L03 Utility

Big picture u Behavioral Postulate: A decisionmaker chooses its most preferred alternative from the set of affordable alternatives. u Budget set = affordable alternatives u To model choice we must have decisionmaker’s preferences.

Preferences: A Reminder u Rational agents rank consumption bundles from the best to the worst u We call such ranking preferences u Preferences satisfy Axioms: completeness and transitivity u Geometric representation: Indifference Curves u Analytical Representation: Utility Function ~ 

Indifference Curves x2x2x2x2 x1x1x1x1

Utility Functions u Preferences satisfying Axioms (+) can be represented by a utility function. u Utility function: formula that assigns a number (utility) for any bundle. u Today: –Geometric representation "mountain” –Utility function and Preferences –Utility function and Indifference curves –Utility function and MRS (next class)

Utility function: Geometry x2x2 x1x1 z

x2x2 x1x1 z

x2x2 x1x1 z

x2x2 x1x1 z Utility 3 5

Utility function: Geometry x2x2 x1x1 z Utility 3 5 U(x1,x2)

Utility Functions and Preferences u A utility function U(x) represents preferences if and only if:  x y U(x) ≥ U(y) x y x  y ~   ~ 

Usefulness of Utility Function u Utility function U(x 1,x 2 ) = x 1 x 2 u What can we say about preferences (2,3), (4,1), (2,2), (1,1), (8,8) u Recover preferences:

Utility Functions & Indiff. Curves u An indifference curve contains equally preferred bundles. u Indifference = the same utility level. u Indifference curve u Hikers: Topographic map with contour lines

Indifference Curves x2x2x2x2 x1x1x1x1 u U(x 1,x 2 ) = x 1 x 2

Ordinality of a Utility Function u Utilitarians: utility = happiness = Problem! (cardinal utility) u Nowadays: utility is ordinal (i.e. ordering) concept u Utility function matters up to the preferences (indifference map) it induces u Q: Are preferences represented by a unique utility function?

Utility Functions   U(x 1,x 2 ) = x 1 x 2 (2,3) (4,1)  (2,2). u Define V = U 2.   V(x 1,x 2 ) = x 1 2 x 2 2 (2,3) (4,1)  (2,2). u V preserves the same order as U and so represents the same preferences.  U=6 U=4 U=4 V= V= V=

Monotone Transformation x2x2x2x2 x1x1x1x1 u U(x 1,x 2 ) = x 1 x 2 u V= U 2

Theorem (Formal Claim) u T: Suppose that (1) U is a utility function that represents some preferences (2) f(U) is a strictly increasing function then V = f(U) represents the same preferences Examples: U(x 1,x 2 ) = x 1 x 2

u Perfect Substitutes (Example: French and Dutch Cheese) u Perfect Complements (Right and Left shoe) u Well-behaved preferences (Ice cream and chocolate) Three Examples

u Two goods that are substituted at the constant rate u Example: French and Dutch Cheese (I like cheese but I cannot distinguish between the two kinds) Example: Perfect substitutes

Perfect Substitutes (Cheese) French Dutch U(x 1,x 2 ) =

Perfect Substitutes (Proportions) x 1 Pack (6 slices) x 2 (1 Slice) U(x 1,x 2 ) =

u Two goods always consumed in the same proportion u Example: Right and Left Shoes u We like to have more of them but always in pairs Perfect complements

Perfect Complements (Shoes) L R U(x 1,x 2 ) =

Perfect Complements (Proportions) Sugar Coffee U(x 1,x 2 ) = 2:1