# John Kennan’s “Open Borders”

This post is going to attempt to do something difficult, namely: bring a contribution to technical economic theory within reach of lay readers. The typical lay reader, or for that matter even an atypically intelligent reader who is not a specialist in economics, could understand little of Kennan’s paper, or for that matter most academic economics papers. I don’t totally understand the paper either, but I think I mostly understand it. I’m pretty sure I understand the main thrust. The work in question is “Open Borders” by John Kennan. If one had to pick one thing to stick in a newspaper headline, it would be Kennan’s prediction that

For the 40 countries in Figure 6 this gives an estimate of \$10,798, per worker (including nonmigrants), per year (in 2012 dollars, adjusted for purchasing power parity). This is a very large number: the average income per worker in these countries is \$8,633, so the gain in (net) income is 125%. For all of the countries in the Penn World Table that are not at the productivity frontier (as deﬁned above), using GDP data to estimate relative wages, the estimated gain is \$10,135, relative to an average income of \$9,079, so the gain is 112%. These are of course just rough estimates, relying on a number of strong simplifying assumptions. But unless these assumptions are extremely far off the mark, the results indicate that the gains from open borders would be enormous.

In other words, open borders could double the income of the world’s most disadvantaged people. Far from causing a “brain drain” effect, harming poor countries by poaching productive people, even nonmigrants would benefit from open borders. Furthermore:

These gains are associated with a relatively small reduction in the real wage in developed countries, and even this effect disappears as the capital-labor ratio adjusts over time; indeed if immigration restrictions are relaxed gradually, allowing time for investment in physical capital to keep pace, there is no implied reduction in real wages.

How does Kennan arrive at this conclusion? Via a theoretical model, calibrated to fit certain real world data. The approach is oversimplified and crude, yet at the same time, in some ways, painstakingly subtle… but that’s economic theory for you. The logic must be impeccable, but economists’ tolerance for departures from realism can be opaque at first, then, once understood, rather shocking. But one has to do it. A question like “what would happen if the world opened its borders?” involves such a large departure from current reality that common sense and experience fail us. Theory can, so to speak, see in the dark. It allows us to keep thinking clearly, at least, about very remote situations. But down to business.

After a short intro, Kennan’s first really substantive paragraph is:

Before proceeding to analyze a world economy with open borders, the ﬁrst question that must be answered is whether restrictions on factor mobility have any real effects. If product prices are the same across countries (because there is free trade and transportation is not costly, for example), and if there are two goods that are produced in two different countries, and if the production technologies (for these two goods) are the same across the two countries, then the factor price equalization theorem applies. That is, real wages and other factor prices are equalized across countries even though factors are immobile, because differences in factor prices are implicitly arbitraged through the product market. The theoretical argument is beautiful, but of course the facts are otherwise. For example, wages in the U.S. are about 2.5 times the Mexican wage, for comparable workers.

This will require some unpacking, especially “factor price equalization” and “differences in factor prices are implicitly arbitraged through the product market.” An important result in international economics is that in the “long run,” given certain fairly standard (albeit apparently unrealistic) assumptions, immigration will not reduce wages in the host country, because the mix of industries in the host country will shift to accommodate the new supply of workers to such an extent that wages will be exactly the same. (See the closely related Rybczynski theorem.) Thus, to use the example from the Feenstra and Taylor textbook that I teach this stuff out of, suppose there are two industries, computers and shoes. Computers are a capital-intensive industry, shoes a labor-intensive industry. If a lot of immigrants enter a country called, say, Home, then Home will start producing more shoes and fewer computers.

Fewer computers? Yes, fewer. Even though there are more workers? Not just proportionally fewer? No, fewer in absolute terms. Think of it this way. There’s the same amount of capital in Home as there was before. But there are now more workers. It’s not surprising that the shoe industry will take the lead in absorbing these workers, since its technology is the more labor-intensive of the two. But as it does so, it will lower the marginal product of labor and raise the marginal product of capital in the shoe industry. Or to try to express it without the jargon, the shoe industry will have trouble finding useful things for so many new workers to do without new machines, structures, and other capital goods for them to work with. Even if the shoe industry can’t increase its capital at all, it could find something for all the new workers to do. It will be able to make more shoes. But not that many more shoes. The greater supply of workers increases the shoe industry’s demand for capital. In fact, it implies that the shoe industry wants capital more, at the margin, than the computer industry does. As capital moves from the computer industry to the shoe industry, workers move too, since the scarcity of machines makes them less productive in computers. Importantly, the relative price of computers and shoes stays the same. This is because prices are pinned down by international trade: any deviation in the relative price would cause arbitrage. Wages don’t fall– again, this is in the long run– because relative prices don’t fall, and wages depend on relative prices. The growth of the shoe industry and the shrinkage of the computer industry raise the economy’s demand for labor relative to capital, exactly canceling out the tendency for a greater supply of labor to reduce its wage. This implies not only that wages shouldn’t be reduced by immigration, but also that wages should be the same in every country.

“But of course the facts are otherwise.” Yes. But the theory narrows down the set of possible stories that are logically valid. It’s not enough to say, for example, that wages are higher in rich countries because they have more capital. In principle, “differences in factor prices are implicitly arbitraged through the goods market,” i.e., capital-poor countries should specialize in labor-intensive industries but shouldn’t pay equivalent workers less. Why doesn’t that happen? It could be that rich-country residents are just more productive because of their own abilities or “human capital,” but then we wouldn’t expect to see immigrants’ wages rise so much when they move. Here, crucially, Kennan assumes that something about countries just makes labor more or less productive. Thus, he writes:

The main novelty in this paper is that the gains from open borders are analyzed in an environment in which income differences are attributed to differences in labor productivity, and the factor price equalization theorem holds.

And

[Empirical] relative wages are used below to measure [I would say: are interpreted as] cross-country differences in labor efficiency – that is, differences in the productivity of a given stock of human capital, when moved from one country to another. For example, the wage of a Mexican worker in Mexico is about 40% of the wage of a worker with the same education and experience who was born and educated in Mexico but who was working in the U.S. in the 2000 Census. This is taken to mean that Mexican workers have 0.4 efficiency units of labor (per unit of human capital), so that a Mexican worker who crosses the U.S. border becomes as productive as 2.5 Mexican workers who stayed at home. The assumption here is that the variables that reduce labor productivity in Mexico (whatever they might be) are specific to the location, and not to the people who work in that location. This is obviously a strong assumption, given that there is no theory of what the relevant variables are.

Kennan doubtless has theories of what variables in fact determine international differences in labor productivity. But articles are written to take their place in a larger intellectual division of labor, and by leaving international differences in labor productivity as a freely-floating, entirely unexplained exogenous factor, he makes his estimates suitable to be included as components in various larger worldviews. As Kennan says, the assumption that wage differences reflect productivity differences is consistent with the factor price equalization theorem: in that sense, it has a good theoretical warrant. It also has a kind of ballpark empirical plausibility: Mexicans do raise their wages a lot, probably indeed by something like 2.5 times, in crossing the US border. Kennan mentions selection bias but cites evidence that it is not large. Kennan also deduces human capital for different countries from the ratio between relative income per worker and wage relative to the US, but this part of the model I did not understand. I can see its relevance but it doesn’t seem essential.

Section 3.1 is brief but very technical. I think I understand it but doubt I can explain it well, but here’s an attempt: under free trade, countries should, in theory, specialize in goods which intensively use the factors in which they are abundant, but since factor abundance depends not only on raw quantities of the factors but also their not easily measurable productivity differences, it may be helpful, instead of testing the theory, to assume it holds and deduce factor productivities from real flows of trade. If this is done, one finds that labor productivity has a particularly strong tendency to vary across countries, capital productivity less so. Kennan proceeds to assume for the purposes of his model that only labor productivity differs across countries.

In Section 3.2, Kennan derives a cost function for a generic good s, points out that free trade (and, simplifying, no transportation costs) implies that the cost function must equal the world price, writes down cost functions for two goods r and s and sets them equal to price, the idea being that these are two goods a country makes, then makes the following assertion…

These equations determine the factor prices in country j. If the marginal rates of technical substitution satisfy a single-crossing condition, the factor prices are uniquely determined.

… as a bridge to a claim that wages in each country must be proportional to labor productivity. What is a “single-crossing condition” and what happens if marginal rates of technical substitution don’t satisfy it? Hmm. A vague graph from some long ago class floats through my mind. I think this has to do with whether a country arrives at some “interior solution,” in which it produces some of each of two different goods, or wholly specializes. In my above example, the “single-crossing condition”– I think— would mean that a country would produce both computers and shoes. There’s a certain range of world prices for which this would be true. Outside that range, the country would completely specialize. One difficulty here is that I think Kennan’s model has the unwelcome implication that each country produces at most two goods from a spectrum of goods ranked by the capital-intensiveness of their production technology. One could appeal to monopolistic competition models of trade to mitigate this unwelcome prediction, but it underlines how tricky a task it can be to make and interpret economic theories. You have to do it if you’re going to study the effects of policy changes, and the more radical the policy changes, generally speaking, the more unrealistic assumptions you need to make. If that makes economic theory seem useless, remember that Ricardo established the principle of comparative advantage using a model with two countries, two goods, and one factor of production, and remember, too, that philosophers like to use far-fetched examples in their arguments to illustrate particular points clearly. And Kennan isn’t just trying to establish the sign of the impact of open borders, but the magnitude as well.

In Section 3.3, the most mathematical section, he derives the following result: that “the elasticity of the factor price ratio with respect to the capital-labor ratio is unity.” That means, that if the capital-labor ratio falls by 1%, the wage falls 1% relative to the rental price of capital. This is a key result because in Kennan’s model, the major direct result of open borders is to increase the world’s effective labor supply. Since international differences in income are attributed to differences in labor productivity, migration is like making more workers. That lowers wages for everyone. This was one of the things that surprised me about Kennan’s model. I had always assumed that open borders would undercut the wages of less-skilled workers in rich countries, but would raise the wages of more-skilled workers. Probably it would do that, in general. But not in Kennan’s model. In Kennan’s model, labor is labor is labor: more human capital just means you have more of it to dispose of. All workers in developed countries, high- and low-skilled, would see their wages fall. That that can make sense in theory makes me wonder: is it true? Of course, that doesn’t mean that all workers in developed countries are worse off. For workers in developed countries may have– nearly always do have– at least a little bit of land and/or capital, too. Collectively, residents of developed countries would be better off. They’d have to face wage cuts, or at any rate smaller raises, but they’d see their stock portfolios and their houses rise in value, and on average this would raise their lifetime wealth. Inasmuch as land and capital are less equally distributed than labor (surely true to some extent though I don’t know how much), open borders would increase inequality. Poor country residents who stay in poor countries would also see wages fall, but rentals on capital and land rise, as migration makes effective labor abundant worldwide. Of course, migrants (always from poor to rich countries in this model) would see substantial increases in their wages because the worldwide fall in wage rates would be more than offset by their increased productivity in the destination country.

In the long run, the fall in wages would be reversed. Wages per effective worker would return to their previous level. Migrants would benefit further from this rise. The long-run wage rise would reflect capital accumulation. The increase in the effective labor supply due to open borders would raise the return on capital, causing investment to exceed depreciation. This would expand the capital stock over time, until it bore the same ratio to the effective labor supply as prior to open borders.

In predicting the volume of migration, Kennan does not assume that humans are strict homini economici who will go wherever they can earn the most. He writes:

One might initially expect that in a world with open borders, everyone would move to the most productive location. But this ignores the strong attachment to home locations that is evident in the data.

He takes this into account by making the migration decision probabilistic, such that the proportion of people who stay in a country is the same as the proportion of the rich-country wage that is paid in that country. For example, if there are open borders between the US and Puerto Rico, and Puerto Rican wages are 2/3 of those in the US, then 2/3 of Puerto Rican adults would stay in Puerto Rico. This roughly fits the data in that particular case, but there is no theoretical motivation for that particular functional form. Relative to a homo economicus model in which everyone who could earn more elsewhere migrated, this assumption causes Kennan to understate the economic benefits of open borders. On the other hand, it also makes Kennan’s version of open borders less scary than it would be if all who stood to gain economically from migration migrated.

Having completed the model, Kennan goes on to map it to the data. One task is to define which countries are on the productivity frontier. He defines \$25,000 as the cutoff for countries to be on the frontier, plus countries (mainly European) that already have open borders with those countries. Fifty-one countries qualify. Counting one US worker as an efficiency unit, the effective world supply today is estimated at 764.1 million workers. Open borders, using Kennan’s procedure for forecasting migration, would raise this 1,507.7 million. Thus open borders would more than double the world’s effectively labor supply. If all the migration occurred instantaneously, this would lead to a 20% drop in wages. A couple of further estimates derived from the model put the gain for people in developing countries at about \$10,000 per worker. That’s an average over all workers, not just migrants.

The goal here is to guess what open borders would look like in practice, and the theoretical work is an instrument for guessing in a methodical and informed way where intuition and experience are of little help. Now, two big things we would like to know about open borders are (a) how many people would move, and (b) how much would world GDP actually increase. If I’m not mistaken, Kennan could easily derive estimates of these things from his model. But he doesn’t. He doesn’t tell us how world GDP would rise under open borders, in the short or the long run. He doesn’t tell us how many people would move, or where they would come from. I think Kennan’s model implies a short-run increase in world GDP of about 65%, and I’m pretty sure in the long run world GDP would double. Since the increase in the effective labor supply comes from growth in the populations of rich countries where labor productivity is high, I think Kennan’s model implies that rich countries’ populations would more than double due to immigration under open borders.

I pretty much believe Kennan’s model, though of course it’s a simplification. The biggest criticism it’s open to is that the factors that make labor productivity high in rich countries depend on the composition of the population in those countries and wouldn’t survive vast influxes of people. Let in a few million immigrants and they’ll assimilate to high labor productivity, but let in tens or hundreds of millions and they’ll drag labor productivity down. Note that while Kennan’s model does not predict that labor productivity in rich countries would fall, it does predict that wages would fall. It wouldn’t surprise me if labor productivity fell a bit too, but I doubt it would fall that much, even if from the perspective of public opinion, the effects of open borders seemed like a catastrophic social upheaval. The case of California seems instructive here: there’s been a huge demographic change, and all sorts of apocalyptic rhetoric, and the state has become a bit less of a beacon than it was, but California’s First World status hasn’t changed at all. Of course, California doesn’t have open borders, but I expect open borders would be a more intense version of the same phenomenon: enormous perceived social upheaval, but no general fall in living standards for natives. It depends a lot on how it’s managed.

There’s plenty more to be said, but I think Kennan’s model is a good guide to the future. By the way, John Kennan could doubtless have written this post better than I just did! If he ever wants to write a guest post, I’m sure we’d welcome it…

Nathan Smith is an assistant professor of economics at Fresno Pacific University. He did his Ph.D. in economics from George Mason University and has also worked for the World Bank. Smith proposed Don’t Restrict Immigration, Tax It, one of the more comprehensive keyhole solution proposals to address concerns surrounding open borders.