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Rubinstein2005-page40

Rubinstein2005-page40 - 12:18 22 master Sheet number 38...

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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 ) = α x 1 + β x 2 with α > 0 and β > 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. d. ( This part is difficult ) Show that if satisfies the three properties, then it has a linear representation. Problem 5. ( Moderate ) Utility is a numerical representation of preferences. One can think about the numerical representation of other abstract concepts. Here, you will try to
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