Unformatted text preview: 1. e (Î» p , u ) = Î» e ( p , u ) (it follows from h (Î» p , u ) = h ( p , u ) ). 2. e ( p , u ) is nondecreasing in p k and strictly increasing in u . 3. e ( p , u ) is continuous in p (this follows from the continuity of h ( p , u ) ). 4. e ( p , u ) is concave in p (not only in p k ). To prove this, let x = h (Î» p 1 + ( 1 âˆ’ Î») p 2 , u âˆ— ) . Since u ( x ) = u âˆ— , e ( p i , u âˆ— ) â‰¤ p i x ; thus e (Î» p 1 + ( 1 âˆ’ Î») p 2 , u âˆ— ) = (Î» p 1 + ( 1 âˆ’ Î») p 2 ) x â‰¥ Î» e ( p 1 , u âˆ— ) + ( 1 âˆ’ Î») e ( p 2 , u âˆ— ) . Claim (the Dual Royâ€™s Equality): The hyperplane H = { ( p , e )  e = ph ( p âˆ— , u âˆ— ) } is tangent to the graph of the function e = e ( p , u âˆ— ) at point p âˆ— ....
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 Fall '10
 aswa
 Derivative, Convex function, Hicksian demand function, Expenditure minimization problem, expenditure function

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