Choice Theory – A Synopsis

14.123 Microeconomic Theory III Muha met Yildiz

Road map

1. Basic Concepts:

1. Choi ce

2. Preference

3. Utility

2. Weak Axiom of Revealed Preferences

3. Preference as a representation of choice

4. Ordinal Utility Representation

5. Co ntin uit y

Basic Concepts

 X = Set of Alternatives

 Mutually exc l usive

 Exhaustive

 A = non-empty set of available alternatives

 Choi ce Function: c : A ↦ c ( A )  A .

 c ( A ) is non-empt y

 Preference: A relation ≽ on X that is

 complete :  x, y  X , eith er x ≽ y o r y ≽ x;

 transitive :  x,y,z  X, [ x ≽ y and y ≽ z]  x ≽ z.

 Utility Function: U : X → R

Choice Function

A

c

c ( A )

 It describes what alternatives DM may choose under each set of constraints

 Feasibility: c ( A )  A .

 Exhaustive: c(A) is non-empty

 Mutuall y excl usi v e: only one alternative is chosen

Preference

 Preference Relation: A relation ≽ on X s.t.

 complete :  x,y  X , either x ≽ y or y ≽ x ;

 transitive :  x,y,z  X , [ x ≽ y and y ≽ z ]  x ≽ z .

 x ≽ y means: DM finds x at least as good as y

 Preferences do not depend on A !

 Strict Preference: x ≻ y  [ x ≽ y and not y ≽ x ]

 Indifference: x ~ y  [ x ≽ y and y ≽ x ].

 Choice induced by preference:

c ≽ ( A ) = {x  A| x ≽ y  y  A}

Weak Axiom of Revealed Preference

Axiom (WARP): For all A , B  X and x , y  A ∩ B , if x  c ( A ) and y  c ( B ), the n x  c ( B ).

 WARP: DM has well-defined preferences

 That govern the choice

 don’t depend on the set A of feasible alternatives

Choice v. Preference

Definition: A choice function c is represented by

≽ iff c = c ≽ .

Theorem: Assume that X is finite. A choice function c is represented by some preference relation ≽ if and only if c satisfies WARP.

Ordinal Utility Representation

Ordinal Representation: U : X → R is an ordinal representation of ≽ iff:

x ≽ y  U ( x ) ≥ U ( y )  x , y  X.

Fa ct: If U represents ≽ and f : R → R is stri ctly increasing, then f ◦ U repres ents ≽ .

Theor em: Assume X is finite (or countabl e). A relation has an or dinal repr esentation if and only if it is complete and transitive.

Example: Lexi cographic preference relation on unit square does not have an ordinal representation.

Continuous Representation

Definition: A preference relation ≽ is said to be continuous iff { y | y ≽ x } and { y | x ≽ y } are closed for every x in X .

Theorem: Assume X is a compact, convex subset of a separable metric space. A preference relation has an ordinal representation if and only if it is continuous.

Indifference Sets of a Continuous Preference

 I (x) = { y | x ~ y }

 I ( x ) is closed.

 If

 x ′ ≻ x ≻ x ′′

 φ :[0,1] → X continuous

 φ (1)= x ′ ; φ (0 ) = x ′′ ,

 Then,  t  [0,1] such that φ ( t ) ~ x .

x