This preview shows page 1. Sign up to view the full content.
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 costminimization 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 contactpoint 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 ...
View
Full
Document
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.
 Fall '09
 Staff

Click to edit the document details