ch09-extreme-values-univariate

ch09-extreme-values-univariate - Extreme Values of...

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Extreme Values of Univariate Functions Professor Erkut Ozbay Economics 300

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Economic models determine optimal outcomes, consumption, interest rates,… Calculate maximum value to identify Highest profit Highest utility Highest tax revenue Calculate minimum value to identify Lowest cost Lowest price Lowest risk
Tax rate and tax revenue 2 200 200 R t t = - ' 200 400 0 R t = - =

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Identifying extreme values Stationary Point x* is a stationary point of a differentiable function f ( x ) if * '( ) 0 f x =
Stationary points * ' 2 8 0 4 '' 2 0 h x x h = - + = = = - < * ' 2 8 0 4 '' 2 0 j x x j = - = = =

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Characterizing extreme values First-Order Condition If f(x) is everywhere differentiable and reaches a maximum or minimum at x* , then f′(x*) = 0 ( x * is stationary) Necessary condition for max or min If max or min, then f′(x*) = 0
A function with two extreme points and its derivative ''( ) 2 1 ''( 1) 0 ''(2) 0 p x x p p = - - <

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A function with an inflection point and its derivative '' 6( 4) ''(4) 0 k x k = - = Neither max nor min, but k’(x)=0
Characterizing extreme values Second-Order Condition If the second derivative of a differentiable function f(x) is negative when evaluated at a stationary point (f (x*) < 0) then x* is a local maximum.

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