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Rubinstein2005-page74

# Rubinstein2005-page74 - x p w is called the demand...

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October 21, 2005 12:18 master Sheet number 72 Page number 56 56 Lecture Five Proof: If x is not a solution, then there is a bundle z such that pz px and z x . By continuity and monotonicity, there is a bundle y = z , with y k z k such that y x and py < pz px . By convexity, any small move in the direction ( y x ) is an improvement and by differentiability, v ( x )( y x ) > 0. Let µ = v k ( x )/ p k for all k with x k > 0. Now, 0 > p ( y x ) = p k ( y k x k ) v k ( x )( y k x k )/µ (since for a good with x k > 0 we have p k = v k ( x )/µ , and for a good k with x k = 0, ( y k x k ) 0 and p k v k ( x )/µ .) Thus, 0 v ( x )( y x ) , a contradiction. The Demand Function We have arrived at an important stage on the way to developing a market model in which we derive demand from preferences. As- sume that the consumer’s preferences are such that for any B ( p , w ) , the consumer’s problem has a unique solution. Let us denote this solution by x ( p , w
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Unformatted text preview: x ( p , w ) is called the demand func-tion . The domain of the demand function is ³ K + 1 ++ whereas its range is ³ K + . Example: Consider a consumer in a world with two commodities having the following lexicographic preference relation, attaching the ﬁrst prior-ity to the sum of the quantities of the goods and the second priority to the quantity of commodity 1: x ± y if x 1 + x 2 > y 1 + y 2 or both x 1 + x 2 = y 1 + y 2 and x 1 ≥ y 1 . This preference relation is strictly convex but not continuous. It induces the following noncontinuous demand function: x (( p 1 , p 2 ) , w ) = ² ( 0, w / p 2 ) if p 2 < p 1 ( w / p 1 , 0 ) if p 2 ≥ p 1 . We now turn to studying some properties of the demand function....
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