Equation 1 · The Unit Economics of a Token

What does this equation mean?

c = p_{\mathrm{in}} n + p_{\mathrm{cw}} w + p_{\mathrm{cr}} r + p_{\mathrm{out}} \left( y + z \right)

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Inputs and operationsp_in n + p_cw w + p_cr r + p_out ( y + z )

Result or conditionc

How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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c

Symbol c

c is part of the quantity the equation computes from the expression on the right.

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p_{\mathrm{in}}

Symbol p_in

$p_i$ n is one of the signed contributions combined to compute the quantity on the left.

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n

Symbol n

the five counts in that order and each p is the corresponding published rate.

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p_{\mathrm{cw}}

Symbol p_cw

$p_c$ w is one of the signed contributions combined to compute the quantity on the left.

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w

Symbol w

the five counts in that order and each p is the corresponding published rate.

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p_{\mathrm{cr}}

Symbol p_cr

$p_c$ r is one of the signed contributions combined to compute the quantity on the left.

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r

Symbol r

the five counts in that order and each p is the corresponding published rate.

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p_{\mathrm{out}}

Symbol p_out

$p_o$ ut is one of the signed contributions combined to compute the quantity on the left.

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y

Symbol y

the five counts in that order and each p is the corresponding published rate.

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z

Symbol z

the five counts in that order and each p is the corresponding published rate.

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=

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

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addition

Add the term after the plus sign to the term or group before it.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

Instrument each request to record five quantities: new input tokens, cache-write tokens, cache-read tokens, visible output tokens, and reasoning tokens. The cost of a single request is then $c = p_{\mathrm{in}} n + p_{\mathrm{cw}} w + p_{\mathrm{cr}} r + p_{\mathrm{out}} \left( y + z \right)$ . where n , w , r , y and z are the five counts in that order and each p is the corresponding published rate. The important feature is the last term: visible output and reasoning share one price, and the reasoning count is the only one of the five you cannot recover by inspecting what came back.

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Sources cited in the article section

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