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R∗=w−r1−rR^{*} = \frac{w - r}{1 - r}

Why this formula appears here

That convergence on roughly one tenth of base input for a cache read — independently verified today across all three platforms [ 2 , 6 , 10 ] — supports a design rule. Let w be the write-price multiplier for the chosen retention window and r ≈\approx 0.1 the read-price multiplier. For a prefix served identically R times within that window — the first serving is the write, the remaining R-1 are reads — caching beats never caching once R∗=w−r1−rR^{*} = \frac{w - r}{1 - r}. is exceeded. At Anthropic’s five-minute rate, w = 1.25 and R∗R^{*} ≈\approx 1.28 : the prefix pays for itself starting with the first read after the write, matching the break-even Anthropic’s pricing page states for that duration [ 6 ] . At the…

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R∗=w−r1−rR^{*} = \frac{w - r}{1 - r}

Equation 5 · Inference Economics

AI Inference Economics in Practice: An Advanced Technical Guide

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

That convergence on roughly one tenth of base input for a cache read — independently verified today across all three platforms [ 2 , 6 , 10 ] — supports a design rule. Let w be the write-price multiplier for the chosen retention window and r ≈\approx 0.1 the read-price multiplier. For a prefix served identically R times within that window — the first serving is the write, the remaining R-1 are reads — caching beats never caching once R∗=w−r1−rR^{*} = \frac{w - r}{1 - r}. is exceeded. At Anthropic’s five-minute rate, w = 1.25 and R∗R^{*} ≈\approx 1.28 : the prefix pays for itself starting with the first read after the write, matching the break-even Anthropic’s pricing page states for that duration [ 6 ] . At the…

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