Dario Amodei has said that their models actually have a good return, even when accounting for training costs [0]. They lose money because of R&D, training the next bigger models, and I assume also investment in other areas like data centers.

Sam Altman has made similar statements, and Chinese companies also often serve their models very cheaply. All of this makes me believe them when they say they are profitable on API usage. Usage on the plans is a bit more unknown.

[0] https://youtu.be/GcqQ1ebBqkc?si=Vs2R4taIhj3uwIyj&t=1088

Their whole company has to be profitable, or at least not run out of money/investors. If you have no cash you can't just point to one part of your business as being profitable, given that it will quickly become hopelessly out-of-date when other models overtake it.