Lecture17-2011

# Lecture17-2011 - Basic Functions Shepard Duality Shephards...

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Basic Functions Shepard Duality Shephard’s Duality Proof - Part II: Lecture XVII Charles B. Moss 1 1 University of Florida October 25, 2011 Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII

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Basic Functions Shepard Duality 1 Basic Functions 2 Shepard Duality Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII
Basic Functions Shepard Duality Basic Functions Following Shephard’s development from the last lecture, we have two basic groups of functions: The distance function, production function, and associated level set. The level set L Φ ( u ) is deﬁned as the set of possible combinations of inputs that can be used to produce the output level u . Given the level set, we can deﬁne a distance function as Ψ ( u , x ) = || x || || λ 0 x 3 : λ 0 = min { λ | λ x L Φ ( u ) } (1) The production function can then be deﬁned as Φ ( x ) = max { u | Ψ ( u , x ) 1 } x D (2) In addition, we can deﬁne the set of eﬃcient input vectors by the distance function E ( u ) = { x | Ψ ( u , x ) = 1 } (3) Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII

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Basic Functions Shepard Duality The function and the cost structure can be deﬁned based on the cost-minimization problem. The cost function is deﬁned as Q ( u , p ) = min x { p 0 x | x L Φ ( u ) } (4) The cost structure is then deﬁned for the set of all possible input prices in a similar way as the level sets of inputs are deﬁned in output space. Speciﬁcally,
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Lecture17-2011 - Basic Functions Shepard Duality Shephards...

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