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Rubinstein2005-page37

# Rubinstein2005-page37 - we have z x and y z and thus by...

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October 21, 2005 12:18 master Sheet number 35 Page number 19 Utility 19 economic theory. For a complete proof of Debreu’s theorem see Debreu 1954, 1960. Here we prove only that continuity guarantees the existence of a utility representation. Lemma: If is a continuous preference relation on a convex set X n , and if x y , then there exists z in X such that x z y . Proof: Assume not. Construct a sequence of points on the interval that connects the points x and y in the following way. First define x 0 = x and y 0 = y . In the inductive step we have two points, x t and y t , on the line that connects x and y , such that x t x and y y t . Consider the middle point between x t and y t and denote it by m . According to the assumption, either m x or y m . In the former case define x t + 1 = m and y t + 1 = y t , and in the latter case define x t + 1 = x t and y t + 1 = m . The sequences { x t } and { y t } are converging, and they must converge to the same point z since the distance between x t and y t converges to zero. By the continuity of
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Unformatted text preview: % we have z % x and y % z and thus, by transitivity, y % x , contradicting the assumption that x Â y . Comment on the Proof: Another proof could be given for the more general case, in which the assumption that the set X is convex is replaced by the assumption that it is a connected subset of < n . Remember that a connected set cannot be covered by two disjoint open sets. If there is no z such that x Â z Â y , then X is the union of two disjoint sets { a | a Â y } and { a | x Â a } , which are open by the continuity of the preference relation. Recall that a set Y ⊆ X is dense in X if in every open subset of X there is an element in Y . For example, the set Y = { x ∈ < n | x k is a rational number for k = 1, .. , n } is a countable dense set in < n ....
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