To estimate the global economic impact of opening the world’s borders to migration involves heroic extrapolation. It is therefore a task in which theory must do most of the heavy lifting, with empirical work limited to determining plausible parameter values. To people who are cynical about economic theory, this makes the double world GDP literature so speculative as to be irrelevant. And cynicism about economic theory is a reasonable attitude to have. After all, economic theories, especially the most abstract ones, tend to start from very questionable assumptions, and they usually make some very implausible predictions, too, if one knows how to tease them out. (Often, theorists hide these.)

I was reminded of this recently by Carl Shulman’s take on the “double world GDP” literature, in which he pointed out something I didn’t notice about John Kennan’s “Open Borders” paper when I tried to summarize it for general audiences a few months ago: the enormous size of predicted migrant flows. Carl writes:

To estimate migrants from a country Kennan multiplies an estimate of a country’s national labor force by 1 minus 1/(the relevant place premium)… In the appendix of his paper, Kennan lists relative wages, labor forces, and other information on 40 less-developed migrant source countries. This leaves out many other countries, but since the world’s most populous ones are included the group would still account for most migrants…

From this sample over 75% of the labor force are predicted to migrate. As noted above, this leaves out many countries, children, and women not in the labor force, among others. If we included family members and other countries the implied number of migrants looks like it would exceed 3 billion.

Kennan doesn’t mention explicitly how many migrants his theory predicts, possibly because it would make his theory too easy to mock. The prediction, once Carl brought it to my attention, struck me as implausible, and provoked me to develop my own model to extrapolate the impact of open borders, which is the topic of this post.

I haven’t calibrated my model to the data yet, so I can’t say in this post whether my theory confirms the “double world GDP” estimates or not, or how many migrants it will predict. But further thinking about Kennan’s model has made me less skeptical of Kennan’s high estimates of how much of the world population would move. Emigration sources like Iowa and Ireland have a fraction of the population that they would, had they retained all their natural increase over the periods when emigration was rife (most of the 20th century for Iowa, the 19th century for Ireland). Granted, the cultural barriers to emigration from Iowa and Ireland were unusually low for their times– Iowa and Ireland are both English-speaking places whose emigrants went to English-speaking places– but (a) I suspect people overestimate cultural barriers to migration, and (b) American-led cultural globalization is such a powerful force these days that I suspect emigration from Tajikistan or Mali to America today faces smaller cultural obstacles than emigration to the US by, say, Russian Jews in the 19th century.

In developing a theoretical model to facilitate extrapolation of the impact of global migration flows, one problem is that “general equilibrium” models do a lousy job of explaining the current global distribution of income, and therefore seem like unreliable guides to the hypothetical global distribution of income under open borders. Some time ago, Robert Lucas pointed out that if the Solow model, still the most influential model of long-run economic growth, is used to explain global income differences, it would also predict vastly higher returns to capital in poor countries than in rich countries, and that if capital is mobile, all new investment should occur in poor countries.  Mankiw, (David) Romer and Weil (1992) “fixed” the Solow model by augmenting it with human capital, only to generate the absurd counter-factual prediction that returns to human capital should be far higher in poor countries, and skilled workers should be migrating from rich countries to poor countries, rather than other way around. Later, Mankiw and (Paul) Romer (1995) had a standoff in which Romer pointed out this glaring weakness in the “human-capital-augmented” Solow model. Romer (1990) has become one of the most cited papers in development economics and the flagship of the “endogenous growth” literature, but Romer’s ideas about ideas only underline the point that since ideas are non-rival and only partially excludable, they can’t do much to explain international differences in income. The conventional wisdom at this moment in time can probably be summed up: “ideas explain long-run growth, institutions explain cross-sectional income differences,” with institutions, about the definition of which there is little agreement and which formal economic theory is mostly unable to elucidate, winning by default because theories that are clear enough to be falsifiable have tended to be falsified. To me, it’s a rather glaring and obvious weakness that totally different theories are invoked to explain international and intertemporal income differences.

My own belief is that the original sin of the growth literature is that it neglects the division of labor, specialization and trade, increasing returns, in short, the first three chapters of The Wealth of Nations. It neglects them because these happen to be inconsistent with general equilibrium and “competitive” markets in the peculiar sense which 20th-century mathematical formalism in economics gave to that word. This blind spot also makes mainstream economics unable to explain why there are cities. And so it is with cities that my model begins.

Warning: from here on, non-economists will have to pick their way through the technical apparatus of a formal economic theory. I’ll explain as I go as best I can, but the content of the model is really explained in the equations. At a later stage, I plan to calibrate this to the data– or perhaps get a co-author to do so– and generate one of those rather spuriously precise estimates of how much open borders would raise GDP, followed by remarks on parameter sensitivities that hardly anyone really reads, etc. More interesting than the final numbers are the reasoning and scenarios one passes through along the way.

The starting place for my model is the city-level production function…

(1)

where Y is the city’s GDP, A is the “total factor productivity” of a city, h is the average level of human capital in the country (not just the city), N is the population of the city, and α and β represent the “output elasticity of capital,” i.e., the % change in output for a % change in capital, holding all else constant, and the “output elasticity of (effective) labor,” i.e., the % change in output for a % change in (effective) labor, that is, the number of workers multiplied by their average human capital.

Importantly, I do not assume that α+β=1. On the contrary, my presumption is that α+β>1, that is, that there are increasing returns at the city level, for the reasons Adam Smith understood well, namely, that the division of labor is limited by the extent of the market. Because of increasing returns, we cannot interpret α and β as Cobb-Douglas exponents. We cannot assume that if capital receives 30% of national income, then α=0.3. I tend to think that, as Paul Romer among others has suggested,  the output elasticity of capital, i.e., α, is more than capital’s share of income, and I suspect that labor may get more than its marginal product, too, due to various political distortions.

By the way, the model can accommodate the case of decreasing returns, α+β<1, as well. This assumption, too, can be plausible at the city level, given the scarcity of land. It would not imply the obvious counter-factual prediction of “backyard capitalism,” with people spreading out evenly over the land, or among the cities, because some cities have higher “total factor productivity” than others. But generally I think increasing returns are the most likely.

Multiplying population by average human capital to get “effective labor,” a method adopted for convenience, tends to downplay potential complementarities between skilled and unskilled labor. In effect, I assume that tools and/or time can substitute for skill. This assumption is not entirely satisfying, but at least it avoids the very counter-factual prediction that skilled labor will emigrate to where it is scarce and can earn more.

“Total factor productivity” deserves comment. This term is borrowed from the large “growth accounting” literature, which grew out of Solow (1957), and I believe the concept is fundamentally flawed, because it assumes away for no good reason what Adam Smith and I think a real understanding of economic growth starts with: increasing returns, gains from specialization and trade. Think of “total factor productivity” as a black box: for reasons we don’t understand, some places/times are more productive than others. However, the fact that I make “total factor productivity” a city-specific parameter has ramifications for its interpretation. A is a kind of pure place premium: it’s the (cultural, historic, geographic, political, whatever) difference between London and San Francisco and New York and Mexico City and Fresno and Gary, Indiana, etc., not the (political) difference between being under US sovereignty and being under British or Mexican or Tibetan or Malawian sovereignty. And because I am determined to accommodate increasing returns, the place premium won’t have to do nearly as much work as it does in some theoretical models.

Imputing “total factor productivity” to cities might seem to give the cities an anomalously large role in determining living standards. After all, aren’t national and regional policies important too? Actually, I am not assuming that productivity is entirely locally determined. If national or regional policies are important to productivity, they would be reflected in higher A for all the cities in the nation or region.

My assumption is that labor is mobile within each country, but not internationally, and the wage in each city must be competitive with other cities. But big cities have to pay higher wages, because there are “congestion disutilities,” which reduce a worker’s utility, as follows:

(2) 

where U is the worker’s utility when w is his wage and N is the population of the city he lives in. The parameter σ regulates the extent of congestion disutilities. For example, if σ=1/3, then if City M is eight times as big as City N, the prevailing wage in City M must be twice as high as that in City N. To be competitive, then, the wage w in a city must be:

(3) 

where w0 is the “base wage” in the nation as a whole. This “base wage” is probably the single best indicator of worker utility in the model. Its definition is a bit subtle, though. It is the wage a worker would earn in a hypothetical city of population 0, where he would suffer no congestion disutilities at all. All workers will actually earn more than this, because they all live in cities with at least some population.

How should “congestion disutilities” be interpreted? Many options here: pollution, crime, the absence of green grass and fresh air, tighter regulation of land use. But probably the best interpretation is: high urban rents. Land markets are difficult to model, and congestion disutilities are an indirect way of taking land scarcity into account. You can build high-rises to economize land, but people tend to prefer houses (with yards) to high-rises, and high-rises are expensive to build.

Note that without the congestion disutilities, the model would become degenerate, because the nation’s whole population would concentrate in one city, except in the special case of decreasing returns. This would occur for two reasons. First, the city with the highest “total factor productivity” would always outbid other cities for population. Second, increasing returns would continually reinforce its advantage as it grew. Congestion disutilities, however, may put a limit on metropolitan growth, as the higher wages the city can pay because of increased productivity are eventually overtaken by workers’ aversion to overcrowding and high rents.

The next step in the model may seem odd, but it is necessitated by the need to accommodate increasing returns. I assume that the city determines labor demand collectively in order to maximize “rents,” that is, to maximize:

(4)  

“Rents,” in a somewhat Ricardian sense, represent the difference between what the city produces and what it has to pay in competitive factor markets for its capital and labor. Who gets these “rents?” One obvious answer is landlords. Those who have the good fortune to own prime urban land in a modern economy can get very rich without doing a whole lot. Another is the government. When a city can offer higher “total factor productivity” than rival cities, or when increasing returns raise the productivity of its labor and capital, then it can afford to extract extra revenues through taxes and regulations which can then be distributed to various political stakeholders. Rents might also be used to benefit broader humanity or posterity, and I think many great cities really have used their extra productivity in generous ways. We all owe much to the intellectual and architectural attainments of the Greek poleis of the golden age, or the Italian cities of the Renaissance, and probably, also, to some great urban universities of our own day, whose work is meant to, and does, enrich many people beyond the narrow confines of the university or the city.

By the way, if it seems odd to treat cities as rent-maximizing corporations, bear in mind that this is part of my strategy for escaping the trap of “general equilibrium” thinking, whose repeated failures I explained above. I think a deeper revolution in economic theory is needed to escape the legacy of general equilibrium and the constant returns assumption. I’m working on that. But in the meantime, one must make do with these awkward expedients.

Whatever the cities do with their R, to maximize R, they should employ capital K*:

(5) 

for any given quantity of labor N; and they should employ labor N*:

(6)   

Since the expression on the right-hand side of (6) is very complex, we can define…

(7)    

… and …

(8)   

… (or “τ” in this font) and rewrite (6) as:

(9) 

So far, then, our theory predicts the size of city i, given the cost of capital (r), the average level of human capital (h), the prevailing wage (w0), city-specific total factor productivity (A1, that is, in the constant returns case, then is simply 1/τ.

The most interesting case, in my view, is where there are city-level increasing returns, but not enough to induce τ<0. For example, if α=0.5, β=0.6, and σ=0.3, then τ=10. See empirical evidence for city-level scale economies here. It seems that moving someone to a city twice as big will typically raise their economic activity of all kinds by 15%, which is consistent with plugging the parameters α=0.5, β=0.6 into equation (12) below. This yields the surprising yet plausible prediction that small differences in total factor productivity can drive huge differences in city size. For example, if City M is 20% more inherently productive than City N, City M would be over six times as large.

We have determined “population demand” at the level of the city, as a function of the base wage. At the national level, population demand must equal population supply, and the base wage will adjust to ensure this. That is, equation (10) must hold…

(10) 

… and w₀ is the variable that must move to make sure it does, since other variables are either exogenous endowments (A and h), or set at the global scale (r, because of the assumption of international mobility of capital).  The base wage that clears the national labor market turns out to be:

(11)

The base wage is a variable of considerable interest, since it is crucial to the living standards of the populace. Equation (11) shows how it is determined. Several points may be made here:

• The base wage is a decreasing function of the global price of capital. This is not too hard to understand. Labor and capital are complements. If it’s cheap to equip workers with machines, bosses will equip them, make them more productive, and pay them more. If capital is expensive, workers will be less well equipped, and will produce and earn less.
• The base wage is an increasing function of average human capital in the nation. This is rather a welcome prediction since, in fact, workers of a given skill level do tend to earn more in places where the average skill level is higher. Part of the “place premium,” then, is explained not by the black box of “total factor productivity,” but by differences in human capital, and increasing returns. This also suggests a reason why wealthy democracies seem to have such a strong bias in favor of “high-skilled” immigration.
• The base wage is a decreasing function of population. It will turn out on further examination that workers do not actually produce or earn less when the population grows; rather, the decline in the base wage reflects increased congestion disutilities. The elasticity of the wage with respect to population is -1/τ, so if τ=10, as I suggested above as a plausible estimate, then the depressing effect of population on the base wage is rather slight.
• The base wage depends a good deal on a special kind of weighted average of the “total factor productivities” of the nation’s cities, for which summation I will suggest a label: “the national endowment.” Think of the national endowment as including many things: a pleasant climate; beautiful beaches; fair landscapes; good institutions; historic art and architecture; the special beauty of a city skyline; the culture, the feel, the ethos, of famous cities. It includes everything about a country that is valued, beloved, and not readily replicable. In various ways, the national endowment will probably be reflected in market prices and measured GDP, and it will certainly affect utility.

By the way, the model is a bit pessimistic about what is non-replicable. Congestion disutilities would be mitigated if open borders would lead to the founding of new towns. Doubtless it would, and the assumption of this model that no new towns can be founded is extreme, but I think founding new towns does tend to be difficult, and new towns can be a bit dull and blank. Cities that have grown up organically over many generations tend to have a charm about them that’s difficult to reproduce. So while “no new towns” is too extreme an assumption, it does take into account something that is worth taking into account.

On the other hand, the model is rather optimistic in that place premia and the national endowment are not easily diluted. A critic of immigration might expect that if one dilutes the population of a place with hordes of foreigners from poor countries, the special assets that made it productive will be diluted. I don’t think history supports that claim, and the model concedes nothing to it. But while immigrants don’t dilute the basic place premium, they do cause congestion, and low-human-capital immigrants cause congestion while offering relatively little compensation in the form of increased economies of scale. Increasing returns depend on effective labor, congestion on mere population. So it’s not irrational for natives to look askance at mass immigration of poorly educated foreigners.

From here, it is fairly easy to calculate city-level GDP…

(12) 

… and national per capita GDP…

(13) 

… which, unlike the base wage, is positively related to national population (at least if α+β>1), I think because a higher population attracts more capital investment and increases the economic rents enjoyed by the nation’s cities.

Now, the way I extrapolate the economic impact of open borders using this model is simply that open borders cause their human capital averages and national endowments to be pooled. If Country C and Country D open their mutual borders, their cities are included in a single list and a joint national endowment calculated, and a new human capital average is calculated, as a population-weighted average of human capital in each country.

The assumption that human capital will average out across the two countries is a rather strong one. Is it plausible? While there are, in fact, differences in education and other human capital measures across US cities, they are nothing compared to the differences in human capital between the US and most developing countries. While complementarities between different skill levels are left out of the production function, they are in a sense reintroduced at this stage. Presumably, the interpretation of human capital averaging is that job availability and wages for different types of workers motivates human capital to move to where it is scarce, within a region of free migration. If the internal open borders of the US are one case where human capital averaging seems to have roughly worked, 19th-century open borders in the northern Atlantic region and internal open borders in the EU are two other cases which, if they don’t strictly confirm the “human capital averaging” assumption, at least lend it plausibility. Average human capital converged between the US and Europe in the 19th century, and it is relatively homogeneous across the contemporary EU.

If  the link below works…

Open Borders Impact Example

… it will give you an Excel simulation of US-Mexico open borders that I made, calibrating the above model with crude numbers from off the top of my head. The simulation isn’t super user friendly, but in principle, you could download it yourself, play with the parameters, and see the results. Rather than trying to state in one number what the model “predicts,” I’d rather summarize my preliminary results in a set of scenarios. In all scenarios, the population of “USA” is 300 million, that of “Mexico” is 100 million.

Scenario 1. (parameters: α=0.5, β=0.6, σ=0.35, h in “USA”=20, h in “Mexico”=12, r=5%)

In this scenario, US GDP per capita starts out at $50,500, and Mexican GDP per capita starts out at $15,686. The base wage in the US is 21.9, and in Mexico, 16.7. Under open borders, net emigration from Mexico is 61,632,000, well over half the Mexican population. Joint GDP for the US and Mexico rises from $16.7 trillion to $17.5 trillion, a 5% increase. In both countries, the new base wage is 20.9. This represents roughly a 5% fall in wages in the US, but a 20% rise in wages in Mexico.

Scenario 2. (parameters α=0.55, β=0.6, σ=0.5, h in “USA”=20, h in “Mexico”=10, r=5%)

In this scenario, US GDP per capita starts out at $51,075, Mexican GDP per capita at $10,811. The base wage in the US is 4.76, in Mexico, 3.49. Under open borders, the base wage becomes 4.47, representing a 6% fall in the US, a 22% rise in Mexico. Net emigration from Mexico is 31,352,000.

Most surprisingly, joint GDP for the US and Mexico actually falls in this case, by a little over 2%, from $16.4 trillion to $16.0 trillion. How can that be? How could the movement of millions of Mexicans to a more productive country reduce world GDP? At first, I was baffled by this, but then I saw why it makes sense: under open borders, there is an emigration of “effective labor” from the US to Mexico, as Americans with relatively high human capital emigrate to Mexican cities to escape urban congestion at home. Human capital averaging makes it possible for Mexico’s population to fall by 31% through emigration, even as “effective labor” in Mexico increases by 20% due to immigration of labor with higher human capital.

This is where theory pays off. It expands your mind. I found this scenario hard to believe at first, but after thinking about it a bit, I decided it was plausible after all. Lots of young 20-somethings like to bounce around Europe for a year or two, or ten. You meet American expatriates all over the world. I’ve been one a few times. What are they looking for? “Adventure,” “culture,” “‘romance,” “experience,” “permanent vacation,” joie de vivre… one could toss out a lot of words groping for it, and of course it varies from person to person and place to place, but in terms of this model, it’s (a) to enjoy another country’s “national endowment,” and (b) to escape congestion disutilities.

I can easily imagine a world in which open borders between the US and Mexico leads, not only to massive emigration of unskilled labor from Mexico, but at the same time, to a large influx of college-educated Americans eager to enjoy the Mexican sunshine and beaches, and to live in historic centers without paying the exorbitant rents of Boston or San Francisco. In a country where college-educated people are relatively scarce, young college-educated Americans could often find good jobs, or start businesses. They would earn somewhat less than at home, but it would be a price worth paying for sunshine, adventure, and history.

A few more comments relevant to the plausibility of this scenario. 1) There is already an American diaspora of maybe 6 million. 2) While the fact that most Americans don’t choose to emigrate might suggest the scenario lacks realism, Americans can’t automatically work in foreign countries. See here for a story about the difficulties faced by an American working in France. 3) Under open borders, emigration would become more attractive for Americans, because wages would rise abroad, and there would be more congestion in American cities and less in foreign cities. 4) If nonetheless large-scale emigration of Americans under open borders seems implausible to you, you can pick parameter values that don’t predict that.

Scenario 3. (parameters α=0.45, β=0.6, σ=0.25, h in “USA”=20, h in “Mexico”=8, r=10%)

In this scenario, Mexico essentially empties out. The initial gap is larger: US GDP per capita is $50,232, Mexican GDP per capita, $6,691. The real US/Mexico gap is not that large, but plenty of other countries are even poorer than that relative to the US. Under open borders, net Mexican emigration is 98,743,000. A mere 1.26 million Mexicans stay in Mexico. The base wage falls by 6% in the US and rises by 53% in Mexico. Joint GDP rises from $15.7 trillion to $17.2 trillion, a 9.5% increase.

Tentative conclusion so far: My sense is that economic models predicting that open borders will “double world GDP” will continue to depend on extremely large movements of people. Again, I will not say that such predictions are unrealistic, upon reflection they seem plausible to me. But we should avoid breezily quoting “double world GDP” predictions while allaying or minimizing people’s fears about epic movements of peoples. It is possible that open borders will prove to be a good less radical in its impact than the available theories suggest. But in that case, it won’t double world GDP, or at least, not in the ways that models like Kennan’s suggest.

Some important benefits of open borders, especially the stimulus it would provide to idea generation and institutional export, are omitted from the extant models, including this one. These factors are difficult to incorporate into theoretical models because there is relatively little agreement about what determines the rate of idea generation, or the quality of institutions. I expect that open borders probably would double world GDP with a mere hundreds of millions, not billions, of people actually migrating, but that may be more than I can say with my economic theorist hat on.