ch09-extreme-values-univariate (1)

ch09-extreme-values-univariate (1) - 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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'' 6( 4) ''(4) 0 k x k = - = Neither max nor min, but k’(x)=0
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This note was uploaded on 02/03/2012 for the course ECON 300 taught by Professor Cramton during the Fall '08 term at Maryland.

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ch09-extreme-values-univariate (1) - Extreme Values of...

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