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Lecture 17-2005

# Lecture 17-2005 - Shephards Duality Proof Part II Lecture...

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Shephard’s Duality Proof: Part II Lecture XVII I. Basic Functions A. Following Shephard’s development from the last lecture, we have two basic groups of functions: 1. The distance function, production function, and associated level set. a. The level set ( ) Lu Φ is defined as the set of possible combinations of inputs that can be used to produce the output level u . b. Given the level set, we can define a distance function as () {} 0 0 ,: m i n x ux x L u x λλ λ Φ Ψ=∋ = c. The production function can then be defined as ( ) { } max , 1 , x uu x x Φ= Ψ D d. In addition, we can define the set of efficient input vectors by the distance function: ( ) ( ) { } ,1 Eu x = Ψ= 2. The function and the cost structure can be defined based on the cost- minimization problem. a. The cost function is defined as ( ) { } ,m i n x Qup pxx L u Φ =∈ b. 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, ( ) ( ) { } , 0 Q up Q u p p Λ= With the equality ( ) ,

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Lecture 17-2005 - Shephards Duality Proof Part II Lecture...

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