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Published equation contexts

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}}

Why this formula appears here

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.

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Δ\Delta

Symbol Δ

Δ appears in the bound of this integral. The bound states where the repeated operation starts, ends, or which values it includes.

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ΔI\Delta I

Symbol Δ I

Δ I appears in the bound of this integral. The bound states where the repeated operation starts, ends, or which values it includes.

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NN

Symbol N

the upper edge of the task space where new tasks appear, and η\eta scales the induced demand from higher output.

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ΔN\Delta N

Symbol Δ N

Δ N appears in the bound of this integral. The bound states where the repeated operation starts, ends, or which values it includes.

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II

Starting index or lower bound: I

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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I+ΔII+\Delta I

Ending index or upper bound: I+Δ I

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

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NN

Starting index or lower bound: N

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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N+ΔNN+\Delta N

Ending index or upper bound: N+Δ N

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

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Published contexts (1)

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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}}

Equation 5 · Labour & Technology

What the Machine Breakers Actually Wanted

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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.

Meanings in this article

  • II: the automation frontier below which capital performs the task.
  • ss: the value-added share of task x.
  • NN: the upper edge of the task space where new tasks appear, and η\eta scales the induced demand from higher output.
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