Lecture 17-2005 - ShephardsDuality Proof: PartII...

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    Shephard’s Duality  Proof:  Part II Lecture XVII
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    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 ( 29 ( 29 { } 0 0 , : min x u x x L u x λ λ λ Φ Ψ = 220d =
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    The production function can then be defined as In addition, we can define the set of efficient input vectors by  the distance function: ( 29 ( 29 { } max , 1 , x u u x x D Φ = Ψ ( 29 ( 29 { } , 1 E u x u x = Ψ =
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    The function and the cost structure can be defined  based on the cost-minimization problem. The cost function is defined as 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, ( 29 ( 29 { } , min x Q u p p x x L u Φ = ( 29 ( 29 { } , 1, 0 Q u p Q u p p Λ =
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Lecture 17-2005 - ShephardsDuality Proof: PartII...

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