Duality
Ye Jianliang
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Duality
? Given the technology,we can obtain the
cost function,are the cost function contains
the same information of the technology
(production function)?
? If the answer is,yes”,then the cost
minimization behavior will indicate the
technology of the firm,
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Duality
? Content,
? Duality in mathematics
? Sufficient condition for cost function
? Factor demand function
? Geometry of duality
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1.Duality in mathematics
? Some concepts and properties,
? Half-spaces,
? Normal vector,
? Hyperplane,
? K is convex closure,
? K is concave,K* is the closed convex hull,
{, }n c? ? ? ?x p xH
n??p
{, }nHc? ? ? ?x p x
a n d,a n d,K x K c c? ? ? ? ? ? ? ? ? ? ?nx p p x p x
* ()
p
KK??H
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1.Duality in mathematics
? Support function,(infimum)
? given an alternative description for K,
? Proposition8,is HD1 and concave,
See the fig,
i n f {, }K K? ? ? ?p p x x?
K??p?
{, f o r e v e r y }n KK ? ? ? ? ? ??x p x p p?
K??p?
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1.Duality in mathematics
? Duality theorem,K is nonempty closure,and
its support function is differentiable
at,then there is only one,that
See the fig,
K??p?
p K?x
a n d KK? ? ?? ? ? ? ?p x = p p x??
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2.S.C,for cost function
? Differentiable function satisfied,
? HD1 of w
? Concave of w
? Non-decreasing of w
? Nonnegative for,
? Then is the cost function definite by
the tech,of
(,)q? w
0,0q??w
(,)q? w
1( ) {, (,) }nV q q???? ? ? ? ?x w x w
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3.what about demand function
? Duality indicate that HD1 and concave is what the
convex tech need for the cost,
? What about the other function,such as factor
demand function,
? is HD0 and is symmetric
negative semidefinite,Then it is the conditional
factor demand function of a certain tech,
(,)qgw
(,)i
j
gq
w
???
??
???
w
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3.what about demand function
? Example,given a cost function,
then what’s the corresponded tech,and
factor demand function?
? Proposition9,elasticity of scale
112(,)c q q w w?? ??w
()(,)
()
A C qe x q
M C q?
111 1 2(,)x q q w w??? ???w 2 1 2(,) ( 1 )x q q w w??? ???w
1112( 1 )q x x? ? ? ???? ? ???
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4.Geometry of duality
x2
x1
isoquant
w1
w2
isocost
x1
w1/w2
factor demand
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4.Geometry of duality
x2
x1
isoquant
2x?
2x?
1x? 1x?
2
1
sl ope ww??
w1
w2
isocost
2w?
1w?
x1
w1/w2
factor demand
2
1
w
w
?
?
1x? 1x?
lecture4 for Chu Kechen Honors College
4.Geometry of duality
x1
x2
isoquant
2x?
1x?
w2
w1
isocost
2w?
2w?
1w? 1w?
2
1
sl o p e xx??
x1
w1/w2
factor demand
2
1
w
w
?
?
1x?
2
1
w
w
?
?
lecture4 for Chu Kechen Honors College
4.Geometry of duality
w1
w2
isocost
2w?
1w?
x2
x1
isoquant
2x?
2x?
1x? 1x?
2
1
sl ope ww??
x1
w1/w2
factor demand
2
1
w
w
?
?
1x? 1x?
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Support function
p
K
x
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Duality theorem
K
x1
x2
p2 is decreasing
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Duality theorem
K
p2
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