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Y=(∫01y(s)σ−1σ ds)σσ−1Y = \left( \int_{0}^{1} y(s)^{\frac{\sigma-1}{\sigma}} \, ds \right)^{\frac{\sigma}{\sigma-1}}

Why this formula appears here

Let total output Y be produced by combining a continuum of specialized tasks s ∈\in [0, 1] according to a constant-elasticity-of-substitution production function: Y=(∫01y(s)σ−1σ ds)σσ−1Y = \left( \int_{0}^{1} y(s)^{\frac{\sigma-1}{\sigma}} \, ds \right)^{\frac{\sigma}{\sigma-1}}. where each task y(s) can be performed by labor or by an automated system if that task falls within the feasible automated set A\mathcal{A} . Under Hulten’s theorem, the first-order aggregate TFP cost savings from automating a subset of tasks is bounded by the expenditure share of the automated tasks multiplied by the average unit cost reduction:

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Ending index or upper bound: 1

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Published contexts (1)

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Y=(∫01y(s)σ−1σ ds)σσ−1,Y = \left( \int_{0}^{1} y(s)^{\frac{\sigma-1}{\sigma}} \, ds \right)^{\frac{\sigma}{\sigma-1}},

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

Let total output Y be produced by combining a continuum of specialized tasks s ∈\in [0, 1] according to a constant-elasticity-of-substitution production function: Y=(∫01y(s)σ−1σ ds)σσ−1Y = \left( \int_{0}^{1} y(s)^{\frac{\sigma-1}{\sigma}} \, ds \right)^{\frac{\sigma}{\sigma-1}}. where each task y(s) can be performed by labor or by an automated system if that task falls within the feasible automated set A\mathcal{A} . Under Hulten’s theorem, the first-order aggregate TFP cost savings from automating a subset of tasks is bounded by the expenditure share of the automated tasks multiplied by the average unit cost reduction:

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