L03 Utility

Quiz How much do you like microeconomics A: I love it (unconditionally) B: I cannot live without it C: I would die for it D: All of the above

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: 1.Geometric interpretation 2.Utility function and Preferences 3.Utility and Indifference curves 4.Important examples

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) u Quiz 1: u A: u B: u C: u D:

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 = 5U.   V(x 1,x 2 ) = 5x 1 x 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= 5U

Theorem (Monotonic Transformation) 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

Preference representations u Utility U(x 1,x 2 ) = x 1 x 2 u Quiz 2: U(x 1,x 2 ) = x 1 +x 2 u A: V = ln(x 1 +x 2 )+5 u B: V=5x 1 +7x 2 u C: V=-2(x 1 +x 2 ) u D: All of the above

u Cobb-Douglas preferences (most goods) u Perfect Substitutes (Pepsi and Coke) u Perfect Complements (Shoes) Three Examples

u Two goods that are substituted at the constant rate u Example: Pepsi and Coke (I like soda but I cannot distinguish between the two kinds) Example: Perfect substitutes

Perfect Substitutes (Soda) Coke Pepsi U(x 1,x 2 ) =

Perfect Substitutes (Proportions) x 1 (6 pack) x 2 (1 can) 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