Equation 11 · Part 6 · The Unit Economics of a Token

=

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

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

Full expression=Article meaning

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 $T_{\mathrm{in}}(k) = \sum_{j=1}^{k} \left( b + j a \right) = k b + a \, \frac{k (k+1)}{2}$ . 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 […

Read this part in the article →

Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

Open the illustrated equality: what the equals sign claims guide →

Sources cited in the article section

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