The wage analogy only works to a certain extent. 

Certain sectors of employment purposely keep wages low, and recruiting efforts minimal,  so that they can then lobby government for the importation of labour. Why did my local urban Tim Hortons and A+W bring in a TFW labour force, when the competition at Starbucks, Waves, Blenz et al, managed to easily find local labour? 

I do agree that America has become reliant on a large shadow workforce, and I don't know how much an increase in wage will attract America's citizens and new immigrants to, say agricultural labour. Perhaps, after giving the market  wage dynamic a go,   the solution lies bringing in a legal temporary workforce.