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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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