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# slide11-sept16 - ECON 401 The Envelope Theorem Siyang Xiong...

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ECON 401: The Envelope Theorem Siyang Xiong Rice University September 15, 2011 Xiong (Rice University) ECON 401 September 15, 2011 1 / 9

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Sensitivity analysis: the Envelope Theorem Consider an unconstrained optimization problem in parametric form: max f . x , s.t. x 2 R n Notation: V D max x 2 R n f . x , x D argmax x 2 R n f . x , Xiong (Rice University) ECON 401 September 15, 2011 2 / 9
Sensitivity analysis: the Envelope Theorem The Envelope Theorem: V 0 D @ f x , ± intuition of the proof: V 0 D ² @ f @ x x , ± ³ ± ² @ x ³ C @ f x , ± D 0 ± ² @ x ³ C @ f x , ± D @ f x , ± Xiong (Rice University) ECON 401 September 15, 2011 3 / 9

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the Envelope Theorem: an example max x 2 C x ± s.t. x 2 R n FOC: 2 x C D 0, ) x D 2 , i.e., x ± D 2 I V ² 2 ³ 2 C 2 D 2 4 Hence, V 0 D 2 Also, by the Envelope Theorem: V 0 D x ± D 2 Xiong (Rice University) ECON 401 September 15, 2011 4 / 9
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slide11-sept16 - ECON 401 The Envelope Theorem Siyang Xiong...

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