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

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Indifference Curves

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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 representation "mountain”–Utility function and Preferences–Utility function and Indifference curves–Utility function and MRS (next class)

Utility function: Geometry

x2

x1

z

Utility function: Geometry

x2

x1

z

Utility function: Geometry

x2

x1

z

Utility function: Geometry

x2

x1

z

Utility

3

5

Utility function: Geometry

x2

x1

z

Utility

3

5

U(x1,x2)

Utility Functions and Preferences

A utility function U(x) represents preferences if and only if:

x y U(x) ≥ U(y)

x y

x y

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Usefulness of Utility Function Utility function U(x1,x2) = x1x2

What can we say about preferences

(2,3), (4,1), (2,2), (1,1) , (8,8)

Recover preferences:

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

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U(x1,x2) = x1x2

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

U(x1,x2) = x1x2 (2,3) (4,1) (2,2). Define V = U2.

V(x1,x2) = x12x2

2 (2,3) (4,1) (2,2).

V preserves the same order as U and

so represents the same preferences.U=6 U=4 U=4

V= V= V=

Monotone Transformation

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U(x1,x2) = x1x2

V= U2

Theorem (Formal Claim) 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(x1,x2) = x1x2

Perfect Substitutes (Example: French and Dutch Cheese)

Perfect Complements (Right and Left shoe)

Well-behaved preferences (Ice cream and chocolate)

Three Examples

Two goods that are substituted at the constant rate

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(x1,x2) =

Perfect Substitutes (Proportions)

x1 Pack (6 slices)

x2 (1 Slice)

U(x1,x2) =

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

Perfect Complements (Shoes)

L

R

U(x1,x2) =

Perfect Complements (Proportions)

Sugar

Coffee

U(x1,x2) =

2:1