Week13 - ECON 117 Mathematical Economics Spring 2008 Week...

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ECON 117 Spring 2008 Mathematical Economics Week 13 3. Optimization with inequality constraints 3.1 The basic idea Q) \\ is convex in () , : yf xf = x Let x solve min ( ) x f x Find the F.O.C. for x x 1 Q) Let x solve: 0 min ( ) .. x f x st x x Find the F.O.C. for x Page 1 of 12
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ECON 117 Spring 2008 Mathematical Economics Theorem) n y \ is covex in x . ( ), : , i.e., , n yf f =→ xx \\ \ Let () 1 ,, n n = x \ solve: 00 0 0 1 min ( ) s.t. where , , n n f ≥= x x x …\ Then, the F.O.C.s for x are: 0 0 ( ) 0 f f •≥ −⋅ = Dx 0 xx Dx Page 2 of 12
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ECON 117 Spring 2008 Mathematical Economics 3.2 The Kuhn-Tucker Theorem (One inequality constraint case) Ex) :, nn fg →→ \\ b \ (1) [ OP : Original Program] Let n x \ solve: max ( ) s.t. g( ) f b x x x i.e., () f f xx s.t. ( ) where ( ) gb g ∀≤ x b (2) [ KTP : Kuhn-Tucker Program] Let ( ) , w h e r e λλ x \ solve: ( ) max min ( , ) ( ) ( ) s.t. 0 fb g λ ≡+ x L x i.e., ( , ) ( , ) given 0 ≤∀ x LL (1) and ( , ) ( , ) 0 given n x \ (2) (3) [ KTC : Kuhn-Tucker Condition] Let ( , ) x satisfy (, ) (, ) 0 0 •= •≥ ⋅≥ x Dx 0 x x Page 3 of 12
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ECON 117 Spring 2008 Mathematical Economics
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Week13 - ECON 117 Mathematical Economics Spring 2008 Week...

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