I find discussions abt immigrants "taking out more than they put in" rather odd. In the UK we run a budget deficit almost every year. So the average Briton "takes out more than she or he puts in". No-one thinks that's some kind of individual moral failing on all our parts, do we?
In a typical Western European economy, an immigrant initially has a net withdrawal from the tax & benefit system that is a bit less than the net withdrawal of the average domestic citizen, because of their age when arriving. As they get older things even up.
The main ways immigrants affect tax-benefits isn't "benefits shopping". One way is that they increase inequality, partly by bidding down wages (esp at the low end) & partly cos their own wages are a little less than the avg domestic citizen's.
However, the *main* way immigrants increase inequality is neither of these (not abt bidding down wages). It's that they increase returns to capital by more than wages (indeed, wages may even fall). Immigrant labour boosts the relative wealth of already-wealthy domestic citizens.
This inequality effect increases pressure for greater social transfers (more benefits & public services spending). That then goes on both poorer domestic citizens & immigrants. (It isn't that the immigrants take more.)
These effects are more than offset in some economies (eg Italy, Germany, I suspect - though I haven't actually calculated it for them) when there is domestic population decline & high legacy debts, cos absent immigration the servicing cost per person on old debts wld rise.
