lecture math econ - 13 Concavity and Inequality-constrained...

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13 Concavity and Inequality-constrained Optima Proposition 13.1 Let U R n be open and convex. Let f : U R and h i : U R be concave C 1 functions, i = 1 , ··· ,l . Suppose also that there exists x U such that h i ( x ) > 0 for all i = 1 , ··· ,l . — Slater’s condition Then, x * is a global maximum of f in the set D = U ∩ { x R n : h i ( x ) 0 , i = 1 , ··· ,l } if and only if there exists λ * = ( λ * 1 , ··· * l ) R l such that (KT1) Df ( x * ) + l X i =1 λ * i Dh i ( x * ) = 0 n (KT2) λ * i 0 for all i = 1 , ··· ,l and l X i =1 λ * i h i ( x * ) = 0 14 Quasi-concavity and Inequality-constrained Optima Proposition 14.1 Let f : D R be a C 1 function where D R n is convex and open. Then, f is quasi-concave if and only if for every x,y D , f ( y ) f ( x ) = Df ( x )( y - x ) 0 . Moreover, if f is quasi-concave, then for all x D with Df ( x ) 6 = 0 and for all y D , f ( y ) > f ( x ) = Df ( x )(
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lecture math econ - 13 Concavity and Inequality-constrained...

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