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# lecture_2 - Utility A preference relation that is complete...

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ECOS2001 Lecture 2 1 Utility A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function. Continuity means that small changes to a consumption bundle cause only small changes to the preference level. A utility function U(x) represents a preference relation if and only if: x’ x” U(x’) > U(x”) x’ x” U(x’) < U(x”) x’ ~ x” U(x’) = U(x”). f °

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ECOS2001 Lecture 2 2 Utility is an ordinal (i.e. ordering) concept Utility Functions & Indiff. Curves An indifference curve contains equally preferred bundles. Equal preference same utility level. Therefore, all bundles in an indifference curve have the same utility level. The collection of all indifference curves for a given preference relation is an indifference map. An indifference map is equivalent to a utility function; each is the other
ECOS2001 Lecture 2 3 There is no unique utility function representation of a preference relation. Suppose U(x 1 ,x 2 ) = x 1 x 2 represents a preference relation. Consider the bundles (4,1), (2,3) and (2,2). U(x 1 ,x 2 ) = x 1 x 2 , so U(2,3) = 6 > U(4,1) = U(2,2) = 4; that is, (2,3) (4,1) ~ (2,2). f

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ECOS2001 Lecture 2 4 U(x 1 ,x 2 ) = x 1 x 2 (2,3) (4,1) ~ (2,2). Define V = U 2 . f
ECOS2001 Lecture 2 5 Then V(x 1 ,x 2 ) = x 1 2 x 2 2 and V(2,3) = 36 > V(4,1) = V(2,2) = 16 so again (2,3) (4,1) ~ (2,2). V preserves the same order as U and so represents the same preferences. f

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ECOS2001 Lecture 2 6 Goods, Bads and Neutrals A good is a commodity unit which increases utility (gives a more preferred bundle). A bad is a commodity unit which decreases utility (gives a less preferred bundle). A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).
ECOS2001 Lecture 2 7 Some Other Utility Functions and Their Indifference Curves Instead of U(x 1 ,x 2 ) = x 1 x 2 consider V(x 1 ,x 2 ) = x 1 + x 2 .

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lecture_2 - Utility A preference relation that is complete...

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