Equation 3 · Part 9 · Selection Accounting: The Price Equation Runs the Economy

=

\Delta \bar z = \frac{\operatorname{Cov}_s(w_i, z_i)}{\bar w} + \frac{\mathbb{E}_s[w_i \, \Delta z_i]}{\bar w}

What this part means

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

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

Full expression=Article meaning

The passage around this formula

Price’s identity, applied to firms with market shares $s_i$ and log-productivity values $z_i$ , reads: $\Delta \bar z = \frac{\operatorname{Cov}_s(w_i, z_i)}{\bar w} + \frac{\mathbb{E}_s[w_i \, \Delta z_i]}{\bar w}$ . The change in mean log productivity splits into a covariance between a firm’s relative fitness — its share-growth factor $w_i$ — 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

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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Sources cited in the article section

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