Unformatted text preview: 1, i.e., x (( p 1 , 2 ) , 1 ) = ( 1 /( 2 − p 1 ) , ( 1 − p 1 )/( 2 − p 1 )) . The above preference relation is monotonic but not continuous. However, we can construct a close continuous preference that leads to demand that is increasing in p 1 in a similar domain. Let α δ ( t ) be a continuous and increasing function on [ 1 − δ , 1 + δ ] where δ > 0, so that α δ ( t ) = 0 for all t ≤ 1 − δ and α δ ( t ) = 1 for all t ≥ 1 + δ . The utility function u δ ( x ) = (α δ ( x 1 + x 2 )( x 1 + 4 x 2 )) + ( 1 − α δ ( x 1 + x 2 )( x 1 + x 2 )) is continuous and monotonic. ±or δ close to 0, the function u δ = u except in a narrow area around the set of bundles for which x 1 + x 2 = 1....
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 Fall '10
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