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Equation 5 · Part 7 · Selection Accounting: The Price Equation Runs the Economy

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dzˉdt∣selection=β Var⁡s(z)\frac{d\bar z}{dt}\Big|_{\text{selection}} = \beta \, \operatorname{Var}_s(z)
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The expressions on both sides represent the same quantity under the stated assumptions.

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The passage around this formula

Add one falsifiable assumption — that market share evolves under replicator dynamics with a linear selection gradient linking a firm’s productivity advantage to its realized share growth — and Proposition 2 states an economic reading of Fisher’s fundamental theorem of natural selection [ 2 ] : dzˉdt∣selection=β Var⁡s(z)\frac{d\bar z}{dt}\Big|_{\text{selection}} = \beta \, \operatorname{Var}_s(z). The selection component of the growth rate of mean log productivity equals a selection-intensity coefficient β\beta times the share-weighted variance of log productivity across firms. The qualitative move is not new: Metcalfe applied Fisher’s principle to competing firms under replicator dynamics in the 1990s [ 4 ] , and Andersen used Price’s equation itself, by name, to split…

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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.

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