$$C_i \;=\; \kappa_i\Big[(1-h_i)\,p^{\text{in}}_i + h_i\, p^{\text{in}}_i\,\gamma_i\Big] \;+\; \rho_i\, p^{\text{out}}_i$$

Equation 10 · Model Evaluation

Comparing Frontier Model Pricing Without Comparing Apples to Oranges

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Collecting every distinction above into one expression makes explicit what a bare price comparison assumes without saying so. For vendor i , let $p^{\text{in}}_i$ and $p^{\text{out}}_i$ be the published input and output rates; let $\kappa_i$ be a tokenizer expansion factor, the number of tokens vendor i ’s own tokenizer needs to encode one fixed reference passage, normalised so $\kappa_i$ = 1 for whichever vendor is used as the baseline; let $h_i$ be the fraction of input served from cache and $\gamma_i$ the cache-read multiplier; and let $\rho_i$ be the ratio of total output tokens generated, visible answer plus hidden reasoning, to the visible answer alone. A workload’s realised cost per completed task…

Meanings in this article

• $h_i$: the fraction of input served from cache and $\gamma_i$ the cache-read multiplier.
• $p^{\text{in}}_i$: the reading.
• $\gamma_i$: the cache-read multiplier.
• $\rho_i$: all equal to each other and to 1 across every vendor in the comparison.

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