What does this equation mean?

Even a comparison that correctly measures $C_i$ is not yet a capability comparison, which is the deeper reason a “cheapest model” headline usually asks the wrong question. Artificial Analysis, an independent benchmarking organisation, maintains a live Intelligence Index alongside pricing across several hundred models from every major vendor, and explicitly plots “Intelligence Index vs. Cost per Task” as a quadrant chart rather than a single ranked list, with a stated methodology that derives “weighted average cost per Intelligence Index task” from “input, cache hit, cache write, reasoning, and answer token prices, divided by task count, and weighted by its Intelligence Index weight” [ 8 ] — which is to say, a capability-adjusted cost figure of exactly the kind the equation above gestures toward, built by someone who actually ran the tasks rather than read the rate card. Epoch AI’s separate analysis of inference price trends makes the same point from the other direction: the price of reaching a fixed capability bar — GPT-4-level performance on PhD-level science questions, specifically — fell by a documented factor of roughly 40 per year, with the overall range across different capability bars running from roughly 9x to 900x per year depending on which bar is chosen, while the authors caution explicitly that “the fastest price drops in that range have occurred in the past year, so it’s less clear that those will persist” [ 7 ] . A decline that large and that uneven overwhelmingly reflects newer, differently designed models reaching a fixed bar more cheaply — substitution between architectures, not a like-for-like discount on one model’s own price.