Lecture 03-2005 - Substituting One Input for Another in...

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1 Substituting One Input for Another in Production: Lecture IV I. Elasticity of Scale and Law of Variable Proportions A. We continue to develop the economics of production through a ray from the origin. 1. Last lecture, we developed the notion of returns to scale by looking at changes in production along a ray from the origin. 1 x 2 x 1 x λ 2 x 2. Following the definition of this ray, we defined the elasticity of scale: () 1 ln ln fx ε = = 11 1 ln ln nn i i ii i xy == =   ∂∂ B. Building on these definitions, we next define the ray average product as ( ) f x RAP = where is a strictly positive scalar. C. In addition, we define the ray marginal product ( ) ( ) 1 n i i i f xf x x x λλ = = To derive this result, substitute:
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AEB 6184 – Production Economics Lecture III Professor Charles B. Moss Fall 2005 2 () ( ) ( ) ( ) 1 n i i i fzx fx xz x zx x x λ λλ = →⇒ = ∂∂ = = By extension of this result ( ) ( ) 2 2 11 nn ij ji x x xx == = ∑∑ D. In order to develop the concept behind these equations, we need to take a slice from the multivariate production function: 1 x 2 x y 1 x 2 x 12 , fxx If we focus on the slice of the production function on the ray from the origin, the production function looks like a univariate production function.
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AEB 6184 – Production Economics Lecture III Professor Charles B. Moss Fall 2005 3 y x λ () yf x =
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Lecture 03-2005 - Substituting One Input for Another in...

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