Rubinstein2005-page77

# Rubinstein2005-page77 - 0 1 2 Â 1 2 0 Similarly x 2 1 1 =...

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October 21, 2005 12:18 master Sheet number 75 Page number 59 Demand: Consumer Choice 59 B ( p , w ) . This defnition is “empty” since any demand Function is consistent with maximizing the “total indiFFerence” preFerence. This is why we usually say that the preFerences % rationalize the demand Function x iF they are monotonic and For any ( p , w ) , the bundle x ( p , w ) is a % maximal bundle within B ( p , w ) . OF course, iF behavior satisfes homogeneity oF degree zero and Walras’s law, it is still not necessarily rationalizable in any oF those senses: Example 1: Consider the demand Function oF a consumer who spends all his wealth on the “more expensive” good: x (( p 1 , p 2 ) , w ) = ½ ( 0, w / p 2 ) if p 2 p 1 ( w / p 1 ,0 ) if p 2 < p 1 . This demand Function is not entirely inconceivable, and yet it is not rationalizable. To see this, assume that it is Fully rationaliz- able or rationalizable by % . Consider the two budget sets B (( 1, 2 ) ,1 ) and B (( 2, 1 ) ,1 ) . Since x (( 1, 2 ) ,1 ) = ( 0, 1 / 2 ) and ( 1 / 2, 0 ) is an internal bundle in B (( 1, 2 ) ,1 ) , by any oF the two defnitions oF rationalizabil- ity, it must be that
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Unformatted text preview: ( 0, 1 / 2 ) Â ( 1 / 2, 0 ) . Similarly, x (( 2, 1 ) , 1 ) = ( 1 / 2, 0 ) and ( 0, 1 / 2 ) is an internal bundle in B (( 2, 1 ) , 1 ) . Thus, ( 0, 1 / 2 ) ≺ ( 1 / 2, 0 ) , a contradiction. Example 2: A consumer chooses a bundle ( z , z , . . . , z ) , where z satisfes z 6 p k = w . This behavior is Fully rationalized by any preFerences according to which the consumer strictly preFers any bundle on the main di-agonal over any bundle that is not (because, For example, he cares primarily about purchasing equal quantities From all sellers oF the K goods), while on the main diagonal his preFerences are according to “the more the better”. These preFerences rationalize his behavior in the frst sense but are not monotonic. This demand Function is also Fully rationalized by the mono-tonic preFerences represented by the utility Function u ( x 1 , . . . , x K ) = min { x 1 , . . . , x K } ....
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