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What it costs is tricky to measure. A large part of the costs are training the model. Once they have the model they are making a ton of profit from what they charge (or so we think - I haven't seen the numbers). However the sunk costs of getting the model need to be paid for and that means an accounting problem where we have to guess how much the model will be used in the future.

Accountants are reasonably good at figuring this out - there are a lot of different things that need a large upfront investment before you can charge anything. People still debate if they are correct in this each case.

Bigger models that Antrophic want to sell cost disproportionately more (e.g. 100% more cost for 5% performance improvement) than small models you would use locally

They have to provide the service at peak scale and high-availability, your local setup doesn't have those extremely expensive requirements.