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Unformatted text preview: Envelope Theorems Let X ⊂ R n for some positive integer n , let T be an open real interval, and let f : X × T → R . 1 Constraint set depends on t Let X = R n , and let g : R n → R m for some positive integer m . Consider the problem V ( t ) = max x ∈ R m f ( x,t ) (1) subject to the constraint that g ( x,t ) ≤ 0. Let L ( x,λ,t ) = f ( x,t )- λ · g ( x,t ). Assume that • The solution x * ( t ) to the problem exists and is unique for every t ∈ T . 1 • The function x * is differentiable and f , g are twice differentiable on their domains. • On a neighborhood of a point t ∈ T , the Kuhn-Tucker necessary conditions for the problem hold at ( x * ( t ) ,t ) with associated multipliers λ * ( t ) = ( λ * 1 ( t ) ,...,λ * m ( t )) ≥ 0. Then V is differentiable at t and V ( t ) = f t ( x * ( t ) ,t )- ∑ m i =1 λ * i ( t ) g i t ( x * ( t ) ,t ), ? where g = ( g 1 ,...,g m ). Or more compactly V ( t ) = L t ( x * ( t ) ,λ * ( t ) ,t )....
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This note was uploaded on 11/19/2010 for the course ECON 202 taught by Professor Schlee during the Spring '10 term at ASU.
- Spring '10