well according to this
it costed $471 with the kinect
now since it doesnt have it it cost about $371(I think the kiect is about $100 back then maybe a couple dollars more than that but around that range)
so I believe they are losing $21 on it(not as much as I thought which is good but they are still losing money by doing this)
That was at release though. Surely some of the components the Xbox One uses have dropped in price since then, the blu-ray drive maybe, or the hard drive. But even if they are losing some on the console they surely make it back on accessory sales, the Xbox One controller costs $15 to manufacture, yet sells for $50+. The chat headsets cost less than $10 manufacture yet sell for $25. Plus they make money from game sales and live subs.