You can draw a straight line from the 13th Amendment to the private prisons of today. After all, the southern states whose economy was ruined by the Civil War had to get cheap labor somehow, so they arrested all those former slaves for random bullshit like loitering, had them convicted by all-white juries, and sent them to work in chain gangs.

Netflix produced a great documentary about all of this - 13th - and it's well worth the watch.

Most prison labor occurs at publicly owned and operated institutions.

Simply because private prisons hold a small portion of the people incarcerated at a state or federal level.

that's somewhat less interesting than a per capita measure

Feel free to figure it out.

My point was that the use of prison labor is not limited to private prisons.