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Unformatted text preview: October 21, 2005 12:18 master Sheet number 38 Page number 22 22 Lecture Two Problem 4. ( Moderate ) The following is a typical example of a utility representation theorem: Let X = < 2 + . Assume that a preference relation % satisfies the following three properties: ADD : ( a 1 , a 2 ) % ( b 1 , b 2 ) implies that ( a 1 + t , a 2 + s ) % ( b 1 + t , b 2 + s ) for all t and s . MON : If a 1 b 1 and a 2 b 2 , then ( a 1 , a 2 ) % ( b 1 , b 2 ) ; in addition, if either a 1 > b 1 or a 2 > b 2 , then ( a 1 , a 2 ) ( b 1 , b 2 ) . CON : Continuity. a. Show that if % has a linear representation (that is, % are represented by a utility function u ( x 1 , x 2 ) = x 1 + x 2 with > 0 and > 0), then % satisfies ADD , MON and CON . b. Suggest circumstances in which ADD makes sense. c. Show that the three properties are necessary for % to have a linear representation. Namely, show that for any pair of the three properties there is a preference relation that does not satisfy the third property.there is a preference relation that does not satisfy the third property....
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This note was uploaded on 12/29/2011 for the course ECO 443 taught by Professor Aswa during the Fall '10 term at SUNY Stony Brook.
- Fall '10