Chapter 2Preference and Choice(高级微观经济学-上海财经大学,沈凌)

Chapter 2: Preference and Choice

? Two approaches to modeling individual choice behavior:

1. Preference-based approach: preference relation

2. Choice-based approach: revealed preference

? A set ( X ) of possible (mutually exclusive) alternatives from which the individual must choose.

2. A: Preference Relations

? Preference relations ? , which is a binary relation on the set of alternatives X We read x ? y as “x is at least as good as y”, where X y x ∈,.

1. x ? y but

not y ? x ? y x f 2. x ? y and y ? x ? x ~ y

? Definition 2.1: The preference relation ? is rational, if it satisfies:

Axiom 1: Completeness. For all X y x ∈,, we have that x ? y or y ? x or both. Axiom 2: Transitivity. For all X z y x ∈,,, if x ? y and y ? z, then x ? z. ? The individual can completely rank any elements in X.

However: Do we really know our own preferences?

Are our preferences really transitive?

Just perceptible differences, the framing problem, collective decision (Condorcet paradox)

2. B: Choice Rules

? A choice structure (B , C(?)), where B is a family of nonempty subsets of X, C(?) is a choice rule.

? The weak axiom of revealed preference : if for some B ∈B with B y x ∈, we have )(B C x ∈, then for any ∈'B B with ',B y x ∈ and )'(B C y ∈, we must also have )'(B C x ∈.

Example: {}z y x X ,,=, and B {}{}{}z y x y x ,,,,=

{}(){}{}(){}x z y x C x y x C C ==,,,,:111 yes {}(){}{}(){}y x z y x C x y x C C ,,,,,:222== no

2. C: The relationship between preference relations and choice rules

? Definition 2.2: Given a choice structure, the revealed preference relation ?* is defined by: x ?* y ? there is some B ∈B such that B y x ∈, and )(B C x ∈. ? Does the revealed preference satisfy Axiom 1 & 2 ?

? Definition 2.3: Given a rational preference, the generated choice structure is (B , C*(?, ?)) such that C*(B, ?) ={x ∈B: x ? y for every y ∈B}.

? Theorem 2.4: Suppose that ? is a rational preference relation. Then the choice structure generated by ?, (B , C*(?, ?)), satisfies the weak axiom.

Proof: Suppose that for some B ∈B we have B y x ∈, and ∈x C*(B, ?). By the definition of C*(B, ?), this implies x ? y .

Suppose that for some B’∈B with ',B y x ∈ we have ∈y C*(B’, ?). This implies that y ? z for all 'B z ∈. Hence, according to transitivity axiom, we have x ? z. We can conclude that ∈x C*(B’, ?). It is precisely the conclusion that the weak axiom demands.

? Theorem 2.5: If (B , C(?)) is a choice structure such that: (1) the weak axiom is satisfied, (2) B includes all subsets of X of up to three elements, then there is one and only one rational preference ? such that C(B)=C*(B, ?) for all B ∈B .

Proof ：it is natural to think that the revealed preference is the candidate. Now we need to proof: i) ?* is rational. ii) C(B)=C*(B, ?*). iii) ?* is the unique preference relation.

i) Completeness: By assumption (2) {}∈y x , B . Thus, either x or y must be an element of C({}y x ,), we must have x ?* y , or y ?* x , or both. Hence, it is complete. Transitivity: let x ?* y and y ?* z. We need to show x ?* z. Consider {}∈z y x ,,B . If {}()z y x C x ,,∈, then x ?* z. If {}()z y x C y ,,∈, then {}()z y x C x ,,∈ by the first assumption, i.e., the weak axiom is satisfied. Suppose {}()z y x C z ,,∈, since y ?* z, the weak axiom yields firstly {}()z y x C y ,,∈, and then {}()z y x C x ,,∈.

ii) C(B)=C*(B, ?*) says that the revealed preference ?* inferred from C(?) actually generates C(?). We show it in two steps. First, suppose that ()B C x ∈. Then x ?* y for all B y ∈. Hence, we have ()??≥∈,B C x . This means ()()??≥?,B C B C . Second, suppose ∈x C*(B, ?*), it implies that x ?* y for all B y ∈. So for each B y ∈, there must exist some set ∈y B B such that y B y x ∈, and ()y B C x ∈. Because ()Φ≠B C , the weak axiom then implies that ()B C x ∈. Hence, C*(B, ?*)()B C ?. Together, we have C*(B, ?*)()B C =.

iii) Uniqueness: simply note that B includes all two-elements subsets of X, the choice behavior in C(?) completely determines the pairwise preference relations. Hence, it is unique.