Are you starting a (graduate) course in international trade this fall? If so, you’ll soon be encountering the so-called Dixit-Stiglitz CES function, a demand system that underlies trade economists’ work on everything from the gravity equation to the organization of multinational enterprises.

Because the specification is so popular, economists frequently skip to its well-known results, like consumers’ Marshallian demand functions and firms’ pricing strategies, without deriving them. And professors introducing the topic sometimes ask students to derive the results without giving them much guidance.

Here’s my introductory guide to the Dixit-Stiglitz demand system. It walks through the most basic derivations step-by-step, in hopes of helping those students so frustrated with wading through tedious algebra and integrals that they’ve turned to Google searching for a very basic introduction to Dixit-Stiglitz lite.

Comments and corrections (!) are most welcome; my email address is on the first page of the pdf.

In a short two-page note, Yale’s Costas Arkolakis shows that you can introduce non-standard transport costs into the Dixit-Stiglitz monopolistic competition set-up and get clean results. You might call the transport costs “variable icebergs”:

t(q) = Τq^a

where a < 0 so that the unit cost of shipping the good is decreasing with the size of the shipment.

Arkolakis shows that consumers’ demands and firms’ mark-ups come out cleanly in the usual CES fashion, so that this formulation is tractable and isomorphic with the usual iceberg approach in some respects. In the Melitz (2003) formulation, you preserve Pareto distributed outcomes, they just have a different coefficient.

Mary Amiti & Donald R. Davis – “Trade, Firms, and Wages: Theory and Evidence” NBER WP 14106

We develop a general equilibrium model which features firm heterogeneity, trade in final and intermediate products, and firm-specific wages. In doing so, it builds on the work on heterogeneous firms of Marc J. Melitz (2003) as amended to allow trade in intermediate goods by Hiroyuki Kasahara and Beverly J. Lapham (2007). Both of these models maintain the assumption of homogeneous labor and a perfect labor market, so that the wages paid by a firm are disconnected from that firm’s performance. We continue to focus on homogeneous labor, but introduce a novel variant of fair wages which links the wages at a firm to the profitability of the firm…

A decline in output tariffs reduces the wages of workers at firms that sell only in the domestic market, but raises the wages of workers at firms that export. A decline in input tariffs raises the wages of workers at firms using imported inputs, but reduces wages at firms that do not import inputs…

We test our model’s hypotheses with a rich data set covering the Indonesian trade liberalization of 1991-2000. The trade liberalization provides us with over 500 price changes per period, covering both input and output tariffs. A distinctive feature of the Indonesian data set is the availability of firm level data on individual inputs, making it possible to construct highly disaggregated input tariffs. This, in turn, enables us to disentangle the effects of output and input tariffs…

A 10 percentage point fall in output tariffs decreases wages by 3 percent in firms oriented exclusively toward the domestic economy. But the same fall in the output tariff increases wages by up to 3 percent in firms that export. A 10 percentage point fall in input tariffs has an insignificant effect on firms that don’t import, but increases wages by up to 12 percent in firms that do import. In short, liberalization along each dimension raises wages for workers at firms which are most globalized and lowers wages at firms oriented to the domestic economy or which are marginal globalizers. Ours is the first paper to show an empirical link between input tariffs and wages, and the first to show differential effects from reducing output tariffs on exporters and non-exporters.

There’s a discussion of international trade at Martin Wolf’s forum.

Larry Summers writes:

The normal argument is that a more rapidly growing global economy benefits workers and companies in an individual country by expanding the market for exports. This is a valid consideration. But it is also true that the success of other countries, and greater global integration, places more competitive pressure on an individual economy. Workers are likely disproportionately to bear the brunt of this pressure.

Lawrence Marsh writes:

We all seem to have forgotten basic economics in general and international trade theory in particular. What happen to the issue of the efficient allocation of resources? Since when do tariffs, trade restrictions and other methods of distorting prices help increase the welfare of society?

Free trade brings in more competition forcing businesses to improve their productivity and efficiency and to lower prices…

Since everyone is a consumer, everyone benefits from lower prices.

Tim Worstall attempts a summary:

I should of course approach the debates amongst those greatly better versed than I in the subject under discussion, the economics of trade and globalisation, with a certain humility… So, without that necessary timidity, here goes…

They’re talking past each other. Summers is putting the American political case for the support of further globalisation. Within the limits of American politics he may well be correct. Marsh is making a very different case: not having been in the political system (Summers was Treasury Secretary under Clinton), only the academic, his view of the matter is more straightforward. We’re economists and this is what economics has to say about trade. As far as I’m aware, what he says is correct too.

Not quite. Why would Larry Summers write a column that merely capitulated to protectionist sentiments contrary to economic insight?

They’re talking past each other because they have different models of international trade in mind. Larry Summers is alluding to a Heckscher-Ohlin-Viner type of model with multiple factor inputs, while Marsh’s story about price competition looks more like an oligopolistic model of trade.

In Summers’ world, the claim that “since everyone is a consumer, everyone benefits from lower prices” is flat out wrong — trade changes relative prices, and in the n &times n HOV model, Stolper-Samuelson holds for at least one factor!

What does Marsh have in mind? In a basic model of oligopolistic competition, à la Brander (1981), there is only one factor of production, so distributional concerns (other than firms vs consumers) don’t apply. Moreover, it’s partial equilibrium, so you can’t really say much about welfare. If Marsh’s story relies on an oligopolistic general equilibrium model of trade, then it’s a long way from “basic economics.”

Most trade produces winners and losers. Not always, but that’s how Paul Krugman would summarize econ 101. That’s why most economists will back Larry Summers over Lawrence Marsh. Sorry, Tim.

[Of course, there can be stories without losers: Krugman (1979) features monopolistic competition, so that moving from trade to autarky simply doubles the size of the economy, leaving all firms the same and improving consumer welfare via gains from variety.]

Mark Thoma sends us to Thomas Palley’s critique of modern trade policy. Frankly, I find much of his post rather confusing.

For example, Palley opens by writing:

The classical theory of comparative advantage has driven US trade policy for the past fifty years. That policy, in combination with technical innovations that have lowered costs of transportation and communication, has opened the global economy. Yet paradoxically, this opening has rendered classical trade theory obsolete. That in turn has left the US economically vulnerable because its trade policy remains stuck in the past and based on ideas that no longer hold.

US trade policy has been driven by the theory comparative advantage? I doubt this. Immediately after WWII, the founding of the GATT was largely motivated by the economic disaster of the interwar period and protectionism’s exacerbation of international tensions in the 1930s. Once the GATT members started cutting tariffs, they largely liberalized industrial sectors, not areas in which poor countries had comparative advantage, like agriculture and textiles.

In fact, the motivation for new trade theory was the inability of the classic theory to explain trade:

For some time now there has been considerable skepticism about the ability of comparative cost theory to explain the actual pattern of international trade. Neither the extensive trade among the indsutrial countries nor the prevalence in this trade of two way exchanges of differentiated products make much sense in terms of standard theory. As a result many people have concluded that a new framework for analyzing trade is needed. [Paul Krugman, Rethinking International Trade, p.22]

Although Ricardian or Hecksher-Ohlin theories of comparative advantage may have constituted the bulk of trade theory prior to 1980, I think it would be very difficult to make the case that they were important determinants of US trade policy.

In his next paragraph, Palley writes:

The logic behind classical free trade is that all can benefit when countries specialize in producing those things in which they have comparative advantage. The necessary requirement is that the means of production (capital and technology) are internationally immobile and stuck in each country. That is what globalization has undone.

While it is true that the basic Hecksher-Ohlin model features K^F and K^H, this simple assumption is not critical to the logic of comparative advantage. Where there are differences in factor endowments and opportunity costs, there are potential gains from trade. If capital is more scarce in poor countries, then capital mobility will result in some rich country capital owners investing in the poor countries to earn higher returns. But only in the extreme case that all countries have the same proportional factor endowments will there be no potential gains from trade. Capital mobility is far from a “necessary requirement” for comparative advantage. See Paul Krugman and Don Boudreaux (pdf) for past encounters with this argument.

The rest of Thomas Palley’s post is largely about empirical issues, but they are either well-worn (the “race to the bottom” arguments) or under-researched at present (trade and income inequality). I’ll leave those for others to tackle.

You have likley already seen “Comparative Advantage and Heterogeneous Firms,” as Andrew Bernard, Stephen Redding, and Peter Schott worked on it for more than four years. If you haven’t, now is the time. The paper was finally published early this year in the Review of Economic Studies (journal pdf; older ungated pdf). It’s a fabulous piece of theory that introduces heterogeneous firms (via a Pareto distributed productivity term) and fixed trade costs to the classic 2x2x2 monopolistic competition model of trade.

Here’s the abstract:

This paper examines how country, industry, and firm characteristics interact in general equilibrium to determine nations’ responses to trade liberalization. When firms possess heterogeneous productivity, countries differ in relative factor abundance, and industries vary in factor intensity, falling trade costs induce reallocations of resources both within and across industries and countries. These reallocations generate substantial job turnover in all sectors, spur relatively more creative destruction in comparative advantage industries than in comparative disadvantage industries, and magnify ex ante comparative advantage to create additional welfare gains from trade. The improvements in aggregate productivity as countries liberalize dampen and can even reverse the real-wage losses of scarce factors.

This is clearly an improvement over existing models of trade with hetereogeneous firms, which often feature a numeraire that pins down the wage and labor as the only input. Now we get to talk about wage effects and comparative advantage! Unfortunately, the resulting mathematical complexity is such that closed-form solutions don’t exist for some of the endogeneous variables. The authors have to resort to numerical simulations to describe some areas of interest. Nonetheless, I really like this paper.

You have likley already seen “Comparative Advantage and Heterogeneous Firms,” as Andrew Bernard, Stephen Redding, and Peter Schott worked on it for more than four years. If you haven’t, now is the time. The paper was finally published early this year in the Review of Economic Studies (journal pdf; older ungated pdf). It’s a fabulous piece of theory that introduces heterogeneous firms (via a Pareto distributed productivity term) and fixed trade costs to the classic 2x2x2 monopolistic competition model of trade.

Here’s the abstract:

This paper examines how country, industry, and firm characteristics interact in general equilibrium to determine nations’ responses to trade liberalization. When firms possess heterogeneous productivity, countries differ in relative factor abundance, and industries vary in factor intensity, falling trade costs induce reallocations of resources both within and across industries and countries. These reallocations generate substantial job turnover in all sectors, spur relatively more creative destruction in comparative advantage industries than in comparative disadvantage industries, and magnify ex ante comparative advantage to create additional welfare gains from trade. The improvements in aggregate productivity as countries liberalize dampen and can even reverse the real-wage losses of scarce factors.

This is clearly an improvement over existing models of trade with hetereogeneous firms, which often feature a numeraire that pins down the wage and labor as the only input. Now we get to talk about wage effects and comparative advantage! Unfortunately, the resulting mathematical complexity is such that closed-form solutions don’t exist for some of the endogeneous variables. The authors have to resort to numerical simulations to describe some areas of interest. Nonetheless, I really like this paper.

UNC’s Karl Smith has posted an argument for protectionist policies (pdf) and invites us to tear it apart. In brief, he reiterates the well-known point that risk-averse agents will forego gambles with positive expected payoff if the gains are sufficiently small. He then applies this behavior to trade:

If the distribution of the gains and losses to trade are uncertain then this imposes a cost on the agents. If the net total gains from trade do not exceed the losses from uncertainty everyone can be worse off.

The document is titled “Losses from Trade,” but if you read the six pages offered by Smith, you’ll realize that this paper is not about trade. His “objections” and “conclusions” sections (the last two pages) don’t even discuss international trade! Smith’s argument applies to all aspects of life – risk aversion simiarly implies that stasis may be preferable to technological advances or institutional shifts that cause small net gains with distributional uncertainty. This line of reasoning is valid as a general argument against dynamism, not trade in particular.

Nonetheless, let’s consider the application. Is trade that uncertain? As Dean Baker has complained, we know that typical US trade liberalization will tend to displace workers in labor-intensive manufacturing, while protecting radiologists and other white collar workers. Globally, French farmers oppose Doha, West African cotton growers favor it, and American doctors have nothing to worry about. The political economy story to investigate is how one interest group defeats another.

Perhaps trade liberalization implies more uncertainty generally. Is there a link to labor market churn? The (Ricardian) gains from trade are a result of reallocating resources across productive opportunities, but such sectoral shifts may be predictable. Do we have reason to believe (from theory or empirics) that lower trade barriers increase the separation rate for all workers?

And does protection imply greater certainty? Perhaps, but I’d have to reflect upon that topic for a while before feeling confident about my answer.

Nonetheless, suppose for now that the government has the ability to preserve the status quo. Risk aversion only rules out trade liberalization that produces small net gains. The implication would then be that we ought to have large spurts of bundled liberalization (WTO rounds, anyone?).

In sum, Dr. Smith has highlighted a potentially promising area of research, but I don’t find his initial sketch very compelling. I’d need to see compelling empirical evidence on the relationships between trade and uncertainty before thinking we needed new theory to explain them.

Julian Sanchez attacks trade protectionism:

Here’s a modest proposal, then: Let’s permit whatever restrictions on trade and globalization people like, but with the “winners” under those rules compensating the “losers” via some sort of special targeted tax. We’ll levy this on the workers and stockholders enriched by whatever form of protection from international competition they want to demand, and cut a check to workers, stockholders, and consumers in other sectors that would have benefited from lower prices or operating costs as a function of trade and outsourcing, in the amount of whatever the benefit to them would have been of less restricted trade…

The key thing to bear in mind here is that there’s nothing morally special about the level of globalization in 2007 or 1990 or 1970 as some kind of special baseline. We talk about “winners” and “losers” relative to some status quo ante where there happened to be a different level and pattern of globalization, but the point of comparison is—from the point of view of justice, if not realpolitik—arbitrary. Which is why compensations from the “winners” to “losers” under protectionism makes as much sense—probably more—than the parallel sort of compensation as globalization increases. The only reason to think otherwise is to suppose we’re specially and permanently entitled to the pattern of holdings we’d have at some particular but arbitrary level and kind of globalization.

Theory aside, given the strength of status quo bias, adjustment assistance or some other form of compensation is politically necessary to make trade liberalization feasible. Paying people to overcome their status quo bias echoes the political necessity of paying people to overcome another bias:

It is tempting to argue… that all changes require adjustment, and that assistance should be provided in a generic fashion… This viewpoint is valid in a cosmopolitan world… However, in the real world, the refusal to accept change – and hence the need to accomodate it and facilitate it through adjustment assistance – is greater when the source of disturbance is foreign… The case for differential adjustment assistance rests on this asymmetry in communities’ attitudes toward change from foreign and domestic sources. [Bhagwati, Protectionism, p.118-9]

Trade adjustment assistance facilitates liberalization by dampening both of these biases. But it might not if the general public knew its effectiveness.

Don Davis & James Harrigan provide theoretical grounding for public worries in “Good Jobs, Bad Jobs, and Trade Liberalization”:

Globalization threatens “good jobs at good wages”, according to overwhelming public sentiment. Yet professional discussion often rules out such concerns a priori. We instead offer a framework to interpret and address these concerns. We develop a model in which monopolistically competitive firms pay efficiency wages, and these firms differ in both their technical capability and their monitoring ability. Heterogeneity in the ability of firms to monitor effort leads to different wages for identical workers – good jobs and bad jobs – as well as equilibrium unemployment. Wage heterogeneity combines with differences in technical capability to generate an equilibrium size distribution of firms. As in Melitz (2003), trade liberalization increases aggregate efficiency through a firm selection effect. This efficiency-enhancing selection effect, however, puts pressure on many “good jobs”, in the sense that the high-wage jobs at any level of technical capability are the least likely to survive trade liberalization. In a central case, trade raises the average real wage but leads to a loss of many “good jobs” and to a steady-state increase in unemployment.

NBER working paper.