Rubinstein2005-page68

# Rubinstein2005-page68 - ous preferences it is equivalent to...

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October 21, 2005 12:18 master Sheet number 66 Page number 50 Problem Set 4 Problem 1. ( Easy ) Characterize the preference relations on the interval [ 0, 1 ] that are contin- uous and strictly convex. Problem 2. ( Easy ) Show that if the preferences ± satisfy continuity and x ± y ± z , then there is a bundle m on the interval connecting x and z such that y m . Problem 3. ( Moderate ) Show that if the preferences ± satisfy continuity and monotonicity, then the function u ( x ) , deﬁned by x ( u ( x ) , ... , u ( x )) , is continuous. Problem 4. ( Moderate ) In a world with two commodities, consider the following condition: The preference relation ± satisﬁes convexity 3 if for all x and ε ( x 1 , x 2 ) ( x 1 ε , x 2 + δ 1 ) ( x 1 2 ε , x 2 + δ 1 + δ 2 ) implies δ 2 δ 1 . Interpret convexity 3 and show that for strong monotonic and continu-
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Unformatted text preview: ous preferences, it is equivalent to the convexity of the preference relation. Problem 5. ( Moderate ) Formulate and prove a proposition of the following type: If the preferences ± are quasi linear in all commodities, continuous, and strongly monotonic, then there is a utility function of the form ( . . . add a condition here . . . ) that represents it. Problem 6. ( Difﬁcult ) Show that for any consumer’s preference relation ± satisfying continuity, monotonicity and quasi-linearity with respect to commodity 1 and for every vector x , there is a number v ( x ) so that x ∼ ( v ( x ) , 0, . . . , 0 ) ....
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