1Envelopes

1Envelopes - Envelope Theorems Let X Rn for some positive...

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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 0 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 0 and V 0 ( t 0 ) = f t ( x * ( t 0 ) , t 0 ) - m i =1 λ * i ( t 0 ) g i t ( x * ( t 0 ) , t 0 ), ? where g = ( g 1 , ..., g m ). Or more compactly V 0 ( t 0 ) = L t ( x * ( t 0 ) , λ * ( t 0 ) , t 0 ). Proof : Differentiate V ( t ) = f ( x * ( t ) , t ) - λ * ( t ) · g ( x * ( t ) , t ) (why is the equality true?) with respect to t , evaluate it at t = t 0 , and use the Kuhn-Tucker conditions.

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