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Written schematically, the change in labour demand decomposes as $\Delta \ln L \;=\; \underbrace{\eta \, \Delta \ln Y}_{\text{productivity}} \;-\; \underbrace{\int_{I}^{I+\Delta I} s(x)\,dx}_{\text{displacement}} \;+\; \underbrace{\int_{N}^{N+\Delta N} s(x)\,dx}_{\text{reinstatement}}$. 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 $\eta$ 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 $\Delta$ $\ln$ L is not determined by the size of $\Delta$ I alone.

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