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Alan V. Deardorff

**1.1 Introduction**

It has long been recognized that bilateral trade patterns are well described empirically by the so-called gravity equation, which relates trade between two countries positively to both of their incomes and negatively to the distance between them, usually with a functional form that is reminiscent of the law of gravity in physics. It also used to be frequently stated that the gravity equation was without theoretical foundation. In particular, it was claimed that the Heckscher-Ohlin (HO) model of international trade was incapable of providing such a foundation, and perhaps even that the HO model was theoretically inconsistent with the gravity equation. In this paper I will take another look at these issues. It is certainly no longer true that the gravity equation is without a theoretical basis, since several of the same authors who noted its absence went on to provide one. I will briefly review their contributions in a moment. Since none of them build directly on an HO base, it might be supposed that the empirical success of the gravity equation is evidence against the HO model, as at least one researcher has implied by using the gravity equation as a test of an alternative model incorporating monopolistic competition. I will argue, however, that the HO model, at least in some of the equilibria that it permits, admits easily of interpretations that accord readily with the gravity equation. At the same time, developing these interpretations can yield additional insights about why bilateral trade patterns in some cases depart from the gravity equation as well.

There are two keys to these results, which once stated may make the rest of the paper obvious to those well-schooled in trade theory. The two keys open doors to two different cases of HO-model equilibria, one with frictionless trade and one without.

With frictionless trade-that is, literally zero barriers to trade of all sorts, including both tariffs and transport costs-the key is that trade is just as cheap, and therefore no less likely, as domestic transactions. Therefore, instead of thinking as we normally do of countries first satisfying demands out of domestic supply and then importing only what is left, we should think of demanders as being indifferent among all equally priced sources of supply, both domestic and foreign. Suppliers likewise should not care about to whom they sell. The HO model (and other models based solely on comparative advantage and perfect competition) is usually examined only for its implications for net trade, and we then jump to the conclusion that gross trade flows are equal to net. But with no trade impediments, there is no reason for trade to be this small. If instead we allow markets to be settled randomly among all possibilities among which producers and consumers are indifferent, then trade flows will generally be larger and will fall naturally into a gravity-equation configuration, in a frictionless form without a role for distance. With identical preferences across countries, this configuration is particularly simple. With nonidentical preferences it is a bit more complex, but it is also more instructive.

The other key is to the case of trade in the presence of trade impediments. If there exist positive impediments to all trade flows, however small, then the HO model cannot have factor price equalization (FPE) between any two countries that trade with each other. For if they did have FPE, then their prices of all goods would be identical and neither could overcome the positive barrier on its exports to the other. Since we do observe trade between every pair of countries that we care about, it follows that the HO equilibria we look at with impeded trade should be ones without FPE between any pair of countries. If we assume also that the number of goods in the world is extremely large compared to the number of factors, it will be true that for almost all goods only one country will be the least-cost producer. With trade barriers this does not imply complete specialization by countries in largely different goods, but it makes such a case more plausible than might have been thought otherwise. In any case, motivated by this observation, I will study bilateral impeded trade under the assumption that each good is produced by only one country. With that assumption, bilateral trade patterns in the HO model are essentially the same as in other models with differentiated products, and it is no surprise that the gravity equation emerges once again. My contribution here will be to derive bilateral trade in terms of incomes and trade barriers in a form that may be more readily interpretable than before.

None of this should be very surprising, although I admit that this is much clearer to me now than it was when I started thinking about it. All that the gravity equation says, after all, aside from its particular functional form, is that bilateral trade should be positively related to the two countries' incomes and negatively related to the distance between them. Transport costs would surely yield the latter in just about any sensible model. And the dependence on incomes would also be hard to avoid. The size of a country obviously puts an upper limit on the amount that it can trade (unless it simply reexports, which one normally excludes), so that small countries necessarily trade little. For income not to be positively related to trade, it would therefore have to be true also that large countries trade very little, at least on average. Therefore, the smaller the smallest countries are, the less must all countries trade in order to avoid getting a positive relationship between size and trade. Looked at in that way, it would therefore be very surprising if some positive relationship between bilateral trade and national incomes did not also emerge from just about any sensible trade model. The HO model has some quirky features, but in this respect, at least, it turns out to be sensible.

As for the functional form, a simple version of the gravity equation-what I will call the standard gravity equation-is typically specified as

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where *[T.sub.ij]* is the value of exports from country *i* to country *j*, the *Y*s are their respective national incomes, *[D.sub.ij]* is a measure of the distance between them, and *A* is a constant of proportionality. While this particular multiplicative functional form may not be obvious, the easiest alternative of a linear equation clearly would not do, for trade between two countries must surely go to zero as the size of either goes to zero. None of this constitutes a derivation of the gravity equation, of course, but it does suggest why one would expect something like it to hold in any plausible model.

I turn in section 1.2 to a brief review of the literature, followed by the two cases just mentioned: frictionless trade in section 1.3 and impeded trade in section 1.4.

**1.2 Theoretical Foundations for the Gravity Equation**

As has been noted many times, the gravity equation for describing trade flows first appeared in the empirical literature without much serious attempt to justify it theoretically. Tinbergen (1962) and Poyhonen (1963) did the first econometric studies of trade flows based on the gravity equation, for which they gave only intuitive justification. Linnemann (1966) added more variables and went further toward a theoretical justification in terms of a Walrasian general equilibrium system, but the Walrasian model tends to include too many explanatory variables for each trade flow to be easily reduced to the gravity equation. Leamer and Stern (1970) followed Savage and Deutsch (1960) in deriving it from a probability model of transactions. Their approach was very similar to what I will suggest below, but they applied it only to trade, not to all transactions, and they did not make any explicit connection with the HO model. Leamer (1974) used both the gravity equation and the HO model to motivate explanatory variables in a regression analysis of trade flows, but he did not integrate the two approaches theoretically.

These contributions were followed by several more formal attempts to derive the gravity equation from models that assumed product differentiation. Anderson (1979) was the first to do so, first assuming Cobb-Douglas preferences and then, in an appendix, constant-elasticity-of-substitution (CES) preferences. In both cases he made what today would be called the Armington assumption, that products were differentiated by country of origin. His framework was in fact very similar to what I will examine here with impeded trade, although I motivate the differentiation among products, as already noted, by the HO model's case of non-FPE and specialization rather than by the Armington assumption. Anderson modeled preferences over only traded goods, while I will assume for simplicity that they hold over all goods. Anderson's primary concern was to examine the econometric properties of the resulting equations, rather than to extract easily interpretable theoretical implications as I seek here.

Finally, Jeffrey Bergstrand has explored the theoretical determination of bilateral trade in a series of papers. In Bergstrand (1985) he, like Anderson, used CES preferences over Armington-differentiated goods to derive a rcduced-form equation for bilateral trade involving price indexes. Using GDP deflators to approximate these price indexes, he estimated his system in order to test his assumptions of product differentiation. For richness his CES preferences were also nested, with a different elasticity of substitution among imports than between imports and domestic goods. His empirical estimates supported the assumption that goods were not perfect substitutes and that imports were closer substitutes for each other than for domestic goods.

In Bergstrand (1989, 1990) he departed even further from the HO model by assuming Dixit-Stiglitz (1977) monopolistic competition, and therefore product differentiation among firms rather than among countries. This was imbedded, however, in a two-sector economy in which each monopolistically competitive sector had different factor proportions, thus being a hybrid of the perfectly competitive HO model and the one-sector monopolistically competitive model of Krugman (1979). In the first paper Bergstrand used this framework to derive yet again a version of the gravity equation, and in the second he examined bilateral intraindustry trade.

Bergstrand's later work therefore serves to bring together the earlier Armington-based approaches to deriving the gravity equation with a second strand of literature in which gravity equations were derived from simple monopolistic competition models. Almost from the start of the new trade theory's attention to such models, it was recognized that they provided an immediate and simple justification for the gravity equation. Indeed, Helpman (1987) used this correspondence between the gravity equation and the monopolistic competition model as the basis for an empirical test of the latter. That is, he interpreted the close fit of the gravity equation with bilateral data on trade as supportive empirical evidence for the monopolistic competition model. For this to be correct, of course, it would need to be true, as Helpman apparently believed, that the gravity equation does not also arise from other models. He remarked that "the factor proportions theory contributes very little to our understanding of the determination of the volume of trade in the world economy, or the volume of trade within groups of countries" (63), and he went on to demonstrate geometrically that the volume of trade under FPE in the 2X2X2 HO model is independent of country sizes. Helpman was, I would like to think, in good company. No less an authority than Deardorff (1984, 500-504) noted several of the empirical regularities that are captured in the gravity equation and pronounced them paradoxes, inconsistent with, or at least not explainable by, the HO model.

Helpman applied his test to data on trade of the Organization for Economic Cooperation and Development (OECD) countries, where most would agree that monopolistic competition is plausibly present. Hummels and Levinsohn (1995) decided to attempt a sort of negative test of the same proposition by looking for the same relationship in the trade among a much wider variety of countries, including ones where monopolistic competition is less plausibly a factor. To their surprise, they found that the test worked just as well for that group of countries, thus leading one to suspect that perhaps the relationship represented by the gravity equation is more ubiquitous, and not unique to the monopolistic competition model. It might be thought that the work by Anderson and Bergstrand cited above would have already suggested this, since they derived gravity equations from a variety of models other than the monopolistic one that Bergstrand eventually incorporated into his analysis. But in fact the versions of the gravity equation that Anderson and Bergstrand obtained were somewhat complex and opaque, and it was not obvious that they would lead to the success of the very simple gravity equation tested by Helpman.

My point in this paper, of course, is that one can get essentially this same simple gravity equation from the HO model properly considered, both with frictionless and with impeded trade. This does not mean that the empirical success of the gravity model lends support to the HO model, any more than it does to the monopolistic competition model. For reasons I have already indicated, I suspect that just about any plausible model of trade would yield something very like the gravity equation, whose empirical success is therefore not evidence of anything, but just a fact of life.

**1.3 Frictionless Trade**

Consider now an HO model with any numbers of goods and factors. In fact, for most of what I will say in this section, the argument is more general and could apply to any perfectly competitive trade model with homogeneous products, including a Ricardian model, a specific-factors model, a model with arbitrary differences in technology, and so forth. For this model, consider a frictionless trade equilibrium-that is, an equilibrium with zero transport costs and no other impediments to trade-with each country a net exporter of some goods to the world market and a net importer of others. This equilibrium need not be unique, as it will not be in the HO model with FPE and more goods than factors. If the model is HO, then there may be FPE among some or all countries, but there need not be. We need merely have some vectors of production, consumption, and therefore net trade in each country that are consistent with maximization by perfectly competitive producers and consumers in all countries, facing the same prices (due to frictionless trade) for all goods, the vectors being such that world markets clear.

It is customary to note that patterns of bilateral trade are not determined in such a model, and indeed they are not. But the reason for this indeterminacy is itself important: both producers and consumers are indifferent, under the assumption of frictionless trade and homogeneous products, among the many possible destinations for their sales and sources for their purchases. Therefore, while it is true that a wide variety of outcomes is possible, we can get an idea of the average outcome by just allowing choices among indifferent outcomes to be made randomly.

Thus, having already found the equilibrium levels of production and consumption, let the actual transactions be determined as follows: producers in each industry put their outputs into a world pool for their industry; consumers then choose randomly their desired levels of consumption from these pools. If consumers draw from these pools in small increments, then the law of large numbers will allow us to predict quite accurately what their total choices will be by using expected values. In general, these expected values will be appropriate averages of the wide variety of outcomes that are in fact possible in the model.

*(Continues...)*

Excerpted fromThe Regionalization of the World EconomyCopyright © 1998 by National Bureau of Economic Research. Excerpted by permission.

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