Should minimum wage laws apply to illegal immigrants?

What if the Mexicans want to eat and live somewhere? Its not as though when Mexicans come here they will still be paying the same costs of living in Mexico. The market will set the wages and immigrants looking to better themselves aren't going to deliberately starve themselves.

If you care about jobs for Americans we should be looking for ways to make American workers more productive and competitive rather than erect enforcement barriers or artificial wage rates.