One of the topics that came up at the Open Borders Bay Area Meetup (Sunday January 12) was the idea of slippery slopes towards open borders. My post is based on ideas raised in the discussions, but since the participants’ comments were largely off the record, I’ll take full responsibility (but not necessarily credit) for everything I write here.

Immediate, unconditional open borders, or even something fairly close to it, is unlikely to materialize in the near future. From a practical perspective, then, the best route to open borders is a long series of gradual steps. One key concern of people who have their eyes set on the end goal of open borders is that the first few steps should make future steps easy to achieve. A path that is ostensibly in the direction of open borders, but where the first few steps might lead to a shift backward (such as a nativist backlash that turns public opinion against more liberal migration) would be a bad path. A better choice would be a slippery slope: one where the initial steps may not necessarily look as “extreme” as open borders to the people who might object to them, but once those steps are taken, the next steps towards open borders become easier to accomplish. (UPDATE written after the post was drafted: Tyler Cowen’s recent post makes this type of backlash argument to explain a Swiss referendum to restrict immigration from the EU; see also responses from Bryan Caplan and Jason Brennan).

Some possible slippery slopes

• Gradual expansion and merging of free migration zones: For instance, the EU is a free migration zone for European countries, and it is gradually adding countries. Suppose the United States and Canada created their own free migration zone, Southeast Asian countries created their own free migration zone, and a few free migration zones emerged in Africa and Latin America (South America). The US-Canada free migration zone could, after some time, add Mexico and the Caribbean Islands, and then merge with the South American free migration zone. The European free migration zone could eventually merge with the new unified American migration zone. Over time, as the threat of terrorism, and the subjective sense of the threat, receded more and more into the past, the Gulf states could merge with the European free migration zone. And so on, until eventually the whole world would be a free migration zone.
• Gradual lowering of admissions standards for migrants: The idea here would be to begin with complete open borders for migrants who meet a particular skill level threshold or income threshold, and then to gradually reduce the threshold over time, either directly, or by failing to index for inflation, or by relying on global economic growth. A plausible argument for why this might work is that high-skilled and high-income migrants lead to fewer problems, both real and perceived, for natives, making the natives somewhat more pro-migrant (based on their experience interacting with migrants so far). It might therefore be easier to cross historical thresholds of migrant proportions in the population without any visible difficulties experienced by natives. Australia, Canada, and Singapore might be examples to look at, and it would be interesting to see the extent to which they avoid the nativist backlash phenomenon through an initial focus on high-skilled migration.
• Gradual increase in quotas: Hard quotas in various migrant categories may be gradually expanded, and new categories that permit more general forms of migration may be added, initially with low numerical caps but with the caps steadily rising. Although this could in principle be huge, it doesn’t seem to have had much slipperiness historically in and of itself.
• Family reunification, with a gradually expanding definition of “family”: Family reunification may begin small, with, for instance, the spouses or children, then it may expand to siblings, parents, or cousins, and finally it may expand to close friends or anybody whom a resident can provide some sort of sponsorship guarantee as a “personal acquaintance.” For a fixed definition of family, too, the amount of “chain migration” can be significant, and although insufficiently slippery to get to open borders all by itself, it can act as an amplifier to any other slippery tendencies. Quantitatively, this has been significant in both the US and Europe, and quite difficult to stop, even though governments have often attempted to crack down on various aspects of family reunification to prevent what they see as abuse.
• A tariff/tax scheme with a gradually declining tariff or tax rate: A DRITI-type scheme that begins with high surtaxes and high mandatory savings, to a level that meets the concerns both of restrictionists and people who want the government to earn revenue. If the revenue-earning part of it goes well, that would create a strong impetus to create and expand the program, and lower the surtaxes and mandatory savings in order to attract more migrants and generate more revenue. Even though staunch restrictionists may oppose that second step, governments and the majority of voters may have become too addicted to the revenue streams and expanded economy size afforded by the program to backtrack from it. Over time, the rates would come down to the point that it would look more like a bureaucratic inconvenience than a program specifically designed to regulate migration. At this stage, general egalitarian concerns, or general philosophical views about taxation, may take over and lead to an abolition of the surtax and of the forced savings program.

Reasons for continued slipping

• Growing tangible benefits (real or perceived) feeding into growing support for more migration at the margin: People see tangible benefits from the expansion so far, and want to keep going further in that direction. This may be because the constituency that has been created by partial expansion of migration is more eager for further expansion than the original set of natives (for instance, recent migrants might favor more liberal migration policies — something that seems empirically borne out). It may be because people get addicted to the revenues from a DRITI-like scheme and feel the need to expand it.
• Support for steady expansion as long as there are no tangible or major harms, based on a presumption that ceteris paribus, allowing more people to migrate is better: People don’t actually see much by way of tangible benefits, but they don’t see any tangible changes at all, so, given that they concede some sort of “presumption in favor of cautious expansion of migration as long as there are no visible drastic damages” they are happy to let it expand. Or perhaps the people aren’t even that aware of the exact state of things, but their representatives in government are willing to let migration gradually expand barring any unexpected developments.
• Slippery slope in people’s perception of the forms of migration that morality or justice require should not be forbidden: Each step in the slippery slope changes people’s perception of the moral necessity of open borders. For instance, people may go from “it’s unjust to forbid somebody from migrating if they have family in the country or are fleeing persecution” to “it’s unjust to forbid somebody from migrating if they are going to be a net fiscal plus for the nation” to “it’s unjust to forbid somebody from migrating if they’re honest and hardworking” to “it’s unjust to forbid somebody from migrating unless it can be demonstrated that they pose a significant threat to the life and liberty of people in the country.”
• Increasing share of the decision-making population that has a personal positive connection with migration: Migrants tend to be somewhat more pro-migration than natives (see the quote below), so increasing the proportion of migrants would increase the proportion of residents with pro-migration views. This may not have much effect alone, because migrants may be ineligible for direct participation in political decision-making, but they could influence the views of others they interact with. Natives who interact with migrants on a personal level may also become more pro-migration simply because of the “human face” of the issue, even if the migrants themselves don’t try to convince them. The children of migrants are also somewhat more pro-migration than natives. A caveat: the support for further migration may be limited to the migrant’s ethnic, racial, or religious group or socio-economic class.

The following quote from Bryan Caplan’s Cato Journal piece is relevant to the assertion made above about migrants having more pro-migration views:

Finally, there is at least one issue where immigrants are sharply more libertarian than natives: immigration itself. Materially, recent immigrants have the most to lose from additional immigration. Ottaviano and Peri (2008: 59) estimate that immigration from 1990–2006 depressed foreign-born workers’ wages by over 7 percent. But immigrants, like human beings generally, do not derive their political philosophies from material self-interest (Mansbridge 1990). The General Social Survey asks respondents to put their views on immigration on a 1–5 scale, with 5 being most hostile. People with two native-born parents have an average response of 3.9, with a
median of 4; people with at least one foreign-born parent have an average response of 3.1, with a median of 3. By way of comparison, people who call themselves “extremely liberal” have an average response of 3.3—versus 4.0 for the “extremely conservative.” People with foreign-born parents rarely favor open borders, but economists
and libertarians aside, no one is less opposed to immigration.

Reasons why we might expect slipping to not happen

Slipping is not an inevitability of nature. Without concerted effort, it may not happen. Some reasons:

• It is very difficult to keep making changes steadily. In general, we should expect change in fits and starts.
• As the proportion of migrants in the population reaches historically unprecedented levels, migration becomes a lot more visible as a phenomenon. Even with a moderate level of anti-foreign bias, people are likely to blame migration for any problems that arise. Also, appear to historical analogy fails to be a powerful argument for further expansion. The possibility of nativist backlash remains strong, and recent pushback to migration in the UK and Australia might be evidence of some natural limits in the amount of migration people will naturally tolerate given their current beliefs about how the world works.

Braking mechanisms

Braking mechanisms could matter for two reasons. First, we may be genuinely unsure of the extent to which open borders would be a blessing, even if we think it’s quite likely to be one. In this case, having a slope that slips by default, but that can be braked if things are turning out badly would be helpful. Just how much importance we place on a braking mechanism depends on what level of badness we deem sufficient to halt further progress. That level could vary between hardcore open borders advocates and people whose support for free migration is highly conditional to specific consequences.

Second, even if we (as open borders advocates) don’t believe that the braking mechanism would be necessary, others who are more skeptical of the change might be reassured by the presence of a braking mechanism. The braking mechanism would be like the chain passengers can pull for emergency-braking in some trains: it is rarely pulled, but its presence adds to the (perception of) safety (though the analogy also highlights that the braking mechanism may end up getting used for spurious reasons, just as miscreants misuse emergency brakes to get down wherever it’s convenient for them).

There could be different senses of braking and reversal:

• Outcome reversal: Additional migrants who have come in through the expansion so far are deported, bringing the situation as close as possible to how it would have been if the expansion had never occurred. Note: This isn’t intended to endorse such “outcome reversal” as either moral or even practically feasible; see the discussion below.
• Policy reversal: The migration policy for new migrants reverts to what it was before we started slipping the slope, but people who already migrated under the temporarily more liberal migration policy are not forced to leave. A slight variant of policy reversal, that I call punitive policy reversal, would be where policy is not just reverted, but made even more strict than it was in the past until such time as we effectively have an outcome reversal. For instance, if quotas are doubled for the next three years, and people then want to reverse the effects, quotas may be set at zero for the three years after that, and then returned to current levels. Note that punitive policy reversal can accomplish outcome reversal only in cases where the original quota was nonzero, and is best suited to numbers-based increases rather than the addition of totally new categories of migration.
• Policy freeze: The policy remains stuck at the new, expanded level, but further expansion becomes difficult.

(For the mathematically inclined, each sense references the derivative of the preceding one with respect to time).

(For an example of confusion that arose because of people in a conversation talking of reversibility in different senses, see the comments starting here).

Outcome reversal is nearly always difficult and often impossible to accomplish directly. Even if it were possible to force all the additional migrants to leave, the level of (visible) coercion and disruption necessary could give pause even to people highly critical of the expanded migration policy. Even those who wish to accomplish outcome reversal may end up suggesting the route of punitive policy reversal. Policy reversal is somewhat easier to accomplish if people feel the expansion has not worked well. Policy freeze, where further liberalization of migration is blocked, might be the easiest in so far as status quo bias argues in favor of it.

1. Gradual expansion of free migration zones can be stopped by the participating countries, even though it would be relatively hard to undo an existing free migration zone. Outcome reversal is near-impossible, policy reversal is very difficult, but policy freeze is relatively easy.
2. Gradual lowering of admissions standards can be stopped, and even reversed, if the results are deemed bad. Outcome reversal is near-impossible, policy reversal is of moderate difficulty (less so than with free migration zones), and policy freeze is relatively easy.
3. Gradual increases in quotas can be stopped, and even reversed. Outcome reversal is near-impossible, policy reversal is of moderate difficulty (perhaps a little easier than with admissions standards, because “how to get in” is anyway a sufficient mystery that people don’t feel they are being deprived of something they were definitively promised), and policy freeze is relatively easy. Note that punitive policy reversal can be accomplished in principle in cases where the original quota was substantially greater than zero anyway, and would be easier than outcome reversal through deportation, but would still be pretty difficult.
4. Family reunification would be difficult to undo. Outcome reversal would be near-impossible, policy reversal would be fairly difficult (though not impossible, and it might be easier to partially reverse at later stages), and policy freeze would be relatively easy.
5. For tariff/tax schemes, a continued reduction in the tariff or tax rate will happen only if there is sufficiently high revenue (so that it’s a quantitatively significant source of revenue, making revenue considerations more important relative to other considerations) plus strong evidence to suggest that lowering the tax or tariff rate would attract even more revenue by increasing the “customer” base considerably. Here, outcome reversal is near-impossible, while policy reversal and policy freeze are relatively easy.

The type of braking mechanism available for the policy might be a consideration in the choice of slippery slope. If you’re fairly confident that the braking mechanism is more likely to be abused than used out of necessity, then it makes sense to choose a slippery slope where braking is harder. If, on the other hand, you anticipate that the initial stages could reveal information that could lead to the need to brake, it makes sense to choose a slippery slope where braking is easier.

The utility of different slippery slopes in relation to estimates of the global impact of open borders

In a recent post, I argued based on Carl Shulman’s post that, since existing estimates of the global economic impact of open borders rely on more people moving than may be realistic, estimates may have to be adjusted downward to 20-60%. Whether this affects the overall utility of a particular partial move towards open borders depends on whether that move is targeted at increasing numerical caps on how many people may move, or at removing other policy barriers to movement (whose effect in terms of the number of people that move at the margin might change once we correct our estimates). In the Drake equation language, we are interested in the product $latex Wxy$, and the downward adjustment of $latex W$ might be accompanied by different compensating adjustments for $latex y$ depending on the sort of measure. The sensitivity of expected utility to uncertain estimates of the global economic impact of open borders might be a consideration in comparing different slippery slopes.

Slippery slopes versus keyhole solutions

On this site, we’ve discussed keyhole solutions a lot. Some people might use “keyhole solutions” and “slippery slopes” interchangeably, but there is a difference:

• A slippery slope involves a series of steps with each step liberalizing migration relative to the previous one, and with the end goal coming reasonably close to open borders. It represents a means or path.
• A keyhole solution is a (typically policy-based) modification of pure open borders that addresses a significant real or perceived problem with open borders while disrupting open borders as little as possible. It represents an end goal.

Slippery slopes and keyhole solutions could be related: the final stages of a slippery slope might resemble “open borders with a keyhole solution” and the initial stages might also be so described if “keyhole” is interpreted very generously. Also, some keyhole solutions with adjustable parameters (such as the surtax and mandatory savings parameters in DRITI) may lend themselves to a slippery slope: begin with highly restrictionist values of the parameters (so restrictionist that it would be a stretch to call it a “keyhole solution”), then gradually move the parameters in the direction of open borders.

PS: Also relevant is our moderate versus radical open borders page and Joel Newman’s blog post If Open Borders Are Instituted Gradually, What Should Be The Initial Number of Immigrants Admitted?
Thanks to Carl Shulman for helpful comments on a draft.

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.