Rubinstein2005-page34

Rubinstein2005-page34 - u a 0 Let q a be a rational number...

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October 21, 2005 12:18 master Sheet number 32 Page number 16 16 Lecture Two Example: Let X be the unit square, i.e., X =[ 0, 1 ]×[ 0, 1 ] . Let x % k y if x k y k . The lexicographic ordering % L induced from % 1 and % 2 is: ( a 1 , a 2 ) % L ( b 1 , b 2 ) if a 1 > b 1 or both a 1 = b 1 and a 2 b 2 . (Thus, in this example, the left component is the primary criterion while the right compo- nent is the secondary criterion.) We will now show that the preferences % L do not have a utility representation. The lack of a utility representation excludes lexico- graphic preferences from the scope of standard economic models in spite of the fact that they constitute a simple and commonly used procedure for preference formation. Claim: The preference relation % L on [ 0, 1 ]×[ 0, 1 ] , which is induced from the relations x % k y if x k y k ( k = 1, 2), does not have a utility rep- resentation. Proof: Assume by contradiction that the function u : X →< represents % L . For any a ∈[ 0, 1 ] , ( a ,1 ) Â L ( a ,0 ) we thus have u ( a ,1 )>
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Unformatted text preview: u ( a , 0 ) . Let q ( a ) be a rational number in the nonempty interval I a = ( u ( a , 0 ) , u ( a , 1 )) . The function q is a function from X into the set of ra-tional numbers. It is a one-to-one function since if b > a then ( b , 0 ) Â L ( a , 1 ) and therefore u ( b , 0 ) > u ( a , 1 ) . It follows that the in-tervals I a and I b are disjoint and thus q ( a ) 6= q ( b ) . But the cardinality of the rational numbers is lower than that of the continuum, a con-tradiction. Continuity of Preferences In economics we often take the set X to be an in±nite subset of a Euclidean space. The following is a condition that will guarantee the existence of a utility representation in such a case. The basic in-tuition, captured by the notion of a continuous preference relation,...
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