Generally they are sold for a loss to begin with early in the generation. But later iterations usually make money. As a whole, successful consoles, usually can make money for the company producing them. That said both PS3 and Xbox 360 lost money for their respective companies. I believe the Wii and the PS2 however were great financial successes though.

Up until the WiiU, Nintendo had sold, if memory serves, every console at a profit.

The issue here is that it's being bought without games. They lose money on the tech, and then don't make it up on the licensing.

Wii, and in general Nintendo consoles are not sold at a loss.

Iirc the Wii sold at a profit because it deliberately used older technology.

Weren't schools buying Playstations for a bit because they were cheaper than computers with similar specs?

I swear that was a thing for a while, but I can't find any sources, so I might be mis-remembering

AFAIK, all of the recent consoles have been profit making on just the hardware alone. Not very much money per unit at first, but none of them shipped as a loss leader.

Wasn't nintendo making money on the Wii pretty much the entire time?