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Equation 11 · Part 10 · The Unit Economics of a Token

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Tin(k)=∑j=1k(b+ja)=kb+a k(k+1)2T_{\mathrm{in}}(k) = \sum_{j=1}^{k} \left( b + j a \right) = k b + a \, \frac{k (k+1)}{2}
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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.

The passage around this formula

The nonlinearity that surprises people does not live in this equation, which is linear. It lives in agent loops. Suppose a loop runs for k steps, each step appends roughly a tokens of tool output and assistant text, and the context started at b tokens. The total input billed across the whole loop is then Tin(k)=∑j=1k(b+ja)=kb+a k(k+1)2T_{\mathrm{in}}(k) = \sum_{j=1}^{k} \left( b + j a \right) = k b + a \, \frac{k (k+1)}{2}. which is quadratic in the number of steps. Doubling the length of an agent trajectory roughly quadruples its input billing. This is why cache hit rate dominates agentic economics: prefix caching converts most of that quadratic term into reads at one tenth the price, and a workload that loses its cache to a time-to-live expiry or an effort change pays the full quadratic […

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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.

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Sources cited in the article section

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