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The passage around this formula

The pricing structure has a feature worth building a mental model around before it costs you money in production: it is not a marginal rate that blends smoothly as a request grows, it is a cliff. Every model’s documented price list carries two tiers — one for requests under 200,000 tokens, and a materially higher one, roughly double across the board, once a request reaches that threshold — and the higher rate applies to the entire request, not just the tokens past the line [ 1 ] . Written as a cost function of input tokens $n_i$ and output tokens $n_o$ against a threshold T = 200{,}000 , with low-tier prices $p_i^{L}$, $p_o^{L}$ and high-tier prices $p_i^{H}$, $p_o^{H}$ : $\text{Cost}(n_i, n_o) = \begin{cases} n_i\, p_i^{L} + n_o\, p_o^{L}, & n_i < T \\ n_i\, p_i^{H} + n_o\, p_o^{H}, & n_i \geq T \end{cases}$. For…

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