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
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1 Convexity and concavity Alternative characterization of convexity. Afunc t ion f, twice di f 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 Atw ice -d i f 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 i sthefunc t ionconcaveove r 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 = 00 x,y = f 00 y,x = 00 y,y = Hessian matrix H : H = Ã f 00 x,x = f 00 x,y = f 00 = f 00 = ! Compute | H 1 | = f 00 x,x and | H 2 | = f 00 x,x f 00 ³ f 00 x,y ´ 2
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Why are convexity and concavity important? Theorem. Consider a twice-di f erentiable concave (convex) function over C R n . If the point x 0 satis f es the f st order conditions, it is a global max- imum (minimum).
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This note was uploaded on 09/05/2009 for the course ECON 101a taught by Professor Staff during the Spring '08 term at University of California, Berkeley.

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

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