If by the competition you mean Sony, then yes.

Not a chance. Sony couldn't afford to do that. I think many people will be shocked over how small this generation jump is going to be, in terms of raw power.

Sony just puts them on there for free? No sort of deal comes out of it? 

@bolded PS Vita hasn't performed well, and you've got R&D for PS4.

So far I've added some of the profitable factors and they'll still take around a billion in losses unless it's a magical year, so it probably isn't possible.

A $200 loss per console would mean the end of Sony's gaming division. It would be virtually impossible to make the money back. The typical game attach rate is 10 for a console, so if you sold 10 games at a $10 profit each, you would still be at a loss at the end of the console's life. The loss-lead strategy Sony and MS have used is a terrible one. There is a reason Nintendo made more money off GameCube than Sony off PS2...which sounds rediculous to say. Having a big loss on hardware only works if you have an alterior motive - such as pushing bluray or the Windows format - where your corporatation would make extra money elsewhere thanks to support from your games division. Unless Sony has such a secret plan, selling PS4 at a major loss would forever doom the division to be in the red, and probably mean it's end.

Also, selling at the same price as WiiU doesn't mean it will get more sales.

MS has something that lets them recover the money, live gold. Most people don't see the huge cashcow live is, and why they won't cancel it on the next generation.

they can do it and if things go wrong they can sell a building or two

Remember, part of the PS3's cost was tied to the attempt to push Blu-Ray. They were willing to lose more money because they anticipated profits in other segments of the company. A lot of the PS3's losses should be applied to Sony as a whole rather than the gaming division. I don't think there is any such consideration this time.

They need to go with breaking even or with a small loss this time. Where they are as a company makes that very important. They also need to stay aggressive with cost-saving redesigns.

I don't think the rumored specs would make the PS4 all that expensive, so I'm pretty hopeful we'll get a reasonable price without Sony taking an immense financial loss.

proof?

Neither Sony or Microsoft will take huge loses.

Sony are in enough trouble as it is and need to turn around all their divisions and start making profit, losing billions is not in their best interest.

Microsoft are probably sick of the Xbox division losing money and will push to make profit from it.