← All parts of this equation

Equation 5 · Part 16 · What the Machine Breakers Actually Wanted

superscript

Δln⁡L  =  η Δln⁡Y⏟productivity  −  ∫II+ΔIs(x) dx⏟displacement  +  ∫NN+ΔNs(x) dx⏟reinstatement\Delta \ln L \;=\; \underbrace{\eta \, \Delta \ln Y}_{\text{productivity}} \;-\; \underbrace{\int_{I}^{I+\Delta I} s(x)\,dx}_{\text{displacement}} \;+\; \underbrace{\int_{N}^{N+\Delta N} s(x)\,dx}_{\text{reinstatement}}
superscript

What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

The passage around this formula

Written schematically, the change in labour demand decomposes as Δln⁡L  =  η Δln⁡Y⏟productivity  −  ∫II+ΔIs(x) dx⏟displacement  +  ∫NN+ΔNs(x) dx⏟reinstatement\Delta \ln L \;=\; \underbrace{\eta \, \Delta \ln Y}_{\text{productivity}} \;-\; \underbrace{\int_{I}^{I+\Delta I} s(x)\,dx}_{\text{displacement}} \;+\; \underbrace{\int_{N}^{N+\Delta N} s(x)\,dx}_{\text{reinstatement}}. where tasks are indexed by x , s(x) is the value-added share of task x , I is the automation frontier below which capital performs the task, N is the upper edge of the task space where new tasks appear, and η\eta scales the induced demand from higher output. The expression is illustrative of the structure rather than a reproduction of any published estimating equation; the substantive claim it encodes is that the sign of Δ\Delta ln⁡\ln L is not determined by the size of Δ\Delta I alone.

Read this part in the article →

Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

Open the illustrated exponents: repeated multiplication and powers guide →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.