Lecture 24-2005 - Factor Bias, Technical Change, and...

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Factor Bias, Technical Change, and Valuing Research Lecture XXIV
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Mathematical Model of Technical Change If we start from the quadratic production function specified as assuming an output price of p and input prices of w 1 and w 2 for inputs x 1 and x 2 , respectively, the derived demands for each input can be expressed as ( 29 ( 29 1 2 0 1 1 2 2 2 2 11 1 12 1 2 22 2 , 1 2 2 f x x a a x a x A x A x x A x = + + + + +
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( 29 ( 29 ( 29 ( 29 * 12 2 22 1 22 1 12 2 1 1 2 2 11 22 12 * 12 1 11 2 12 1 11 2 2 1 2 2 11 22 12 , , , , A a p A a p A w A w x p w w p A A A A a p A a p A w A w x p w w p A A A - + - = - - - + = -
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In order to analyze the possible effect of technological change, we hypothesize an input augmenting technical change similar to the general form of technological innovation introduced by Hayami and Ruttan.
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Specifically, we introduce two functions where γ 1 (Ψ) and γ 2 (Ψ) are augmentation factors and Ψ is a technological change ( 29 ( 29 * 1 1 1 * 2 2 2 x x x x = γ ψ = γ ψ
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Hence, γ 1 (Ψ), γ 2 (Ψ)≥1 for any Ψ. Thus, technological change increases the output created by each unit of input. Integrating these increases into the forgoing production framework, the derived demands for each input becomes:
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( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 * 1 1 2 12 2 1 2 22 1 1 2 22 2 1 12 1 2 2 2 11 22 12 1 2 * 2 1 2 12 1 1 2 11 2 1 2 12 2 1 22 1 2 2 11 22 12 1 2 , , , , , , x p w w A a p A a p A w A w p A A A x p w w A a p A a p A w A w p A A A ψ = γ ψ γ ψ - γ ψ γ ψ + γ ψ - γ ψ - γ ψ γ ψ ψ = γ ψ γ ψ - γ ψ γ ψ - γ ψ + γ ψ - γ ψ γ ψ
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In order to simplify our discussion, we assume that the new technology does not affect the effectiveness of x 2 , or
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Lecture 24-2005 - Factor Bias, Technical Change, and...

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