I think we're agreeing with each other. There's no pure, efficient, "labor market" but instead lots of small, loosely connected "labor markets" with a variety

I think we're agreeing with each other. There's no pure, efficient, "labor market" but instead lots of small, loosely connected "labor markets" with a variety of inefficiencies - lock-in, friction, imperfect information. An employer can exploit illegal immigrant labor because it tends to be concentrated in lower-skilled, relatively interchangeable jobs, and because illegal immigrants are in a difficult spot.