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Lecture 1
Chapter 1 Preference and Choice
We may study consumer choice behavior bad on rational preference or consistent choices. Preference relations
In the preference-bad approach, the objectives of the decision maker are summarized in a binary preference relation. A binary relation , R, on a t X , is simply a rule such that for each x and y in X , we can determine whether x R y, y R x, or neither, or both. Examples: “>”, “=”, “≤”, etc. Let G be a binary relation on a t X . We say that G is:
1. Complete iff: xGy X y x :),(∈∀ or yGx or both.
2. Reflexive iff: xGx X x :)(∈∀.
3. Irreflexive iff xGx X x ¬∈∀:)(.
4. Symmetric iff yGx xGy X y x ⇒∈∀:),(.
5. asymmetric iff yGx xGy X y x ¬⇒∈∀:),(.
6. transitive iff: xGy X z y x [:),,(∈∀ & xGz yGz ⇒]. A preference relation, which is often denoted by ;, is a binary relation on the t of alternatives X . We read x ;y as “x is at least as good as y .” We define “x y ;⇔x y ; & y x ¬;” and “y x ~⇔x y ; & y x ;.”
武装直升机怎么画Definition B1 A preference relation is rational if it is complete and transitive .
Proposition B1 If relation ; is rational, then (i) Relation ; is irreflexive and transitive; (ii) Relation ~ is reflexive, transitive and symmetric; (iii) If x y ;z ;, then x z ;.
Definition B2 A function :u X R → is a utility function reprenting preference relation ; if, for all ,x y X ∈, x y ;()()u x u y ⇔≥.
Proposition B2 If a preference relation can be reprented by a utility function, then the relation must be rational.
Not all rational preference relation can be reprented by a utility function, unless X is finite. Examples where transitivity violated: just perceptible differences; framing problem.
Choice Rules
In the choice approach to the theory of decision-making, choice behavior itlf is taken to be the primitive object of the theory. Choice behavior is reprented by choice structure (B , C (.)): (i) B is a family of nonempty subts of X , which is often referred as budget ts;
(ii) C (.) is a choice rule that assigns a nonempty t of chon elements C (B )⊂B , for every budget t ∈B B .
Definition C1 The choice structure (B , C(.)) satisfies the weak axiom of revealed preference if the following property holds: If B ∃∈B with ,x y B ∈ & ()x C B ∈, then 'B ∀∈B with ,'x y B ∈ & (')y C B ∈, we must have (')x C B ∈.
Definition C2 Suppo that a choice structure (B , C(.)) satisfies the weak axiom of revealed preference. We define the revealed preference relation *; by: *
x y ; if B ∃∈B such that ,x y B ∈ & ()x C B ∈.
烤鳕鱼的做法The weak axiom can be restated as: “If x is revealed at least as good as y , then y cannot be revealed preferred to x .” Note that the revealed preference relation may not be either complete or transitive.
The relationship between Preference Relations and Choice Rules
Two questions regarding the relationship between the two approaches are:
1. If a decision marker has a rational preference ordering, do her decisions when facing choices from budget ts in B necessarily generate a choices structure that satisfies the weak axiom?
小鸟学飞2. If an individual’s choice behavior for a family of budget ts B is captured by a choice structure (B , C(.)) satisfying the weak axiom, is there necessarily a rational preference relation that is
consistent with the choices? Suppo that an individual has a rational preference relation ; on X . If this individual faces a nonempty subt of alternatives B X ⊂, her preference-maximizing behavior is to choo any one of the elements in the t: *(,){:,}C B x B y B x y =∈∀∈;;. We only consider preference ; and families of budget ts B such that *(,)C B ; is nonempty for all B ∈B. We say that the rational preference ; generates choices structure (B , *(,)C B ;). Proposition D1 Suppo that ; is a rational preference relation. Then the choice structure generated by ;, (B , *(,)C B ;), satisfies the weak axiom.
Definition D1 Given a choice structure (B , C (.)), we say that rational preference relation ; rationalize
s C (.) relative to B if C (B )=*(,)C B ; for all B ∈B , that is, if ; generates the choice structure (B , C (.)).
骑操
Proposition D2 If (B , C (.)) is a choice structure such that (i) the weak axiom is satisfied, and (ii) B includes all subts of X of up to three elements, then there is a rational preference relation ; that rationalizes C (.) relative to B . Furthermore, this relational preference relation is the only preference relation that does so.
Chapter 2 Consumer Choice
Consider consumer demand in a market economy . The decision problem faced by the consumer in a market economy is to choo consumption levels of the various goods and rvices that are available for purcha in the market. We call the goods and rvices commodities . For simplicity, we assume that the number of commodities is finite and equal to L (indexed by l =1,…, L ).
A consumption t is a subt of the commodity space L R , denoted by L X R ⊂, who elements are the consumption bundles that the individual can conceivably consume given the physical constraints impod by his environment. For simplicity, we consider the simplest sort of
consumption t: {:0,1,...,}L L l X R x R x l L +==∈≥∀=. A feature of t L R +
is its convexity.
Much of the theory to be developed applies for general convex consumption ts as well as for L R +.意气奋发
In addition to the physical constraints, the consumer also faces an economic constraint: his consumption choice is limited to tho commodity bundles that he can afford. We assume that the prices of the commodities are publicly quoted and the consumers are price-takers.
Definition D1 The (competitive) budget t ,{:}L p w B x R p x w +=∈⋅≤ is the t of all feasible
consumption bundles for the consumer who faces market prices p and has wealth w .
Demand Function
A consumer’s Walrasian (or market) demand correspondence (,)x p w assigns a t of chon consumption bundles for each price-wealth pair (,)p w . When x(p,w) is single-valued, we call it a demand function .
Definition E1 The Walrasian demand correspondence (,)x p w is homogeneous of degree zero if (,)(,)x p w x p w αα= for any p, w and 0α>.
Definition E2 The Walrasian demand correspondence (,)x p w satisfies Walras’ law if for every太清洞
p>>0 and w>0, we have p x w ⋅= for
all (,)x x p w ∈ Comparative Statics
Wealth effects : For fixed prices p , the function of wealth (,)x p w is called the consumer’s Engel
function . A commodity l is normal at (p,w ) if (,)0l x p w w
∂≥∂; It is inferior at (p,w ) if (,)0l x p w w
∂<∂. If every commodity is normal at all (p,w ), then we say that the demand is normal . Price effects: Keeping wealth and other prices fixed, the locus of points demanded in L R + as we
range over all possible values of the price of a commodity is called an offer curve of the commodity. More generally, the derivative (,)l k
x p w p ∂∂ is known as the price effect of price k p on the demand for good l . Good l is a Giffen good at (p,w ) if (,)0l l
x p w p ∂>∂.
2x 2x
Income effect 1x Price effect 1x Proposition E1 If a Walrasian demand function (,)x p w is homogeneous of degree zero, then for any price vector p and wealth w , we have
1(,)(,)0L
l l k k k x p w x p w p w p w =∂∂+=∂∂∑ (or 1//0//L l l l l k k k x x x x p p w w =∂∂+=∂∂∑), for 1,...,l L =. Proposition E2 If demand function (,)x p w satisfies Walras’ law, then for any p and w , we have
1(,)(,)0L l l k l k x p w p x p w p =∂+=∂∑, for k=1, …, L, and ∑==∂∂L
l l l p w w p x 11),(. The Weak Axiom of Revealed Preference
锦句Assume that x (p,w ) is single-valued, homogeneous of degree zero, and satisfies Walras’s law. Definition F1 The Walrasian demand function (,)x p w satisfies the weak axiom of revealed preference if the following property holds for any two prices wealth situations (,)p w and (',')p w :
If (
',')p x p w w ⋅≤ and (',')(,)x p w x p w ≠, then '(,)'p x p w w ⋅>. (Or, if (',')(,)x p w x p w ≠, then either (',')p x p w w ⋅≤ or '(,)'p x p w w ⋅≤ is
untrue.) Lemma The Weak Axiom holds if and only if for any two prices wealth situations (,)p w and (',')p w : If (',')p x p w w ⋅= and (',')(,)x p w x p w ≠, then '(,)'p x p w w ⋅>.
Homework : Page 15-16, 1.B.5 and 1.D.3; Page 37, 2.E.5 and 2.F.3.
