Equation 17 · Part 3 · Comparing Frontier Model Pricing Without Comparing Apples to Oranges

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p^{\text{in}}_i

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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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The passage around this formula

Reading $p^{\text{in}}_i$ and $p^{\text{out}}_i$ off a rate card and comparing them directly across vendors is equivalent to assuming $\kappa_i$ , $\gamma_i$ , and $\rho_i$ are all equal to each other and to 1 across every vendor in the comparison. This article has just shown that one of those three, $\gamma_i$ , genuinely does converge close to 0.1 across all three companies’ published cards — the one term a naive comparison happens to get right by accident. The other two do not converge, are not published as clean multipliers by any vendor, and vary by content type and by task in ways that only measurement on the caller’s own workload can pin down. A price comparison that reports $p^{\text{in}}_i$ and…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

Open the illustrated exponents: repeated multiplication and powers guide →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.