L03 Utility

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

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

Indifference Curves x2 x1

Utility Functions Preferences satisfying Axioms (+) can be represented by a utility function. Utility function: formula that assigns a number (utility) for any bundle. Today: Geometric interpretation Utility function and Preferences Utility and Indifference curves Important examples

Utility function: Geometry x2 z All bundles in I1 are strictly preferred to all in I2. x1

Utility function: Geometry x2 z All bundles in I1 are strictly preferred to all in I2. x1

Utility function: Geometry x2 z All bundles in I1 are strictly preferred to all in I2. x1

Utility function: Geometry 5 x2 3 All bundles in I1 are strictly preferred to all in I2. z x1

Utility function: Geometry U(x1,x2) Utility 5 x2 3 All bundles in I1 are strictly preferred to all in I2. z x1

Utility Functions and Preferences A utility function U(x) represents preferences if x y U(x) ≥ U(y) x y x ~ y ~ f ~ f p

Usefulness of Utility Function Utility function U(x1,x2) = x1x2 (2,3), (4,1), (2,2) Quiz 1: U represents preferences A: B: C: D:

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

Indifference Curves U(x1,x2) = x1x2 x2 x1

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

Utility Functions p U=6 U=4 U=4 U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2). Define V = 5U. V(x1,x2) = 5x1x2 (2,3) (4,1) ~ (2,2). V preserves the same order as U and so represents the same preferences. p V= V= V=

Monotone Transformation U(x1,x2) = x1x2 V= 5U x2 x1

Theorem (Monotonic Transformation) T: Suppose that U is a utility function that represents some preferences f(U) is a strictly increasing function then V = f(U) represents the same preferences

Preference representations Utility U(x1,x2) = x1x2 Quiz 2: U(x1,x2) = x1 +x2 A: V = ln(x1 +x2)+5 B: V=5x1 +7x2 C: V=-2(x1 +x2) D: All of the above

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

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

Perfect Substitutes (Soda) Pepsi U(x1,x2) = Coke

Perfect Substitutes (Proportions) x2 (1 can) U(x1,x2) = x1 (6 pack)

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

Perfect Complements (Shoes) U(x1,x2) = L

Perfect Complements (Proportions) Coffee 2:1 U(x1,x2) = Sugar