fraction

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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