Rubinstein2005-page92

# Rubinstein2005-page92 - 1 e(Î p u = Î e p u(it follows from...

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October 21, 2005 12:18 master Sheet number 90 Page number 74 74 Lecture Six Here are some properties of the Hicksian demand function: 1. h p , u ) = h ( p , u ) .I f x is a solution to the problem D ( p , u ) , it is also a solution to the problem D p , u ) . The function λ px is a positive linear transformation of px ; thus, the prob- lem min x { λ px | u ( x ) u } has the same solution as the problem min x { px | u ( x ) u } . 2. h k ( p , u ) is nonincreasing in p k . Note that for every p 0 ph ( p , u ) ph ( p 0 , u ) since h ( p 0 , u ) also satisFes the constraint of achieving a utility level of at least u and h ( p , u ) is the cheapest bundle satisfying the constraint. Similarly, p 0 h ( p 0 , u ) p 0 h ( p , u ) . Thus, ( p p 0 )( h ( p , u ) h ( p 0 , u )) = p [ h ( p , u ) h ( p 0 , u ) ] + p 0 [ h ( p 0 , u ) h ( p , u ) ]≤ 0 . When p p 0 = ( 0, ... , ² , ... ,0 ) we get that h k ( p , u ) h k ( p 0 , u ) 0. Thus, increasing the price of commodity k has a nonpositive effect on Hicksian demand. 3. h ( p , u ) is continuous in p (verify!). DeFne e ( p , u ) = ph ( p , u ) to be the expenditure function . This concept is analogous to the indirect utility function in the prime problem. Here are some properties of the expenditure function:
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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 non-decreasing 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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