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101A_lec4

# 101A_lec4 - Economics 101A(Lecture 4 Stefano DellaVigna...

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Economics 101A (Lecture 4) Stefano DellaVigna January 29, 2009

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Outline 1. Convexity and concavity II 2. Constrained Maximization 3. Envelope Theorem II 4. Preferences
1 Convexity and concavity Alternative characterization of convexity. A function f, twice di ff erentiable, is concave if and only if for all x the subdeterminants | H i | of the Hessian matrix have the property | H 1 | 0 , | H 2 | 0 , | H 3 | 0 , and so on. For the univariate case, this reduces to f 00 0 for all x For the bivariate case, this reduces to f 00 x,x 0 and f 00 x,x f 00 y,y ³ f 00 x,y ´ 2 0 A twice-di ff erentiable function is strictly concave if the same property holds with strict inequalities.

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Examples. 1. For which values of a, b, and c is f ( x ) = ax 3 + bx 2 + cx + d is the function concave over R ? Strictly concave? Convex? 2. Is f ( x, y ) = x 2 y 2 concave? For Example 2, compute the Hessian matrix f 0 x = , f 0 y = f 00 x,x = , f 00 x,y = f 00 y,x = , f 00 y,y = Hessian matrix H : H = Ã f 00 x,x = f 00 x,y = f 00 y,x = f 00 y,y = ! Compute | H 1 | = f 00 x,x and | H 2 | = f 00 x,x f 00 y,y ³ f 00 x,y ´ 2
Why are convexity and concavity important? Theorem. Consider a twice-di ff erentiable concave (convex) function over C R n . If the point x 0 satis fi es the fi st order conditions, it is a global max- imum (minimum).

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101A_lec4 - Economics 101A(Lecture 4 Stefano DellaVigna...

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