *k*kJi/p7. Finally, for trade I again use the value of net exports at world prices: (42) i j ¥ i i Now let 0, and t be vectors of length N—lmn whose elements are u>hi] = u>hj for all /', 6hij, and Thij = 'rij for all h. The elements in w are repeated m times and those in t are repeated / times, in order to conform to the dimension of 0. I can then state the following corollary: COROLLARY 2: // the factor requirements matrix, G'J, has the property that S'J = G'JT'J satisfies both (11) and (12), and if Assumptions 1-10 are also satisfied, then (43) com(u, 0,t)>O. PROOF: It is easily derived from (38) that N (44) com(x,y,z) = 2 -^J^,-\-2Nxyz -x(yz)-y(x-z)-z(x-y). From (39) and (40) it follows that co = 0, while f = 0 from either (8) or (10). Thus all but two terms in (44) disappear for evaluating (43), which becomes / m n (45) com(u,0,r)= 2 2 2 "hu6huThU h=\i=\j=\ I m n - I m n 2 2 2 ( h=M=\j=\ Wh~Wh )gkW = 2^2^-2 2 . h=\ j=\ j=i/i=i The first term here is nonnegative, from (12), while the sum over h in the second term is the inner product that was shown to be negative in the theorem of Section III. Thus, from (12) and (20), n (47) com(o},e,r)> - 2 waJS'J>0. y = i To interpret this result, rewrite it in terms of the definition in equation (38): (48) com(o3,e,r) = 2 (uk-a)(ek-e)(Tk-T)>o, k = \ where the triple subscript has been replaced by a single subscript k. Each k represents a particular factor-good-country combination for which there are abundance, intensity, and trade observations. For the summation in (48) to be positive, the TV terms must on average be positive. And for the product that is each term to be positive, either all three factors must be positive or exactly two must be negative. Consider what this means for the pattern of trade. Goods that are exported (ta.>t = 0), must on average use relatively intensively (dk>6) those factors in relative abundance ) 692 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1982 (coA:>aJ) and use relatively unintensively (6k<6) those factors that are relatively scarce (w/t