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Rubinstein2005-page91

Rubinstein2005-page91 - P p w ∗ then there exists an x...

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October 21, 2005 12:18 master Sheet number 89 Page number 73 Choice over Budget Sets and the Dual Problem 73 The dual problem D ( p , u ) Find a bundle minimizing the expenses needed to obtain a level of utility u , that is, min x { px | u ( x ) u } . Claim: 1. If x is the solution to the problem P ( p , w ) , then it is also the solution to the dual problem D ( p , u ( x )) . 2. If x is a solution to the dual problem D ( p , u ) , then it is also the solution to the problem P ( p , px ) . Proof: 1. If x is not a solution to the dual problem D ( p , u ( x )) , then there exists a strictly cheaper bundle x for which u ( x ) u ( x ) . For some positive vector ε (that is, ε k > 0 for all k ), it still holds that p ( x + ε) < px w . By monotonicity u ( x + ε) > u ( x ) u ( x ) , contradicting the assumption that x is a solution to P ( p , w ) . 2. If x is not a solution to the problem
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Unformatted text preview: P ( p , w ∗ ) , then there exists an x such that px ≤ px ∗ and u ( x ) > u ( x ∗ ) ≥ u ∗ . By continuity, for some nonnegative vector ε 6= 0, x − ε is a bundle such that u ( x − ε) > u ∗ and p ( x − ε) < px ∗ , contradicting the assumption that x ∗ is a solution to D ( p , u ∗ ) . The Hicksian Demand Function Assume that the dual problem D ( p , u ) has a unique solution. This is the case, for example, if u represents strictly convex continuous preferences. The Hicksian demand function h ( p , u ) is the solution to D ( p , u ) . This concept is analogous to the demand function in the prime problem....
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