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

What does this equation mean?

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

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Start withCov_s(w_i, z_i)

Divide bybar w

This relates toΔ bar z

How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

Read it piece by piece

\Delta

Symbol Δ

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

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\bar z

Symbol bar z

bar z is computed from the expected values combined on the right.

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s

Symbol s

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

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w_i

Symbol w_i

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

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z_i

Symbol z_i

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

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\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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\mathbb{E}_s

Symbol E_s

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

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\Delta z_i

Symbol Δ z_i

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

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=

=

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

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fraction

fraction

Divide the expression above the line by the one below it.

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addition

addition

Add the term after the plus sign to the term or group before it.

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change

change

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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\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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\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.

What the article says around this equation

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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Sources cited in the article section

These citations give research context. Read each source to check which claims it supports.

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