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Lecture 16-2005 - Shephards Duality Proof Part I Lecture...

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Shephard’s Duality Proof: Part I Lecture XVI I. Definition of the Distance Function A. As a starting point of the discussion of the distance function, Shephard defines the space of possible inputs over the nonnegative domain . Then the space of all possible input bundles is segmented into a sequence of regions: D 1. The origin: { } 0 . 2. Interior points of : D { } 1 0 D x x = > . 3. The boundary points of excluding the origin: D 2 1 0, 0 n i i D x x x = = = 2 D is further subdivided into two regions a. ( ) ( ) { } 2 2 , for some 0, D x x D x L u u λ λ Φ ′ = > > 0 b. ( ) ( ) { } 2 2 for some 0, 0 D x x D x L u u λ λ Φ ′′ = > > B. The last segregation segregates into those points that are on a level set ( 2 D ( ) L u Φ ) and those points not on a level set. 1. Looking at the definition if 2 1 0, 0 n i i x D x x = = at least one of the i x must be equal to 0. Thus, x λ such that ( ) 0, λ represents all possible s x such that one of the i x is equal to zero.
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