What does this equation mean?

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 ] : $\frac{d\bar z}{dt}\Big|_{\text{selection}} = \beta \, \operatorname{Var}_s(z)$. The selection component of the growth rate of mean log productivity equals a selection-intensity coefficient $\beta$ 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 economic change into selection and innovation effects in 2004 [ 5 ] . What the paper adds is a discipline on $\beta$ : because realized market selection acts on profitability rather than on physical output alone, per Foster, Haltiwanger, and Syverson’s finding that price and physical productivity pull in opposite directions across surviving plants [ 14 ] , $\beta$ is never a constant of nature. It must be estimated market by market, and any estimate built on revenue-based productivity will confound genuine efficiency with market power. A third result, Proposition 3, iterates the same identity over nested levels of the economy and reads Hsieh and Klenow’s cross-country misallocation measurements [ 10 ] as a statement about which level is doing the selecting and which level has its selection suppressed — misallocation, on this reading, is not a residual but selection a higher level of aggregation is failing to let through. The persistence that scales $\beta$ into a real forecast is itself already measured: regressing a firm’s productivity on its own lagged value yields autoregressive coefficients on the order of 0.6 to 0.8 in the plant- and firm-level literature [ 9 ] , the direct economic analogue of the narrow-sense heritability that scales Fisher’s original biological theorem.