How Many People Would Want To Immigrate to the U.S.?

I haven't blogged much about immigration issues; it's a difficult empirical question on which I have little expertise, and my sense is that it would take more time than I now have to acquire enough expertise to be able to say something useful. So let me ask this question, stressing that it's just a question and not an answer:

Say that we consider largely removing limits on immigration, as I've heard some people suggest, and as was indeed the law throughout much of the nation's history. (Let's set aside narrow limits, such as on people with criminal records, terrorist connections, or easily communicable diseases.) Say also that we will offer these now largely legal immigrants those social welfare benefits that are in fact politically likely — not the bare minimum that some libertarians might like, nor the vast amount that some welfare-state proponents might suggest, but those benefits that are likely: Public education for their children, emergency health care, whatever nonemergency health care is likely to be broadly offered under either current health subsidy plans or under whatever plans are likely in the future, and the like.

Say also that we maintain the traditional liberties that we have long offered legal immigrants, such as the right to settle wherever they want (which likely means where they can find relatives, people from their home country, and jobs), rather than requiring them to stick to (say) South Dakota and avoid already crowded places like New York City or Southern California. And say that we likewise maintain the various traditional restraints that are in fact unlikely to be politically changed, such as imperfectly enforced minimum wage laws, imperfectly enforced tax laws, imperfectly enforced housing laws, and the like.

Of the world population of 6 billion, how many people are likely to want to come to the U.S., and stay for the indefinite future, under this model? Ten million? One hundred million? More? Less? Is it even possible to estimate this? Relatedly, if one goal of allowing broader immigration would be to avoid the problems caused by illegal immigration, would capping the limit at some number (2 million per year? 5 million per year?) dramatically decrease illegal immigration by offering the prospect of legal immigration to those who stand in line? Or would it increase illegal immigration, as people see more of their acquaintances and family members living in the U.S., and thus conclude that moving here illegally would be easier and less painful than it once was?