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Published equation contexts

Δzˉ=Cov⁡s(wi,zi)wˉ+Es[wi Δzi]wˉ\Delta \bar z = \frac{\operatorname{Cov}_s(w_i, z_i)}{\bar w} + \frac{\mathbb{E}_s[w_i \, \Delta z_i]}{\bar w}

Why this formula appears here

Price’s identity, applied to firms with market shares sis_i and log-productivity values ziz_i , reads: Δzˉ=Cov⁡s(wi,zi)wˉ+Es[wi Δzi]wˉ\Delta \bar z = \frac{\operatorname{Cov}_s(w_i, z_i)}{\bar w} + \frac{\mathbb{E}_s[w_i \, \Delta z_i]}{\bar w}. The change in mean log productivity splits into a covariance between a firm’s relative fitness — its share-growth factor wiw_i — and its productivity, plus a share-weighted average of each firm’s own productivity change, weighted by how much it grew. The first term is selection: value reallocated toward whoever is already ahead. The second is transmission: value created inside units that already exist. This holds for any partition of any economy in any period. It is exact bookkeeping, not a behavioral model, and it carries no error term to hide behind.

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wˉ\bar w

Symbol bar w

bar w occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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Es\mathbb{E}_s

Symbol E_s

EsE_s occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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Δzi\Delta z_i

Symbol Δ z_i

Δ ziz_i occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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Cov⁡s(wi,zi)\operatorname{Cov}_s(w_i, z_i)

Numerator: Cov_s(w_i, z_i)

The complete quantity above the fraction bar.

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Es[wi Δzi]\mathbb{E}_s[w_i \, \Delta z_i]

Numerator: E_s[w_i Δ z_i]

The complete quantity above the fraction bar.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

Δzˉ=Cov⁡s(wi,zi)wˉ+Es[wi Δzi]wˉ\Delta \bar z = \frac{\operatorname{Cov}_s(w_i, z_i)}{\bar w} + \frac{\mathbb{E}_s[w_i \, \Delta z_i]}{\bar w}

Equation 3 · Evolutionary Economics

Selection Accounting: The Price Equation Runs the Economy

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

Price’s identity, applied to firms with market shares sis_i and log-productivity values ziz_i , reads: Δzˉ=Cov⁡s(wi,zi)wˉ+Es[wi Δzi]wˉ\Delta \bar z = \frac{\operatorname{Cov}_s(w_i, z_i)}{\bar w} + \frac{\mathbb{E}_s[w_i \, \Delta z_i]}{\bar w}. The change in mean log productivity splits into a covariance between a firm’s relative fitness — its share-growth factor wiw_i — and its productivity, plus a share-weighted average of each firm’s own productivity change, weighted by how much it grew. The first term is selection: value reallocated toward whoever is already ahead. The second is transmission: value created inside units that already exist. This holds for any partition of any economy in any period. It is exact bookkeeping, not a behavioral model, and it carries no error term to hide behind.

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