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

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Factor Bias, Technical Change, and Valuing Research Lecture XXIV I. Mathematical Model of Technical Change A. If we start from the quadratic production function specified as () ( ) 22 1 2 0 1 1 2 2 11 1 12 1 2 22 2 1 ,2 2 fxx a ax ax Ax Axx Ax =+ + + + + assuming an output price of p and input prices of and for inputs 1 w 2 w 1 x and 2 x , respectively, the derived demands for each input can be expressed as * 12 2 22 1 22 1 12 2 11 2 2 11 22 12 * 12 1 11 2 12 1 11 2 21 2 2 11 22 12 ,, Aap Aap Aw Aw xpww pAA A A +− = −− + = B. 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. 1. Specifically, we introduce two functions ( ) * * 1 2 x x x x =γ ψ where and 1 γψ ( ) 2 are augmentation factors and is a technological change. ψ 2. Hence, ( ) 12 , γψγψ≥ 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: ( ) ( ) ( ) ( )( () () ) () () () () () () ( ) () () 12 2 1 2 22 1 1 2 22 2 1 12 1 2 * 2 11 22 12 1 2 12 1 1 2 11 2 1 2 12 2 1 22 1 2 * 2 2 11 22 12 1 2 ,,, Aa p Aa p A w A w A w A γψγψ − γψγψ + γψ − γψ ψ= −γ ψ γ ψ γψγψ − γψ + γψ ψ γ ψ
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3. In order to simplify our discussion, we assume that the new technology does not affect the effectiveness of 2 x , or ( ) 2 1 γψ→ . Under this assumption the derived demand for each input becomes () ( ) ( )( ) () () () ( ) 12 2 1 22 1 1 22 1 12 1 2 * 11 2 22 11 22 12 1 12 1 1 11 2 1 12 1 22 1 2 * 21 2 11 22 12 1 ,,, Aa p Aa p Aw A w xpww pAA A w A γψ − γψ + − γψ ψ= −γ ψ + γψ ψ 4. In order to examine the effect of the technological change on each derived demand, we take the derivative of each of the demand curves in equation 5 with respect to ψ as ( ) 1 1 yielding ( )() ( ) ( ) ( ) 1 1 * 1 1 2 12 2 22 1 22 1 12 2 1 2 11 22 12 1 12 2 22 1 12 2 1 2 11 22 12 * 2 1 2 1 2 1 1 1 2 1 2 11 22 12 1 12 1 11 2 11 2 1 2 11 22 12 2 A a pAa pAw Aw AA A p Aap Aap Aw A a pA a ∂ψ + γ =− ∂ψ −−γ ψ + + γ ∂ψ −+γ ψ + ψ ψ II.
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Lecture 24-2005 - Factor Bias, Technical Change, and...

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