Utility - Utility Preferences - A Reminder x y: x is...

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Utility
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Preferences - A Reminder x y: x is preferred strictly to y. x y: x and y are equally preferred. x y: x is preferred at least as much as is y. ~
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Utility Functions 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.
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Utility Functions A utility function U(x) represents a preference relation if and only if: x y U(x) > U(y) x y U(x) ≥ U(y) x y U(x) = U(y). ~ ~
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Utility Functions Utility is an ordinal (i.e. ordering) concept. E.g . if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to bundle y. But x is not preferred three times as much as is y.
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Utility Functions 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. Again consider the bundles (4,1), (2,3) and (2,2).
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Utility Functions 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).
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Utility Functions U(x 1 ,x 2 ) = x 1 x 2 (2,3) (4,1) (2,2). Define V = U 2 .
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Utility Functions U(x 1 ,x 2 ) = x 1 x 2 (2,3) (4,1) (2,2). Define V = U 2 . 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.
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Utility Functions U(x 1 ,x 2 ) = x 1 x 2 (2,3) (4,1) (2,2). Define W = 2U + 10.
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Utility Functions U(x 1 ,x 2 ) = x 1 x 2 (2,3) (4,1) (2,2). Define W = 2U + 10. Then W(x 1 ,x 2 ) = 2x 1 x 2 +10 so W(2,3) = 22 > W(4,1) = W(2,2) = 18. Again, (2,3) (4,1) (2,2). W preserves the same order as U and V and so represents the same preferences.
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Utility Functions If U is a utility function that represents a preference relation and f is a strictly increasing function, then V = f(U) is also a utility function representing . ~ ~
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Consider the bundles (4,1), (2,3) and (2,2). Suppose (2,3) (4,1) (2,2). Assign to these bundles any numbers that preserve the preference ordering; e.g. u(2,3) = 6 > u(4,1) = u(2,2) = 4. Call these numbers utility levels .
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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.
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So the bundles (4,1) and (2,2) are in the indiff. curve with utility level U ≡ 4 But the bundle (2,3) is in the indiff. curve with utility level U
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Utility - Utility Preferences - A Reminder x y: x is...

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