This essay has its origins as a free-response exam question for a class on the history of economic thought. I wrote this in an hour without the aid of the internet, but believe it good enough to share here. I hope you enjoy it!

Modern trade theory begins in earnest with Dornbusch-Fisher-Samuelson, and by extension, the Dixit-Stiglitz model of monopolistic competition. The two-good, two-country Ricardian models so familiar to economists from undergraduate study is largely unusable outside of toy examples; the Heckscher-Ohlin models, which relates trade to factor endowments, delivers predictions at odds with observed trade flows. DFS borrow the tricks of Dixit-Stiglitz in order to make the model solvable; namely, there are a continuum of goods (such that we can abstract away from strategic considerations), there is a constant elasticity of substitution between goods, and every firm produces one and only one good. Now you can define comparative advantage as continuous functions, and the problem reduces to that of a supply and demand graph. It also gives us predictions of the size of trade flows as a function of trade costs and size. These gravity equations will allow us to actually estimate these models.

Krugman builds upon the Dixit-Stiglitz foundation to consider why intra-industry trade occurs, in particular in his 1980 paper. (His 1979 paper allows for variable markups, thus ruling out CES functions). Firms are homogenous with the cost structure A + Bx, and firms charge a fixed markup over price. Thus, we can explain trade between countries with similar factor endowments – they produce similar, but differentiated, goods, and are better off for it. Krugman (1981) gives a practical application for this – if we allow for there to be multiple factors of production, which cannot be frictionlessly converted from one into another, then trade can create winners and losers. The owners of the scarce factors will be uncompetitive to foreign imports and may have their wages fall, if trade is due to differences in endowments; if trade is due to intraindustry trade, then everyone necessarily benefits. We can see this intuitively in the home-market effect of Krugman (1980) – with one factor of production, L, wages are necessarily higher in the country which is larger. Having trade with an identical country is just the same as if our country were to double in size.

The next stage of trade theory builds upon one or the other as their pillars. Eaton-Kortum (2002) is the successor of DFS, while Melitz is the successor to Krugman. Both of them allow for random draws from a distribution, which allows them to better characterize the real world. EK allows for countries to receive a random draw of productivities and technology from a Frechet distribution, and then estimate parameters. Meanwhile, Melitz, instead of having everyone share their marginal costs, has firms receive their marginal costs B from a known distribution, after paying the fixed cost to enter A. The advantage of EK is that it can actually be applied to any economic activity, anywhere in space. Autor-Dorn-Hanson is an example of applied EK, just as “Railroads of the Raj” is, or any other paper dealing with the distribution of economic activity in space.

Melitz allows for there to be reallocation of business to the more competitive firms. The way he does this in the 2003 paper is something of a trick, however, and is unsatisfying to me. Firms do not know their marginal product when they enter domestically, but when access to a foreign market comes, they do now know their marginal costs. Thus, only the most productive firms choose to export. What makes it unsatisfying is that Melitz gives a micro-foundation for firms not knowing their future productivities by saying that they won’t know their own demand. But then, wouldn’t they not know their own demand when they try to export to a foreign market? This does, nevertheless, give us some features observed in the data. Firms vary in their cost, new firms are less productive and faster to exit, and reduced trade barriers benefit the most productive firms, while inducing the least productive firms to exit.

The assumption of CES, as in Melitz (2003), rules out the possibility of pro-competitive effects of trade. Since markups can’t change, there’s no way for more competition to reduce the distortion from markups. Nevertheless, this can be changed (generally using a translog demand function – please do not make me define that without access to the internet), and given an elegant intuitive explanation. Expanding the size of the market is equivalent to shifting the demand curve out, while increasing competition is equivalent to making the demand curve have a flatter slope. The net effect is as though the demand curve were twisting. High marginal cost firms are more hurt by the change in slope than they are by the expansion of the markups, and vice versa for low-cost firms. An easy modification to tie it more closely to observed trade flows is to allow for an increasing marginal cost to find consumers, as Arkolakis (2010) does. Thus, we can explain the trade of small quantities of goods across borders, and give an easily verified prediction that a fall in trade costs will lead to a faster increase in the trade of the previously least traded goods.

Of course, there has been some questioning of the relevance of these changes. Arkolakis, Costinot, and Rodriguez-Clare (2012) show that all of these models deliver equivalent results, given some fairly restrictive but common assumptions. Specifically, you can find the total gains and losses from trade using only the share spent on imports, and the elasticity of trade with respect to trade costs. (Admittedly, their treatment of heterogenous firms requires an unbounded distribution of possible firm productivities. Capping this will increase the possible gains from trade, as it will make demand less elastic. Cf. Melitz and Redding (2015)). ACR 2012 is only for CES models; however, a strikingly common result is that allowing for variable markups actually reduces the expected gains from trade. With non-homothetic preferences, consumers reallocate their purchases more to firms with higher markups. The distribution of markups remains the same, but the consumption bundles change, as Arkolakis, Costinot, Donaldson, and Rodriguez-Clare.

Before turning to the future questions, we should deal with the early attempts to model oligopoly in trade theory. Brander and Krugman (1983) is the key paper – if we assume Cournot competition and restrictions on entry, then if we begin with one firm in each country opening trade between them will reduce output, benefitting the consumer but hurting the firms even more. This is because the first good sold in the other country is selfishly profitable, but they have to pay trade costs on it (unlike if they sold from home). As a wag put it, “The US exports sugar cookies to Denmark, and they export sugar cookies to us. Wouldn’t it just be cheaper if we swapped recipes?” (It’s in a Krugman essay somewhere). As with Krugman (1981), the key for creating winners and losers are frictions.

The questions that remain unanswered are largely those of accounting for varying elasticities of substitution. Assuming CES makes things extremely tractable, as we can extrapolate from observed changes in trade costs all the way to unobserved changes in trade costs. However, if we allow for some goods to be somewhat substitutable for small changes, but absolutely not substitutable for large changes, then the gains from trade become much larger. In essence, we will be applying the advances in network production and in the study of oligopolies to trade. What was once abandoned due to excessive sensitivity to the form of production assumed can be returned to with better data. I think the future of trade theory is an increasingly close union of industrial organization with trade data, and it is not just me who feels this way. While at the AEA meeting this winter, I attended a session on oligopoly in international trade, and read Pol Antras’ preparatory paper on the subject. The next steps in international trade are using Census data, integrating trade flows with domestic firms, and using input-output tables to find the real effects of trade. In particular, it will build upon Baqaee and Farhi. We will be able to discard the implicit Hulten’s theorem which underlies empirical trade work.

We also need more work on the flow of ideas. “Free trade” in the minds of economists is not just literally the exchange of goods, but a bundle of ideas relating to openness. We cannot easily measure the flow of ideas. The best, and most recent, work on the subject is certainly Marta Prato’s recent paper on innovation, where people migrating to the United States made their previous co-inventors who did not migrate more productive.

I expect that more work will also be done in applying trade theory to arbitrary geographies. The vast majority of economic activity occurs within countries. I would cite in particular the work of Treb Allen and Costas Arkolakis here. Much of the interest of economists in trade has been driven by the simple availability of data. Nevertheless, new and better datasets are available for those who look for them now. The proliferation of smartphones is an incredible guide to where people are and how they spend their time. We can use cellphones to directly estimate the elasticity of travel behavior, for example. Similar clever datasets include using the universe of Uber drivers to estimate an exact willingness to pay for road repaving and the use of millions of London Tube riders to estimate the value of forced experimentation during a labor strike.

Lastly, we need a better treatment of trade costs. Trade costs are traditionally presumed to vary with the value of the good, so that our predictions are invariant to the cost of the good or to the income of a country. Breaking this, and having a fixed cost to exporting, distorts the mixture of products shipped. No one, to my knowledge, has been able to really tractably incorporate this (though some papers have noted the problem, and tried to quantity barriers which take on the form of fixed costs).