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Rubinstein2005-page39

# Rubinstein2005-page39 - y | y Â x and y | x Â y are open...

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October 21, 2005 12:18 master Sheet number 37 Page number 21 Problem Set 2 Problem 1. ( Easy ) The purpose of this problem is to make sure that you fully understand the basic concepts of utility representation and continuous preferences. a. Is the statement “if both U and V represent then there is a strictly monotonic function f : such that V ( x ) = f ( U ( x )) ” correct? b. Can a continuous preference be represented by a discontinuous func- tion? c. Show that in the case of X = , the preference relation that is rep- resented by the discontinuous utility function u ( x ) = [ x ] (the largest integer n such that x n ) is not a continuous relation. d. Show that the two definitions of a continuous preference relation (C1 and C2) are equivalent to Definition C3: For any x X , the upper and lower contours { y | y x } and { y | x y } are closed sets in X , and to Definition C4: For any
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Unformatted text preview: { y | y Â x } and { y | x Â y } are open sets in X . Problem 2. ( Moderate ) Give an example of preferences over a countable set in which the preferences cannot be represented by a utility function that returns only integers as values. Problem 3. ( Moderate ) Consider the sequence of preference relations ( % n ) n = 1,2, .. , deFned on < 2 + where % n is represented by the utility function u n ( x 1 , x 2 ) = x n 1 + x n 2 . We will say that the sequence % n converges to the preferences % ∗ if for every x and y , such that x Â ∗ y , there is an N such that for every n > N we have x Â n y . Show that the sequence of preference relations % n converges to the prefer-ences Â ∗ which are represented by the function max { x 1 , x 2 } ....
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