Macroeconomics Exam Review 142

Macroeconomics Exam Review 142 - 3.122 Assume is convex...

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Unformatted text preview: 3.122 Assume is convex which implies epi is convex. The points ( x , ( x )) epi . Since epi is convex 1 ( x 1 , ( x 1 )) + 2 ( x 1 , ( x 1 )) + + ( x , ( x )) epi that is ( 1 x 1 + 2 x 2 + + x ) 1 ( x 1 ) + 2 ( x 1 ) + + ( x )) Conversely, letting = 2 and = 1 , (3.25) implies that ( x 1 + (1 ) x 2 ) ( x 1 ) + (1 ) ( x 2 ) Jensens inequality can also be proved by induction from the definition of a convex function (see for example Sydsaeter + Hammond 1995; p.624). 3.123 For each , let = log so that = = Since is convex (Example 3.41) 1 1 2 2 ... > 0 = exp( ) = exp (...
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