The question of predicting what will happen under open borders involves trying to make predictions both about legal (authorized) migration flows and about illegal (unauthorized) migration flows. In this post, I’ll be discussing immigration mainly to attractive immigrant destinations such as the US. Also, for simplicity, I will gloss over the distinction between temporary and long-run migration, although your predictions may differ for these two categories.

Open borders advocates, restrictionists, and economic determinists would be led to different predictions by their different theoretical frameworks for understanding migration:

1. The hardline economic determinist holds that migration levels are determined, not by the laws for and against migration, but by economic conditions. According to the hardline economic determinist, then, the total amount of migration under open borders would remain roughly the same as under the status quo, but probably a lot of currently “illegal” migration would become legal. In this view, there is one-to-one substitutability between legal and illegal immigrants. While I doubt that too many people are hardline economic determinists, this is a plausible reading of some of their writings, such as this post by Hein de Haas and this article by Jagdish Bhagwati.
2. Most open borders advocates, as well as moderate economic determinists, hold that under more open borders, the total amount of migration will increase, and the level of illegal immigration will decrease (to near-zero levels — i.e., in the few thousands — under open borders), but the increase in legal immigration levels would more than offset this decline. In other words, legal immigration does substitute for illegal immigration, but it’s not a one-to-one substitutability and total levels of migration are affected by migration policies. For those who take double world GDP estimates as serious, this is the position that is most consistent with such estimates.
3. Some restrictionists, however, have argued that increases in legal immigration could lead to increases in, or at any rate are unlikely to lead to decreases in, illegal immigration. An article by Washington Watcher for VDARE titled Legal Si, Illegal No? The Treason Lobby Says Immigration Is Inevitable So We Should “Relax And Enjoy It” makes this kind of argument (though it is not central to the post). Washington Watcher first concedes that under truly open borders, illegal immigration would drop to near-zero levels, but still argues that for a halfway solution (considerably expanded, but not unlimited, legal immigration) the quantity of illegal immigration would increase. Washington Watcher identifies two relevant phenomena — camouflage (whereby large quantities of legal migration make illegal migrants less conspicuous) and chain migration (where legal migrants are able to assist friends/relatives in migrating illegally, something that would not be possible if the legal migrants weren’t in the destination country in the first place). The relevant paras:
   

   A writer I know once had the chance to ask Griswold just how many visas we should give out. He replied that we should give the same number of additional visas to the total number of illegal aliens and that would satisfy demand.

   But this would not stop illegal immigration—in fact. it would increase it.

   Why? One reason is “Say’s Law” , one of the classic economic doctrines. It states that supply creates its own demand. With an unlimited supply of cheap labor, many jobs that would not exist in an advanced economy exist anyway. For example, in Third World countries, someone making what would be considered a middle class income in America can afford several servants. Because labor is more expensive in the U.Ss, servants only work for the very wealthy. But if labor prices went down, more people would hire servants. For the same reason, cheap labor undercuts the development of labor-saving technological innovations.

   Legal immigration also makes it easier for illegal aliens to live in America without detection. In 1960, 99% of the population outside of the Southwest was White or African American. Were it not for the fact that we admitted legal Braceros, then a farmer with hundreds of Mexican laborers would obviously be hiring illegal aliens. But the legal Braceros allowed for the illegal alien Mexican workers to blend in.

   Most illegal aliens would not even think of coming here to begin with were it not for legal immigration. When a Third World peasant sees that a friend or family member who came here legally can live a relatively extravagant lifestyle, they are going to want to come too—regardless of whether they can get a legal visa.

   Finally, there are a lot of people who have no intention of coming into this country illegally, either because they come from law-abiding societies and respect our laws too and/or they are falsely concerned that they might be deported or be unable to get a job.

Which of these comes closest to reality? I’m inclined to support (2), but then, that’s where my bias lies. I think that the phenomena observed by Washington Watcher in support of (3) are real and genuine phenomena, but I don’t think that these would be quantitatively sufficient to override the effects of (2). Particularly in the context of the United States, considering that the US already has large numbers of people of different ethnicities, camouflage is already almost fully operational, and so the marginal gains in camouflage through additional legal migration are probably not too high.

That being said, the phenomena observed by Washington Watcher might be important in the context of a country that has extremely closed borders, like Japan, to the point that there could be a significant increase in camouflage effects through increases in legal migration. If, for instance, Japan allowed substantially more legal migration from Vietnam, then the camouflage and chain migration effects might lead to more illegal migration from Vietnam as well.