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

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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. ~
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). ~ ~
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).
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 .
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.
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 . ~ ~
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.
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 preferred...

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