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Equation 3 · Part 14 · Selection Accounting: The Price Equation Runs the Economy

Numerator: Cov_s(w_i, z_i)

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

What this part means

The complete quantity above the fraction bar.

Its job in the formula

Covs(wi,zi)v_s(w_i, z_i) occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

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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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the article section

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