You also need money, materials, and space to build housing though and I doubt all immigrants are carpenters, electricians, plumbers, and all the other professionals needed to build homes.
in my country that's exactly what is happening, they are taking the simpler jobs for much cheaper, and lot of our "native" people has/had jobs like this.
ironically, this country is among the loudest in anti-immigration in the EU, all the while they are immigrating people from neighboring countries exactly for cheap labor.