lecture_03.pdf - Preference and Utility Econ 2100 Fall 2017...

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Preference and Utility Econ 2100 Fall 2017 Lecture 3, 5 September Problem Set 1 is due in Kelly°s mailbox by 5pm today Outline 1 Existence of Utility Functions 2 Continuous Preferences 3 Debreu°s Representation Theorem
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De°nitions From Last Week A binary relation % on X is a preference relation if it is a weak order, i.e. complete and transitive. The upper contour set of x is % ( x ) = f y 2 X : y % x g . The lower contour set of x is - ( x ) = f y 2 X : x % y g . The utility function u : X ! R represents the binary relation % on X if x % y , u ( x ) ° u ( y ) : Question: Under what assumptions can a preference relation be represented by a utility function? We know we need transitivity and continuity. Are they enough?
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Existence of a Utility: Alternative De°nition The following provides an alternative way to show that that a preference is represented by some function (very useful). Question 1, Problem Set 2 (due Tuesday September 12). Let % be a preference relation. Prove that u : X ! R represents % if and only if: x % y ) u ( x ) ° u ( y ); and x ± y ) u ( x ) > u ( y ) : This result is useful as it makes it (sometimes) easier to show some utility function represents % .
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Existence of a Utility Function When X is Finite Theorem Suppose X is °nite. Then % is a preference relation if and only if there exists some utility function u : X ! R that represents % . When the consumption set is ±nite ±nding a utility function that represents any given preferences is easy. The proof is constructive: a function that works is the one that counts the number of elements that are not as good as the one in question. This function is well de±ned since the are only ±nitely many items that can be worse than something. In other words, the utility function is u ( x ) = j - ( x ) j where - ( x ) = f y 2 X : x % y g is the lower contour set of x , and j²j denotes the cardinality of ² . Notice that we only need to prove that this function represents - ( x ) because from a previous result we know that if that is the case then - is a preference relation.
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Existence of a Utility Function When X is Finite If X is ±nite and % is a preference relation ) 9 u : X ! R that represents % . Proof. Let u ( x ) = j - ( x ) j . Since X is ±nite, u ( x ) is ±nite and therefore well de±ned. Suppose x % y . I claim this implies u ( x ) ° u ( y ) . Let z 2 - ( y ) , i.e. y % z . By transitivity, x % z , i.e. z 2 - ( x ) . Thus - ( y ) ° - ( x ) . Therefore j - ( y ) j ± j - ( x ) j . By de±nition, this means u ( y ) ± u ( x ) . Now suppose x ± y . I claim this implies u ( x ) > u ( y ) . x ² y implies x % y , so the argument above implies - ( y ) ° - ( x ) . x % x by completeness, so x 2 - ( x ) . Also, x ² y implies x = 2 - ( y ) . Hence - ( y ) and f x g are disjoint, and both subsets of - ( x ) . Then - ( y ) [ f x g ° - ( x ) j - ( y ) [ f x gj ± j - ( x ) j j - ( y ) j + jf x gj ± j - ( x ) j u ( y ) + 1 ± u ( x ) u ( y ) < u ( x ) This proves x % y ) u ( x ) ° u ( y ) x ± y ) u ( x ) > u ( y ) and thus we are done.
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Utility Function for a Countable Space Existence of a utility function can also be proven for a countable space.
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