Lecture17-2011 - Basic Functions Shepard Duality...

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Unformatted text preview: Basic Functions Shepard Duality Shephard’s Duality Proof - Part II: Lecture XVII Charles B. Moss1 1 University of Florida October 25, 2011 Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII 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 LΦ (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 ||x | | ￿: λ0 = min { λ |λx ∈ LΦ (u ) } ||λ0 x | | The production function can then be defined as Ψ (u , x ) = (1) Φ (x ) = max { u |Ψ (u , x ) ≥ 1 } x ∈ D (2) E ( u ) = { x |Ψ ( u , x ) = 1 } (3) In addition, we can define the set of efficient input vectors by the distance function Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII 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 { p ￿ x |x ∈ LΦ (u ) } x (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, Λ Q ( u ) = { p | Q ( u , p ) ≥ 1, p ≥ 0 } (5) With the equality Q (u , p ) = 1 being established by the normalization of p ||p0 | | (6) By semmetry, we can define a set of efficient prices for E (u ). Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII Basic Functions Shepard Duality Shephard Duality The claim of Shephard duality is then ￿ ￿ Q (u , p ) = min p ￿ x |Ψ (u , x ) ≥ 1 , u ≥ 0, p ≥ 0 x￿ ￿ Ψ (u , x ) = inf p ￿ x |Q (u , p ) ≥ 1 , u ≥ 0, x ≥ 0 (7) p Geometric Relationship between Dual Cost and Production Structures “There is a rather simple and elegant geometric relationship between the isoquants of the sets LΦ (u ) of the production structure and the isoquants of the sets ΛQ (u ) of the cost structure, for any positive output rate u .” The proof of the correspondence involves showing the relationship between the efficient sets E (u ) defined from the distance function and E (u ) defined from the cost function. Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII Basic Functions Shepard Duality x0 x0 p x0 p0 u, x 0 u 0 Q u, p Eu 0 u, x p 0 x Q u, p 0 p 0 Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII Basic Functions Shepard Duality That is that a price vector p 0 and a level of output u determine the level of input use x0 on the cost surface ˆ ￿ ￿ 0. Q u, p Put differently ￿ ￿ x0 = Q u , p 0 ˆ (8) This point is not necessarily unique. Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII Basic Functions Shepard Duality Based on this relationship, we define the raty p0 = ˆ Working backwards Q ( u , p0 ) = Q ˆ p0 ￿ Q u, p0 ￿ ￿ p0 ￿ u, ￿ Q u, p0 (9) ￿ =1 (10) p0 ∈ ΛQ (u ) since the cost function is the distance function ˆ for set } Q (u )} . Further, ￿ 0￿ ￿| ￿ ￿ 0￿ ￿|p ￿ | = ￿p | ￿ ˆ (11) Q u, p0 Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII Basic Functions Shepard Duality x0 x0 p x0 p0 u, x 0 u 0 Q u, p Eu 0 u, x p 0 x Q u, p 0 p 0 Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII Basic Functions Shepard Duality Letting η0 be the intersection between the cost function at ￿ ￿ ￿ ￿ u , p 0 and the ray θp 0 |θ ≥ 0 ￿ ￿ ￿ 0￿ ￿ 0￿ ￿ ￿ Q u, p0 0 ￿| p ￿ | ￿ | θ p ￿ | = Q u , p ⇒ θ = ￿ ￿ ˆ ￿| p 0 ￿ | 2 ￿ ￿ 0 (12) Q u, p 1 ⇒ ||η0 | | = ￿ 0 ￿ = ￿| p ￿ | ||p0 | | ˆ ￿ ￿ 0￿ 0 p x = Q u, p Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII Basic Functions Shepard Duality x0 x0 p x0 p0 u, x 0 u 0 Q u, p Eu 0 u, x p 0 x Q u, p 0 p 0 Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII Basic Functions Shepard Duality Working from the other side, we define support of ΛQ (u ) as ￿ ￿ x 0p = Ψ u, x 0 In this direction, p0 is the contact￿point￿ between the ray ˆ defined by x 0 and the level set Ψ u , x 0 . x0 = ˆ Charles B. Moss x0 ￿ Ψ u, x 0 ￿ (13) (14) Shephard’s Duality Proof - Part II: Lecture XVII Basic Functions Shepard Duality x0 x0 p x0 u, x 0 p0 u Q u, p Eu 0 0 u, x p 0 x Q u, p 0 p0 Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII Basic Functions Shepard Duality Therefore Because ￿ ￿ x 0 ￿ p 0 = Ψ u , x 0 ⇒ x0 ￿ p 0 = 1 ˆ ˆ (15) ￿ ￿ ￿ ￿ p 0￿ x0 = Q u , p 0 ⇒ Ψ￿ u , x 0 = 1 ˆ ˆ ￿0 ￿| ￿ ￿ 0￿ (16) ￿|x ￿ | = ￿x | ￿ ˆ 0 Ψ u, x Back to the distance function representations, we let ξ0 be the ￿ ￿ intersection of the ray λx 0 |λ ≥ 0 with the hyperplane ￿ ￿ x 0￿ p = Ψ u , x 0 . Hence, ￿ ￿ 0 ￿ 0￿ ￿ 0 0￿ ￿ ￿ ￿|x ￿ | ￿|λ x ￿ | = Ψ u , x 0 ⇒ λ0 = Ψ u ,￿x ￿02 ￿| x ￿ | ￿ ￿ (17) 0 ￿ 0￿ ￿￿ x 1 ￿|ξ ￿ | = λ0 ￿|x 0 ￿ | = Ψ￿ u , ￿ ￿ 0￿ ⇒ = ￿| x 0 ￿ | ￿| x ￿ | ˆ Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII Basic Functions Shepard Duality x0 x0 p x0 p0 u, x 0 u 0 Q u, p Eu 0 u, x p 0 x Q u, p 0 p 0 Charles B. Moss Shephard’s Duality Proof - Part II: Lecture XVII ...
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This note was uploaded on 02/01/2012 for the course AEB 6184 taught by Professor Staff during the Fall '09 term at University of Florida.

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