Lecture 1 1
1 Lecture 1
1.Choice Theory Possible observations (choices that people make)denoted x,y,z are members of a space X. X is a set of possible(mutually exclusive) alternatives from which an individual can choose. If there are L commodities then xcR. The space X(the consumption space)is determined by technological limits (MWG, pp.18-22). Budget sets will be subsets of X. Preferences will rank elements of X
2 1.Choice Theory • Possible observations (choices that people make) denoted x,y,z are members of a space X. • X is a set of possible (mutually exclusive) alternatives from which an individual can choose. • If there are L commodities then . • The space X (the consumption space) is determined by technological limits (MWG, pp.18-22). • Budget sets will be subsets of X. • Preferences will rank elements of X. L X R
1.Choice Theory (Continued) Preference relations. We start with x.yex,and denote a preference relation. ·Ne read x穰as x is at least as good as,or weakly preferred to,y. Strict preference relations xA implies x%x but not y%x. We read this one x is strongly preferred to y. Indifference relation x~ydefined as x%yand y%x.we read this one as x is indifferent to y
3 1.Choice Theory (Continued) • Preference relations. • We start with , and denote a preference relation . • We read as x is at least as good as, or weakly preferred to, y. • Strict preference relations implies but not . • We read this one x is strongly preferred to y. • Indifference relation defined as and . we read this one as x is indifferent to y. x y X , x y 穣 y · x y  x y % y x % x y ∼ x y % y x %
2.Rationality Definition 1.B.1:The preference relationship is rational if it possesses the following two properties: (i)Completeness:for all x.x,we have that xor y%x or both. ·(i)Transitivity:for all x,y,zeX,ifx穰y andy穰then x穰·
4 2.Rationality • Definition 1.B.1: The preference relationship is rational if it possesses the following two properties: • (i) Completeness: for all , we have that or or both. • (ii) Transitivity: for all , if and then . · x y X , x y 穣 y y x % x y z X , , x y 穣 y z 穣 x z 穣
2.Rationality (continued) Rationality implies(proposition 1.B.1) that: (i)A is both irreflexive (i.exA x never holds)and transitivexA(ie.yA and implies (ii)~is reflexive (i.e.x-x for all x), transitive and symmetric (i.e.xy impliesx). .(iii)if xA yand y%=then xA
5 2.Rationality (continued) • Rationality implies (proposition 1.B.1) that: • (i) is both irreflexive (i.e. never holds) and transitive (i.e. and implies ) • (ii) is reflexive (i.e. for all x), transitive and symmetric (i.e. implies ). • (iii) if and then .  x x  x y  y z  x z  ∼ x x x y ∼ y x ∼ x y  y z % x z Â
3.Weak Axiom of Revealed Preference B is a family of subsets,denoted B,of X,i.e. every element of B is a set BX. C(.)is a correspondence mapping B to X, such that C(B)B.It is a choice rule that chooses a set C(B)for every set B E B such that C(B)B,for all Be B. A choice structure is the combination: （B,C()》
6 3. Weak Axiom of Revealed Preference • is a family of subsets, denoted B, of X, i.e. every element of is a set B ⊆ X. • C(.) is a correspondence mapping to X, such that C(B) ⊆B . It is a choice rule that chooses a set C(B) for every set B ∈ , such that C(B) ⊆B , for all B∈ . • A choice structure is the combination: ( , C(.)). B B B B B B
3.WARP (continued) The choice structure(B.c())satisfies the weak axiom of revealed preference if the following property holds: ·If for some B.∈B with,yeB,we have x∈C(B), then for any geB withxyeB,and yec(B),we must also have x∈c(B')· The choice function is allowed to have more than one Element (i.e.such that c(B)B.VBEB If xec()then the agent must either prefer x to any other element in c()or the agent must be indifferent between x and any other element in c(B)
7 3. WARP (continued) • The choice structure satisfies the weak axiom of revealed preference if the following property holds: • If for some with , we have , • then for any with , and , we must also have . • # The choice function is allowed to have more than one Element (i.e. such that ) • # If then the agent must either prefer x to any other element in or the agent must be indifferent between x and any other element in (B ，C(•)) BB x y B , x B C( ) BB x y B , y C B ( ) x B C( ) C(B B B ) , B x C B ( ) C(B) C(B)
3.WARP (continued) ·A new definition: 。C(B,%)={x∈B:y∈B,x%y}the elements in(B,%) are the agent's preferred alternatives in B. Definition 1.D.1:The choice structure (B,c()) is rationalized by the preference relation relative to B if c(B)=c(B.%)for VBeB In other words the choice structure (B.c()is rationalized by the preference relation relative to B ifc()is a set of maximizers of over allx∈Bfor VBEB. It is possible to satisfy the weak axiom and yet for there not to exist a preference relation that seems to be maximized by the observed choices
8 3. WARP (continued) • A new definition: • the elements in are the agent’s preferred alternatives in B. • Definition 1.D.1: The choice structure is rationalized by the preference relation relative to if for . • In other words the choice structure is rationalized by the preference relation relative to if is a set of maximizers of over all for . • It is possible to satisfy the weak axiom and yet for there not to exist a preference relation that seems to be maximized by the observed choices. C B x B y B x y ( , : , ) % % = C B( , ) % (B ，C(•)) · B C B C B ( ) ( , ) = % B B (B ，C(•)) B · C B( ) · x B B B
3.WARP (continued) MWG Proposition 1.D.2: ·If(B,c()satisfies: (i)the weak axiom of revealed preference. (ii)includes all subsets of X with two or three elements, then there is a unique rationalizing preference relation,i.e.c(B)=c(8%) for VB∈B
9 3. WARP (continued) • MWG Proposition 1.D.2: • If satisfies: (i) the weak axiom of revealed preference, (ii) includes all subsets of X with two or three elements, • then there is a unique rationalizing preference relation, i.e. for . (B ，C(•)) B C B C B ( ) ( , ) = % B B
4.An Application Step 1:the first player puts forward an allocation of 10 dollars which can be any integer amount. 。 Step 2:the second player either accepts the allocation,in which case both players get the allotted money,or rejects the allocation in which case,neither player gets any money. 。 What would you predict should happen? ·What would“economics”predict should happen? What does happen is that: (1)most offers are either 4 or 5 dollars,and ·(②)those are accepted,and (3)offers that are three dollars or less,are rejected very frequently. 10
10 4.An Application • Step 1: the first player puts forward an allocation of 10 dollars which can be any integer amount. • Step 2: the second player either accepts the allocation, in which case both players get the allotted money, or rejects the allocation in which case, neither player gets any money. • What would you predict should happen? • What would “economics” predict should happen? • What does happen is that: • (1) most offers are either 4 or 5 dollars, and • (2) those are accepted, and • (3) offers that are three dollars or less, are rejected very frequently