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Lecture 16-2005

# Lecture 16-2005 - Shephard's Duality Proof Part I Lecture...

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Shephard’s Duality Proof: Part I Lecture XVI

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Definition of the Distance Function As a starting point of the discussion of the distance function, Shephard defines the space of possible inputs over the nonnegative domain D . Then the space of all possible input bundles is segmented into a sequence of regions: The origin: {0} Interior points of D : The boundary points of D excluding the origin: { } 1 0 D x x = 2 1 0, 0 n i i D x x x = = = Υ
D 2 is further subdivided into two regions The last segregation segregates D 2 into those points that are on a level set ( L Φ ( u ) ) and those points not on a level set ( 29 ( 29 { } 2 2 , for some 0, 0 D x x D x L u u λ λ Φ = ( 29 ( 29 { } 2 2 for some 0, 0 D x x D x L u u λ λ Φ =

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Looking at the definition if at least one x i is equal to zero, but production is still possible (i.e., the input is not strongly necessary). Thus, 2 1 0, 0 n i i x D x x = == Υ ( 29 : 0 x L u u λ Φ 220d
The second possibility is that inputs are strictly necessary.

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Lecture 16-2005 - Shephard's Duality Proof Part I Lecture...

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