“An Elementary Theory of Comparative Advantage,” A. Costinot (2009)

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.

Final 2009 ECTA (IDEAS version). Big thumbs up to Costinot for putting the final, published version of his papers on his website.

“Gains From Trade Without Lump-Sum Compensation,” A. Dixit & V. Norman (1986)

Let’s take a brief respite from the job market; I’ve noticed some foolish things said about free trade in the news recently. Yes, Ricardo showed that trade in two goods generates surplus for both countries under free trade. Samuelson later gave a more formal, general proof of the benefits of Ricardian trade for a nation, though his theorem with Stopler explains which individuals may be made worse off. Samuelson also showed that even when individuals are worse off, there is enough surplus that transfers can be made to the harmed individuals such that free trade is a Pareto improvement on autarky. Note that the last sentence is absolutely not implied by Ricardo, and how could it have been: he didn’t have the apparatus of ordinal utility nor the concept of Pareto improvement nor the idea of the Hicksian demand curve.

All of the above is true, but it does not justify the critique that, since we don’t always redistribute gains from trade to the losers, free trade may make us worse off under some social welfare functions. There turn out to be many ways, aside from direct income redistribution, to generate the Pareto improvement. Dixit and Norman, in a 1986 JIE, give a great example of one. Disallow transfers, but allow for arbitrary taxation of goods.

It is easy to show that free trade plus commodity taxes can leave everybody with exactly the same welfare as under autarky. Let consumers demand x0 under autarky. Under free trade, a new price vector for producers leads to equilibrium changes in production for Ricardian reasons. The government then sets a commodity tax such that consumers face the same prices they faced in autarky, with the government using the tax revenue to buy the excess supply of goods and then burning them. Everyone is exactly as well off as they were before. Now, rather than burning surplus, if there are any goods where some consumers are either all net sellers or all net buyers, then (using the case where all consumers are buyers) use the tax on that good to give it a price slightly below the autarky price. All the net buyers are better off, and those who were indifferent between buying and selling are also indirectly better off. In a non-exchange economy, we can always find at least one good where all the consumers are net on one side of the market.

Later work expanded this idea into a more general model; in particular, with limits on factor mobility, we don’t get a Pareto improvement, so direct payments to immobile factors (such as job market assistance to fired workers) may still be necessary. And of course, learning-by-doing and other forms of increasing returns to scale in production have all of the usual caveats that New Trade Theorists have mentioned. Nonetheless, as a policy perspective, if you are worried about the equity problems from increased openness to trade on the consumption side, taxing the goods that become cheaper as a result of trade is a simple fix. And, unlike direct redistribution, in a large economy it is more or less incentive compatible!

Final 1986 JIE copy (IDEAS version). As an aside, I remembered this old Dixit article when reading his recollections of the great Paul Samuelson in the newest Annual Rewiew of Economics. Dixit mentions a story I hadn’t come across before. In the late 70s, after the Cambridge Capital Controversy had died down, Samuelson joked in the faculty lounge that the whole debate had been nothing but a neoclassical conspiracy to keep far left-wing attackers so busy they don’t notice the rest of our schemes! (Joking aside, Dixit has a nicely pithy summation of the reswitching debate, which sums up well what you should know about it if you are a working economist: the rate of interest is not unique and is uninformative about many interesting phenomena, but turnpike theorems and other similar statements mean that a lot of the rest of growth and general equilibrium theory is still safe.)