文档介绍：Duality
Ye Jianliang
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for Chu Kechen Honors College
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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in mathematics
Some concepts and properties:
Half-spaces:
Normal vector:
Hyperplane:
K is convex closure:
K is concave, K* is the closed convex hull:
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in mathematics
Support function: (infimum)
given an alternative description for K.
Proposition8: is HD1 and concave.
See the fig.
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in mathematics
Duality theorem: K is nonempty closure, and its support function is differentiable at , then there is only one , that
See the fig.
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. 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
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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.
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about demand function
Example: given a cost function , then what’s the corresponded tech. and factor demand function?
Proposition9: elasticity of scale
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of duality
x2
x1
isoquant
w1
w2
isocost
x1
w1/w2
factor demand
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for Chu Kechen Honors College