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 ,,∈.