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Chapter 4

# Chapter 4 - Chapter Four Utility Preferences A Reminder x y...

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Chapter Four Utility

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Preferences - A Reminder x y: x is preferred strictly to y. x a y: x and y are equally preferred. x y: x is preferred at least as much as is y. E ~ ;
Preferences - A Reminder Completeness : For any two bundles x and y it is always possible to state either that x y or that y x. ~ ; ~ ;

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Preferences - A Reminder Reflexivity : Any bundle x is always at least as preferred as itself; i.e. x x. ~ ;
Preferences - A Reminder Transitivity : If x is at least as preferred as y, and y is at least as preferred as z, then x is at least as preferred as z; i.e. x y and y z x z. ~ ; ~ ; ~ ;

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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.
Utility Functions A utility function U(x) represents a preference relation if and only if: 8±[¶² ! 8±[´² 8±[¶² ³ 8±[´² a 8±[¶² 8±[´²µ ~ ; E E

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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.
Utility Functions & Indiff. Curves Consider the bundles (4,1), (2,3) and (2,2). Suppose (2,3) (4,1) a (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 . E

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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.
Utility Functions & Indiff. Curves So the bundles (4,1) and (2,2) are in the indiff. curve with utility level U { ± But the bundle (2,3) is in the indiff. curve with utility level U { 6. On an indifference curve diagram, this preference information looks as follows:

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Utility Functions & Indiff. Curves U { 6 U { 4 (2,3) (2,2) a (4,1) x 1 x 2 E
Utility Functions & Indiff. Curves Another way to visualize this same information is to plot the utility level on a vertical axis.

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U(2,3) = 6 U(2,2) = 4 U(4,1) = 4 Utility Functions & Indiff. Curves 3D plot of consumption & utility levels for 3 bundles x 1 x 2 Utility
Utility Functions & Indiff. Curves This 3D visualization of preferences can be made more informative by adding into it the two indifference curves.

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Utility Functions & Indiff. Curves U { ± U { ² Higher indifference curves contain more preferred bundles.
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Chapter 4 - Chapter Four Utility Preferences A Reminder x y...

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