The Heckscher-Ohlin Vanek Theorem

The Hescher-Ohlin-Vanek Theorem

The Heckscher-Ohlin model was designed to predict the pattern of trade between countries. Imports are produced in the foreign country using their labor and capital inputs. Thus, importing foreign goods amount to importing foreign labor and capital inputs.

Jaroslav Vanek (“The Factor Proportions Theory: The n-factor Case” 1968, Kyklos) was not much interested in predicting the pattern of goods trade. He viewed international trade of goods as an indirect or disguised means of trading factors or factor services embodied in traded baskets of goods. Thus, when Chinese products are imported, in effect Chinese laborers and capital equipment under the disguise of imports are entering the United States, much like people and equipment under a Chinese dragon during the annual (Chinese) New Year festival. Unlike domestic residents, the disguised Chinese capital and labor would not occupy any real estate or consume American goods.

What can we say about the indirect trade in factors? Vanek argued that A labor abundant country exports labor services, while a capital abundant country exports capital inputs.

As Choi (“Implications of Many Industries in the Heckscher-Ohlin Trade Model,” Handbook of International Trade, 2003) and others noted, the production vector is difficult to predict when goods outnumber factors. In most, if not all, empirical studies of trade, the number of goods was far greater than that of factors.

For instance, in the original Leontief Paradox, there were 38 traded sectors using only two factors of production, capital and labor. In this case, the output vector cannot be uniquely solved from a given factor endowment vector. If n is the number of goods and m is that of factors, this model has (n-m) degrees of freedom, and hence there are too many solutions. That is, the output vector is indeterminate. Consequently, the trade vector cannot be predicted.

Leamer (1984) offered a solution to this indeterminacy by suggesting that a country will allocate resources between industries to maximize the total value of outputs. In his model, output prices are fixed and arbitrarily determined in the world market. This model allows positive outputs only in m sectors, and the remaining (n-m) sectors produce zero output. However, Choi (2003) suggests that this is not a realistic solution because in practice most of n industries record positive output levels, which implies output prices move together, ensuring zero profits and indeterminate outputs in those industries.

As a result, the Heckscher-Ohlin-Vanek (HOV) Theorem, which predicts the factor content of trade, becomes more relevant. Even though trade vector is indeterminate, if factor prices are equalized, the factor content of trade is unique.

Let α denote the income share of a country, i.e.,

α = C/Cw,

where C and C* denotes the incomes of the home and the foreign country, respectively. (If trade is not balanced, C and C* should denote expenditures to predict actual trade.)

Given factor price equalization, the home country is abundant in labor if

L/Lw > α.

Then the HOV Theorem states that the country exports its abundant factor. For example, China is expected to export labor services hidden in its net export basket of goods. Of course, China imports American capital and labor in the form of imports of goods from the United States. If these are subtracted from the Chinese exports of labor and capital, then China, being a labor abundant country will export labor services and import capital input.

Similarly, the United States, a capital abundant country, will export capital input. According to the World Bank's data, the United States accounts for a little less than 4.6% of the world population, but the GDP of the United States is about $12 trillion, which is about 1/5 of the world GDP (about $60 trillion) in 2005. Thus, the United States is a labor-poor (and capital abundant) country. Accordingly, the HOV theorem suggests that the United States will export capital services and import labor inputs. With the expansion of global trade and foreign direct investment, the current trend of outsourcing by US multinational companies further expedites capital exports.

The country’s export services will be zero if the labor endowment is Lo = α Lw, but positive if L > Lo. Thus, in the HOV model, the predicted amount of labor exported indirectly is

ELt = L – α Lw,

Where E is the expectation operator. Similarly, the predicted amount of capital services is

EKt = K – α Kw

where E is an expectation operator. Whether the amounts of factor services imported or exported will be equal to the predicted amounts is an empirical question. Trefler noted that the variance of factors traded accounted for only about 3% of the variance of factor endowments, calling it a case of missing trade, i.e., the actual trade volume is much smaller than what the HOV theorem predicts. This empirical evidence suggests among others that factor prices are not equalized.

Choi (2003) suggests factor contents of trade is immaterial when factor prices are different, because the sum of factors a country claims to have exported will differ from the amounts of factors the other country claims to have imported, and hence the sum of bilateral trade volume will not add up to zero. Moreover, if a factor intensity reversal occurs, it is possible for both countries to claim that they exported the same factor. For instance, if the US exports rice to China and imports TVs (and factor intensities are reversed in the two countries), the US claims it exported capital to China, but China will claim it imported labor services. Unless factor prices are similar (as between the US and EU), factor content (volume) of trade is meaningless. However, the direction of trade is clear in the absence of factor intensity reversals. That is, a labor abundant country (a low wage country) will export labor services through commodity trade.


Choi, E. Kwan, “Implications of Many Industries in the Heckscher-Ohlin Trade Model,” in Choi, E. Kwan and James Harrigan (eds.), Handbook of International Trade, Volume 1, Oxford: Blackwell Publishing, 2003.

Trefler, Daniel, "The Case of the Missing Trade and Other Mysteries," American Economic Review 85 (1995), 1029-46.

Davis, Donald R. and David E. Weinstein, “The Factor Content of Trade,” in Choi, E. Kwan and James Harrigan (eds.), Handbook of International Trade, Volume 1, Oxford: Blackwell Publishing, 2003, 119-145.

Leamer, Edward E., Sources of International Comparative Advantage, Cambridge, Mass: MIT Press, 1984.