# Replacing the autarky factor prices with the corresponding post-trade prices: an extension of Deardorff's factor-proportions theory of trade.

1. Introduction

In his 1982 article in The American Economic Review, Alan Deardorff has developed a generalized version of the Heckscher-Ohlin model of trade for the l-factor, m-commodity, and n-country case. One should be cautioned, however, that the Deardoff's results are limited in their empirical applicability. These limitations have to do with the measurability of the autarky factor prices which are used to calculate the different countries' factor abundance and the value of their factor intensity variables.

The above mentioned problem would also limit the credibility of the "regression" versions of the Deardoffs model which are presented by Helpman (1984) and Forstner (1985).

Deardorff's model, which is based on the proposition that autarky factor prices are related to the physical abundance of the corresponding factors of production, shows: 1) a systematic relation between the autarky prices of the factors of production and the factor content of trade and 2) that countries tend to export those goods which use more intensively (sparingly) those factors with relatively cheap (expensive) autarky prices and import commodities which use intensively (sparingly) those productive factors with relatively expensive (cheap) autarky prices.

To prove the two above-mentioned propositions, Deardorff has developed a model which is summarized in the following part.

2. Deardorff's Model

Suppose a country's production, consumption, and trade are described by the following vectors:

(|L.sup.j~, |X.sup.j~) = (|L.sup.j~, |C.sup.j~ + |T.sup.j~), (1)

where |Mathematical Expression Omitted~ is an l-vector standing for country j's employment of the of the l factors, |Mathematical Expression Omitted~ is an m-vector of net outputs of goods in country j, |Mathematical Expression Omitted~ is an m-vector of final demand for goods in country j, and |Mathematical Expression Omitted~ is an m-vector of country j's net exports. Elements of vector |T.sup.j~ will be negative (positive) for goods which are imported (exported).

Following the theoretical framework of the H-O model, Deardorff's model assumes technology to be common to all countries, and it is characterized by a set H of all feasible pairs (L, X). This, of course, directs attention to factor endowments rather than technology as the incentive for trade. This model also assumes that the vector demand function, |D.sup.j~ (|p.sup.j~, |Y.sup.j~) is homogeneous of degree zero in prices (|p.sup.j~) and income (|Y.sup.j~). Finally, the conditions necessary for the existence of the equilibria are implicitly assumed.(1)

By using eleven different (but somehow plausible) assumptions, Deardorff proves the following Theorems and corollaries:

Theorem 1

|W.sup.aj~|S.sup.tj~ |is less than~ 0 (2)

where |W.sup.aj~ is a vector of length l containing the autarky factor prices of country j, and |S.sup.tj~ represents a vector of the same length containing country j's net export of each factor, arranged in the same order.

Corollary 1

Let |Z.sup.a~ be an |(n x l) x 1~ vector containing the autarky factor prices of all countries and factors.

|Mathematical Expression Omitted~.

Also let |E.sup.s~ be a vector of the same length containing the net exports by each country of each factor, arranged in the same order as |Z.sup.a~.

|Mathematical Expression Omitted~.

Corollary 1 states that, if the world contains n countries, all satisfying the model's assumptions, then there exists the following relationship between the two vectors:

Cor (|Z.sup.a~, |E.sup.s~) |is less than~ 0, (5)

where Cor (|center dot~) is the simple correlation between the elements of the two vectors that are its arguments.

Since it is normally the case that autarky factor prices in a country are inversely related to the physical abundance of the corresponding factors, this corollary can be interpreted as saying that countries will on an average tend to be net exporters (importers) of their abundant (scarce) factors. This is, of course, a general statement of the H-O Theorem when factor content of trade is under consideration.

Note that this model requires factor content of imports to be measured by the actual techniques used in their production. Since factor-price equalization is not assumed to hold in general, the actual techniques used abroad in the production of imported goods may differ from those applied in the production of the same commodities produced at home. This approach is, of course, different from what has usually been suggested in the previous empirical studies of the H-O Theorem. For example, Wassily Leontief's test of the H-O model (1953) and others like it have used domestic input-output tables to find the factor content of imports as well as exports.

Corollary 2

Com (|Gamma~, |Theta~, |Tau~) |is greater than~ 0, (6)

where |Gamma~ is a vector having N = l x m x n elements, whose m-time-repeated elements ||Gamma~.sub.hj~ represent the abundance of factor h in country j in terms of the relative deviation of the autarky price in country j, |Mathematical Expression Omitted~, from the world average autarky price, |Mathematical Expression Omitted~,

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~,

and |Theta~ is a vector of the same length as |Gamma~ whose elements, ||Theta~.sub.hij~, represent the share of factor h in one dollar value of good i when this good is produced in region j and the average value of the autarky prices of factor h, |Mathematical Expression Omitted~, is used as per unit cost of this factor. Using |Mathematical Expression Omitted~ as the commodity i's world price, ||Theta~.sub.hij~ can be formally introduced as: |Mathematical Expression Omitted~,

and |Tau~ stands for a vector having the same length as |Gamma~ and |Theta~ with elements |Mathematical Expression Omitted~ which are repeated l times and represent the value of net export of commodity i in country j. Finally, the concept of comvariance, Com (|center dot~), which is a generalization of the concept of covariance, is defined as:

|Mathematical Expression Omitted~

where the triple subscript has been replaced by a single subscript k.(2)

This corollary suggests that on average exported goods |Mathematical Expression Omitted~ must use relatively intensively |Mathematical Expression Omitted~ those factors in relative abundance |Mathematical Expression Omitted~ and use relatively sparingly |Mathematical Expression Omitted~ those factors that are relatively scarce |Mathematical Expression Omitted~. On the other hand, imported goods, on average, use scarce factors intensively and abundant factors sparingly.

This corollary gives mathematical substance (though in an average sense) to the H-O Theorem as an explanation of the pattern of commodity trade.

3. Further Theorems and Corollaries

Theorem 2 (Bertrand)(3)

When trade opens up, each country, through its commodity trade, will be a net importer of its relatively scarce factors. In this theorem, relative (physical) abundance of a productive factor is measured by the comparative ratio of total domestic supply to total foreign supply of the factor.

Proof

See Bertrand for a rigorous proof of the general form of this theorem. Contrary to the original proof of this theorem, given by Vanek(4), factor price equalization is not assumed in Bertrand's model.

Theorem 3 (Ohlin)(5)

Free international trade brings about a tendency (and only a tendency) towards factor price equalization.

Proof

The partial factor price equalization argument has been originated by Ohlin (1933). The proof of this argument can be found in Stolper and Samuelson (1941).(6)

If all assumptions of Deardorff model hold, then

|W.sup.tj~|S.sup.tj~ |is greater than~ |W.sup.aj~|S.sup.tj~, (11)

where |S.sup.tj~ is a vector of country j's actual factor content of trade, and |W.sup.aj~ and |W.sup.tj~ are the vectors of autarky and post-trade factor prices in country j, respectively. The last two vectors are arranged in the same way as |S.sup.tj~.

Proof

By applying Theorem 2, we can argue that if the autarky price of a factor is relatively cheap (expensive), then that factor is abundant (scarce) in the Ohlin's sense, and free trade causes that factor to be exported (imported) through a country's commodity trade. Further, according to Theorem 3, free trade causes the price of an abundant (scarce) factor to increase (decrease) relative to its original value.

Using |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, and |Mathematical Expression Omitted~ as the i-th element of vectors |W.sup.tj~, |S.sup.tj~, and |W.sup.aj~, respectively, we can prove that the following inequality holds:

|Mathematical Expression Omitted~.

This inequality holds, because when a factor is abundant (scarce), |Mathematical Expression Omitted~ is greater (smaller) than |Mathematical Expression Omitted~, and free trade causes the abundant (scarce) factor to be exported (imported) through the commodity trade, and, therefore, |Mathematical Expression Omitted~ is positive (negative). This means that |Mathematical Expression Omitted~, i.e., (12) is proven.

Given (12), (11) follows directly.

Q.E.D.

This theorem simply suggests that the total value of the factor content of trade, when measured by the post-trade factor prices, is greater than the total value of the same amounts of factor when measured by the autarky prices. Corollary 3

If

Cor (|Z.sup.t~, |E.sup.s~) |is less than~ 0,

then

Cor (|Z.sup.a~, |E.sup.s~) |is less than~ 0,

where |Z.sup.a~, |E.sup.s~ are defined as before, and |Z.sup.t~ is a vector of length (n x l) containing the post-trade prices of all countries and factors.

i.e.:

|Mathematical Expression Omitted~.

Proof

The sign of Cor (|Z.sup.t~, |E.sup.s~) is the same as that of the corresponding covariance. Using the same procedure which was used to show Corollary 1, we find:

|Mathematical Expression Omitted~.

We know that |Mathematical Expression Omitted~ (mean of the elements of vector |Z.sup.t~) must be positive for the model to be meaningful, and since from the last assumption of the Deardorff model we have |Mathematical Expression Omitted~ (mean of the elements of vector |E.sup.s~) equal to zero, we can conclude that the second term in (13) is zero. But since the inner product of |Z.sup.t~ and |E.sup.s~ is equal to the sum of the n inner products of |W.sup.tj~|S.sup.tj~ (that were shown in Theorem 4 to be creater than corresponding |W.sup.aj~|S.sup.tj~'s), one can conclude the following:

if

Cov (|Z.sup.t~, |E.sup.s~) |is less than~ 0,

then

Cov (|Z.sup.a~, |E.sup.s~) |is less than~ 0,

Q.E.D.

NOTE: Since autarky factor prices are in reality unobservable, Corollary 3 becomes important in making Deardoff's Corollary 1 a testable hypothesis. The result of any test conducted on Corollary 3 may be used only to validate (and not to falsify) Deardorff's Corollary 1. Reason being that, since Corollary 3 provides an upper bound to the condition of Corollary 1, any confirmation test result conducted on the former corollary is a sufficient condition (and not a necessary condition) for the validity of the latter corollary.

If it is assumed that the mean of the autarky prices of any productive factor, |Mathematical Expression Omitted~, is equal to the mean of the post-trade prices of the corresponding factor, |Mathematical Expression Omitted~, then the following inequality can be proved:

Com (|Gamma~|prime~, |Theta~|prime~, |Tau~) |is less than~ Com (|Gamma~, |Theta~, |Tau~), (14)

where

|Mathematical Expression Omitted~,

and

|Mathematical Expression Omitted~.

All variables on the right-hand side of the inequality have been defined previously, and the lefthand side variables are the same as the right-hand side values except for ||Gamma~|prime~.sub.hj~ and ||Theta~|prime~.sub.hij~ (and their means |Mathematical Expression Omitted~ and |Theta~|prime~).

Variable ||Gamma~.sub.hj~, an element of vector |Gamma~|prime~, can be defined by the same equation that was used to calculate ||Gamma~.sub.hj~, using the post-trade factor prices instead of autarky prices.

|Mathematical Expression Omitted~

Variable ||Theta~|prime~.sub.hij~, an element of vector |Theta~|prime~, can be defined similar to that of ||Theta~.sub.hij~, except the average of the post-trade factor prices replaces the corresponding average of the autarky factor prices.

|Mathematical Expression Omitted~

Proof

Using the basic assumption of this theorem (i.e., the assumption that |Mathematical Expression Omitted~ we can redefine ||Gamma~|prime~.sub.hj~ as follows:

|Mathematical Expression Omitted~.

In order to compare ||Gamma~|prime~.sub.hj~ and ||Gamma~.sub.hj~ we use Theorem 3, which suggests that the free international trade brings about a tendency towards factor price equalization. (This means that post-trade prices of each factor are closer to their mean when compared to the deviations of the pre-trade prices of the corresponding factor from their mean which is assumed to be the same as that of the post-trade.) According to this theorem, we know that: |Mathematical Expression Omitted~ whenever |Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~ whenever |Mathematical Expression Omitted~,

but this implies that:

|absolute value of~ ||Gamma~|prime~.sub.hj~ |is less than~ |absolute value of~||Gamma~.sub.hj~ h = 1,..., l.

Furthermore, following Theorem 3, since the post-trade prices of abundant (scarce) factors (when the relative abundance of these factors is measured based on their autarky price) are not believed to be greater (smaller) than their corresponding means, the sign of ||Gamma~|prime~.sub.hj~ would be the same as ||Gamma~.sub.hj~.

Variables ||Theta~|prime~.sub.hij~ and ||Theta~.sub.hij~ become equal, based on the assumption made about the equality of the average of post-trade and autarky prices of each productive factor.

|Mathematical Expression Omitted~

Since it was proven in Corollary 2 that the sign of Com (|Gamma~, |Theta~, |Tau~) is positive, and it has been shown that to obtain a positive comvariance, on average, either all three components are positive or exactly two are negative, we can conclude that having |absolute value of~ ||Gamma~|prime~.sub.hj~ |is less than~ ||Gamma~.sub.tj~, coupled with the fact that the signs of ||Gamma~|prime~.sub.hj~ and ||Gamma~.sub.hj~ are the same, implies that:

|Mathematical Expression Omitted~

According to the Corollary 2, the above is equivalent to:

Com (|Gamma~|prime~, |Theta~|prime~, |Tau~) |is less than~ Com (|Gamma~, |Theta~, |Tau~).

Q.E.D.

Corollary 4

If

Com (|Gamma~|prime~, |Theta~|prime~, |Tau~) |is greater than~ 0,

then

Com (|Gamma~, |Theta~, |Tau~) |is greater than~ 0.

Proof

Since it has already been shown by Theorem 5 that:

Com (|Gamma~|prime~, |Theta~|prime~, |Tau~) |is less than~ Com (|Gamma~, |Theta~, |Tau~), (18)

it becomes obvious that if the left-hand side of the inequality is greater than zero, so would be the right-hand side.

Q.E.D.

NOTE: Since |Gamma~ and |Theta~ are defined based on autarky factor prices and therefore are unobservable, Corollary 4 will be used to turn Deardorff's Corollary 2 into a testable proposition. The result of any test conducted on Corollary 4 should be, however, interpreted carefully because of the following two reasons:

1. The test result is a sufficient condition (and not a necessary condition) for the validity of Deardorff's Corollary 2. This means that we can provide some supportive evidence for Deardorff's theorem but are unable to disprove it. 2. Even though there is no theoretical reason to believe that the mean of the autarky prices of any productive factor is systematically greater or smaller than the mean of the post-trade prices of the corresponding factor, we cannot provide support for the assumption that these two means are equal. This, of course, makes the results of our test somewhat dependent on the veracity of our claim which itself is not verifiable due to the unobservability of the pre-trade prices.

Notes

1. Set H must be closed, convex, and for any given |L.sup.j~ bounded form above. 2. Each k represents a particular factor-good-country combination for which there are factor abundance, factor intensity, and trade observations.

3. T. J. Bertrand, "An Extension of the N-Factor Case of the Factor-Proportions Theory," Kyklos, July, 1972, 25, pp. 592-596.

4. Jaroslav Vanek, "The Factor Proportions Theory: The N-Factor Case," Kyklos, July, 1968, 305, pp. 749-756.

5. B. Ohlin, Interregional and International Trade, Harvard University Press, Cambridge, Massachusetts, 1933, Chapters 1 to 6, App. 3.

6. For further discussion of the partial equalization argument, see Miltiades Chacholiades (1990), pp. 79-84.

References

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Bertrand, T. J., "An Extension of the N-Factor Case of Factor Proportions Theory," Kyklos (July, 1972), 592-596.

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Bowen, H. P., E. E. Leamer, and L. Sveikauskas, "Multicountry Multifactor Tests on the Factor Abundance Theory," American Economic Review (December, 1987) 791-809.

Chacholides, Miltides, International Economics (New York: McGraw-Hill Publishing Company, 1990).

Deardorff, A. V., "The General Validity of the Heckscher-Ohlin Theorem," American Economic Review (September, 1982) 683-694.

-----, and Robert W. Staiger, "An Interpretation of the Factor Content of Trade," Journal of International Economics (February, 1988) 93-107.

Ethier, Wilfred J., and Lars E. O. Swensson," The Theorems of International Trade with Factor Mobility," Journal of International Economics (February, 1986) 21-42.

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Helpman, E., "The Factor Content of Foreign Trade," The Economic Journal (March, 1984) 84-94.

Johnson, H. G., "Factor Endowment, International Trade, and Factor Prices," Manchester School of Economic and Social Studies (September, 1957) 270-283.

Jones, R. W., "Factor Proportions and the Heckscher-Ohlin Theorem," Review of Economic Studies (October, 1956) 1-10.

Leontief, W. W., "Domestic Production and Foreign Trade: The American Capital Position Re-Examined," Proceedings of the American Philosophical Society 97 (September, 1953) 332-349.

-----, "Factor Proportions and Structure of American Trade: Further Theoretical and Empirical Analysis," Review of Economics and Statistics 38 (November, 1956) 386-407.

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* Assistant Professor of Economics, Lycoming College. I have benefitted from comments by Alan Deardorff and two anonymous referees.