Rubinstein2005-page81

# Rubinstein2005-page81 - 1 i.e x p 1 2 1 = 1 2 − p 1 1 −...

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October 21, 2005 12:18 master Sheet number 79 Page number 63 Demand: Consumer Choice 63 Figure 5.3 These preferences might reﬂect reasoning of the following type: “In the bundle x there are x 1 + x 2 units of vitamin A and x 1 + 4 x 2 units of vitamin B. My Frst priority is to get enough vitamin A . However, once I satisfy my need for 1 unit of vitamin A , I move on to my second priority, which is to consume as much as possible of vitamin B .” (See Fg 5.3.) Consider x (( p 1 ,2 ) ,1 ) . Changing p 1 is like rotating the budget lines around the pivot bundle ( 0, 1 / 2 ) . At a high price p 1 (as long as p 1 > 2 ) , the consumer demands ( 0, 1 / 2 ) . If the price is reduced to within the range 2 > p 1 > 1, the consumer chooses the bundle ( 1 / p 1 ,0 ) . So far, the demand for the Frst commodity indeed increased when its price fell. However, in the range 1 > p 1 > 1 / 2 we encounter an anomaly: the consumer buys as much as possible from the second good subject to the “constraint” that the sum of the goods is at least
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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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