What does this equation mean?

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 grok-4.6 that means a 199,999-token prompt is billed at two dollars per million input tokens, and a 200,001-token prompt — one token longer — is billed at four dollars per million input tokens and twelve dollars per million output tokens for the whole request, not a blended rate [ 1 ] . A retrieval-augmented pipeline that occasionally pads a prompt with one extra chunk of context can cross that line without any change in what the request is actually asking for, and the bill for that request roughly doubles. This is a discontinuity, not a slope, and it is the kind of detail a cost estimate built from a single “price per million tokens” figure will get wrong in exactly the requests that matter most, because long-context requests are disproportionately the expensive ones to mis-price.