Why restricting immigrants' access to benefits might not save the Treasury any money

It has been argued that immigrants in the UK ought not to be able to claim unemployment benefits for some period of time after their arrival, or perhaps indefinitely. Some people make this argument by appealing to a notion of fairness: they contend that immigrants should not be able to receive such benefits until they have first paid into the transfer system through their taxes. Others make the argument with reference to projected savings for the Treasury: they contend that immigrants will be deterred from coming to the UK if they know they will not be able to claim such benefits upon arrival. 

Regarding the latter argument, government figures indicate that if all immigrants who are currently receiving unemployment benefits in the UK stopped receiving them tomorrow, and nothing else changed, the amount of money saved by the Treasury would be not be particularly large. Yet even then, it is unlikely that nothing else would change under such circumstances; in fact, changes would likely occur that would render the amount of money saved by the Treasury smaller than the total sum currently being paid to immigrants. 

Many immigrants to the UK originate in countries where wages for low-skilled workers and average standards of living are much lower than in the UK. It stands to reason that even if they were not able to claim unemployment benefits, low-skilled immigrants from such countries would still find it desirable to come to the UK, albeit somewhat less desirable than at present. More importantly however, reducing immigrants' access to unemployment benefits would presumably make them willing to accept jobs at lower wages. Indeed, it might make them willing to accept jobs at wages that some native workers would find less attractive than unemployment benefits.

In other words, one effect of reducing immigrants' access to unemployment benefits could be to bid down wages to the point where some native workers would shift onto unemployment benefits, thereby curtailing any projected savings for the Treasury.