Equation 9 · Comparing Frontier Model Pricing Without Comparing Apples to Oranges

What does this equation mean?

\rho_i

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

all equal to each other and to 1 across every vendor in the comparison. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

Read it piece by piece

\rho_i

Symbol rho_i

all equal to each other and to 1 across every vendor in the comparison.

Understand this part →

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Understand this part →

How to interpret it

Read this expression with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

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…

Read the full surrounding passage

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 is then approximately

Read the equation in its article →

Sources cited in the article section

These citations give research context. Read each source to check which claims it supports.

Return to Comparing Frontier Model Pricing Without Comparing Apples to Oranges

Browse the mathematical compendium →