n11 - Atsushi Inoue ECG 765: Mathematical Methods for...

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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 11 Concave and Convex Functions So far we have focused on finding local optima using first-order and second- order conditions. Local optima are not necessarily global optima in general. Under what conditions are local optima global ones? When the objective function is concave [convex], a local maximum [local minimum] is a global maximum [global minimum]. We will review various characterizations of cocave and convex functions. Outline A. Concave and Convex Functions B. Differentiable Concave and Convex Functions C. Twice-Differentiable Concave and Convex Functions D. Quasi-Concave and Quasi-Convex Functions A. Concave and Convex Functions Definition. A set X < n is convex if, for every x,y X and every such that 0 1, x + (1- ) y X. Definition. Let f : X < where X < n is convex. f is said to be concave if f ( x + (1- ) y ) f ( x ) + (1- ) f ( y ) for all x,y X and all such that 0 1. f is said to be convex if f ( x + (1- ) y ) f ( x ) + (1- ) f ( y ) 1 for all x,y X and all such that 0 1. Definition. Let f : X < where X < n is convex. f is said to be strictly concave if f ( x + (1- ) y ) > f ( x ) + (1- ) f ( y ) for all x,y X such that x 6 = y and all such that 0 < < 1. f is said to be strictly convex if f ( x + (1- ) y ) < f ( x ) + (1...
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n11 - Atsushi Inoue ECG 765: Mathematical Methods for...

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