Capitalism here means getting the highest price per inmate, and getting the most inmates you can.

Efficiencies of scale are profits, not cost savings

Prison is not a free market and is not subject to market efficiencies. The purpose is to extract the maximum possible value from human suffering. They do this by charging them obscene prices for every basic necessity, double charging the government for their needs, renting them out as slave labor (mandatory work at wages of a couple of dollars per day), and then lobbying fiercely to keep their position. The private prisons want it to be more costly, beacuse they have monopoly contracts and guaranteed sales.

The price per prisoner isn't the relevant metric. Total price to society, from the cost of maintaining and manning prisons, from the cost of lost human connections with those imprisoned, and from lost productivity from those imprisoned, is a relevant metric. A prison system with a 36% 2-year recidivism rate (United States) is a failure when compared to a prison system with only 20% 2-year recidivism rate (Norway), no matter how cheaply it may be managed.

The goal of a functioning prison system is to manage itself out of existence.

Prisoners, and taxpayers, do not benefit from economies of scale.