Arnaud Costinot is one of many young economists doing interesting work in trade theory. In this 2009 Econometrica, he uses a mathematical technique familiar to any auction theorist – log-supermodularity – to derive a number of general results about trade which have long been seen as intractable, using few assumptions other than free trade and immobile factors of production.
Take two standard reasons for the existence of trade. First is differences in factor productivity. Country A ought produce good 1 and Country B good 2 if A has higher relative productivity in good 1 than B, f(1,A)/f(2,A) > f(1,B)/f(2,B). This is simply Ricardo’s law of comparative advantage. Ricardo showed that comparative advantage in good 1 by country A means that under (efficient) free trade, country A will actually produce more of good A than country B. The problem is when you have a large number of countries and a large number of goods; the simple algebra of Ricardo is no longer sufficient. Here’s the trick, then. Note that the 2-country, 2-good condition just says that the production function f is log-supermodular in countries and goods; “higher” countries are relatively more productive producing “higher” goods, under an appropriate ranking (for instance, more educated workforce countries might be “higher” and more complicated products might be “higher”; all that matters is that such an order exists). If the production function is log-supermodular, then aggregate production is also log-supermodular in goods and countries. Why? In this elementary model, each country specializes in producing only one good. If aggregate production is not log-supermodular, then maximizing behavior by countries means the marginal return to factors of production for a “low” good must be high in the “high” countries and low in the “low” countries. This cannot happen if countries are maximizing their incomes since each country can move factors of production around to different goods as they like and the production function is log-supermodular. What does this theorem tell me? It tells me that under trade with any number of countries and goods, there is a technology ladder, where “higher” countries produce “higher” goods. The proof is literally one paragraph, but it is impossible without the use of mathematics of lattices and supermodularity. Nice!
Consider an alternative model, Heckscher-Ohlin’s trade model which suggests that differences in factor endowments, not differences in technological or institutional capabilities which generate Ricardian comparative advantage, are what drives trade. Let the set of factors of production be distributed across countries according to F, and let technology vary across countries but only in a Hicks-neutral way (i.e., “technology” is just a parameter that scales aggregate production up or down, regardless of how that production is created or what that production happens to be). Let the production function, then, be A(c)h(g,p); that is, a country-specific technology parameter A(c) times a log-supermodular function of the goods produced g and the factors of production p. Assume further that factors are distributed such that “high” countries are relatively more-endowed with “high” factors of production, according to some order; many common distribution functions will give you this property. Under these assumptions, again, “high” countries produce “high” goods in a technology ladder. Why? Efficiency requires that each country assign “high” factors of production to “high” goods. The distributional assumption tells me that “high” factors are more likely to appear in “high” countries. Hence it can be proven using some simple results from lattice theory that “high” countries produce more “high” goods.
There are many further extensions, the most interesting one being that even though the extensions of Ricardo and Heckscher-Ohlin both suggest a ladder of “higher” and “lower” goods, these ladders might not be the same, and hence if both effects are important, we need more restrictive assumptions on the production function to generate interesting results about the worldwide distribution of trade. Costinot also points out that the basic three type (country, good, factor of production) model with log-supermodularity assumptions fits many other fields, since all it roughly says is that heterogeneous agents (countries) with some density of characteristics (goods and factors of productions) then sort into outcomes according to some payoff function of the three types; e.g., heterogeneous firms may be choosing different financial instruments depending on heterogeneous productivity. Ordinal discussion of which types of productivity lead firms to choose which types of financial instruments (or any similar problem) are often far, far easier using log-supermodularity arguments that using functional forms plus derivatives.