Week11 - ECON 117 Mathematical Economics Spring 2008 Week...

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ECON 117 Spring 2008 Mathematical Economics Week 11 2. Optimization with equality constraints 2.1 One constraint case Ex) : : n n f g b \\ \ \ \ , : continuous & differentiable fg [ OP : Original Program] ( ) () 12 ,,, max , , , . . g , , , n n xx x n fxx x s tx = b ( Meaning ) Let solve OP, then * n x \ ( ) ( ) ** ff < x ( ) ( ) satisfying where n gb g ∀Δ ≠ ∈ +Δ = = x0 x x x \ b Page 1 of 8
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ECON 117 Spring 2008 Mathematical Economics [ LP : Lagrange Program] () () () max min fb g λ =+ x xx L where \ i.e., 1 : n + \\ (Meaning) Let ( , oo ) x solve LP where , on o x Then, ( ) , , 0 o o λλ >+ Δ Δ x x \ LL n and ( ) , , 0 < ∀ Δ ≠ ∈ \ Theorem) . *0 in OP in LP nn ∈= Page 2 of 8
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ECON 117 Spring 2008 Mathematical Economics Def) If ( ) 2* () , ()0 o λ Δ⋅ Δ< x xD x x L , ( ) * (small) satisfying 0 n g ∀Δ Δ x0 Dx x \ = We call ( ) , o x Dx is negative definite for ( ) * satisfying 0 n g Δ ≠∈ Δ= \ . Theorem) ( ) , o x is negative definite for satisfying n Δ≠∈ \ * ⋅Δ =
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This note was uploaded on 09/15/2008 for the course ECON 324 taught by Professor Kim during the Spring '08 term at HKUST.

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Week11 - ECON 117 Mathematical Economics Spring 2008 Week...

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