Revealed Preferences Lecture 2, 2 September Econ 2100

Fall 2015

Outline 1

Weak Axiom of Revealed Preference

2

Sen’s

3

Equivalence between Axioms and Rationalizable Choices.

4

An Application: the Law of Compensated Demand.

and

Axioms.

De…nitions From Last Class

A preference relation % on X is a complete and transitive binary relation (weak order). describes DM’s taste over all possible pairs of options (these are hypothetical choices).

A choice rule for X is a correspondence C : 2X n f;g ! X such that C(A) A for all A X . Given a binary relation %, the induced choice rule C% is de…ned by C% (A) = fx 2 A : x % y for all y 2 Ag: A choice rule C is rationalized by % if C = C% and % is a preference relation. A choice rule C is rationalizable if there exists a % such that C = C% . Given a choice rule C, its revealed preference relation %C is de…ned by x %C y if there exists some A such that x; y 2 A and x 2 C (A): If x is chosen y is available, then x is revealed preferred to y .

Proposition If C is rationalized by %, then % = %C . Goal for Today: Assumptions on C that characterize being rationalizable. We will see two sets of equivalent conditions.

Weak Axiom of Revealed Preference Axiom (WARP) A choice rule for X satis…es the weak axiom of revealed preference x; y 2 A \ B, If x 2 C(A), and ) x 2 C(B): y 2 C(B) This is also known as Houthakker Axiom. If x could be chosen from A (when y was also available) and y could be chosen from B (when x was also available) then it must be that x could also be chosen from B. In other words, if x is was revealed at least as good as y , then y cannot be revealed strictly preferred to x.

Consequences of WARP WARP: If x; y 2 A \ B, x 2 C(A), and y 2 C(B),

then

x 2 C(B):

Exercise Verify that WARP is equivalent to the following: If A \ C(B) 6= ;,

then

C(A) \ B

C(B):

Question 6, Problem Set 1. Prove that if C is rationalizable, then it satis…es WARP. Exercise Suppose X = fa; b; cg and assume C (fa; bg) = fag, C (fb; cg) = fbg, and C (fa; cg) = fcg. Prove that if C is nonempty, then it must violate WARP. [Hint: Is there any value for C(fa; b; cg) which will work?]

Independence of Irrelevant Alternatives Axiom (Sen’s

or Independence of Irrelevant Alternatives (IIA))

A choice rule for X satis…es Sen’s x 2 B A, and If x 2 C(A),

then

x 2 C(B):

If x has been chosen from A then it must be chosen from any subset of A that contains it. Example: If Pitt is one of the best universities in the USA, then Pitt is one of the best universities in Pennsylvania. Exercise Verify that IIA is equivalent to the following: If B

A,

then

C(A) \ B

C(B):

Exercise Suppose that C satis…es IIA. Verify that if C(B) = B, then C(A) = A for all A

B.

Expansion Consistency Axiom (Sen’s ) A choice rule for X satis…es Sen’s x; y 2 C(A), If A B, and y 2 C(B),

then

x 2 C(B):

If x and y are chosen from some set, and y is also chosen from a larger set, then x must also be chosen from this larger set. Example: If Pitt and Penn State are the best universities in Pennsylvania, and Pitt is one of the best in the USA, then Penn State is also one of the best universities in the USA. Exercise Verify that Sen’s If A Sen’s

is equivalent to the following: B, and C(A) \ C(B) 6= ;,

then

C(A)

C(B):

is sometimes called ‘expansion consistency’because of this expression.

Sen’s Axioms, Nonempty Choice Rules, and WARP Sen’s axioms are not equivalent. Exercise Suppose X = fa; b; cg. Construct a non-empty choice rule that satis…es Sen’s Construct a non-empty choice rule that satis…es Sen’s

but violates Sen’s . but violates Sen’s .

If C is nonempty and satis…es WARP, then C satis…es Sen’s

and Sen’s .

Exercise Prove that if C is non-empty and satis…es WARP, then C satis…es Sen’s . Exercise Prove that if C is non-empty and satis…es WARP, then C satis…es Sen’s .

Sen’s Axioms, WARP, and Rationalizable Preferences Theorem Suppose C is nonempty. Then the following are equivalent: 1 2 3

C satis…es Sen’s

and ;

C satis…es WARP;

C is rationalizable. This gives necessary and su¢ cient conditions for a choice rule to look “as if” the decision maker is using a preference relation to generate her choice behavior via the induced choice rule. This preference relation is a revealed preference by last class’proposition stating that if a choice rule is rationalized by some %, then this preference is a revealed preference.

REMARK Rationality is equivalent to WARP; thus, one can verify whether or not DM is rational by verifying whether or not her choices obey WARP (or Sen’s axioms). Proof strategy: We will show (1) implies (2) implies (3) implies (1).

Sen’s Alpha and Beta Imply WARP

Proof. Suppose C satis…es Sen’s

and .

Assume x; y 2 A \ B, x 2 C(A), and y 2 C(B).

We need to show that x 2 C(B) to establish WARP. Since A \ B

A and x 2 C(A), Sen’s

Similarly, A \ B

implies x 2 C(A \ B).

B and y 2 C(B) imply y 2 C(A \ B).

Since x; y 2 C(A \ B) and y 2 C(B), Sen’s

implies x 2 C(B).

WARP Implies Rationalizable I Step 1: show that if WARP holds then %C is a preference (remember, x %C y if 9A s.t. x; y 2 A and x 2 C (A)). Proof. Show that %C is complete and transitive, so it is a preference order. Let x; y 2 X . Since C is nonempty, either x 2 C(fx; y g) or y 2 C(fx; y g). Then either x %C y or y %C x . This proves %C is complete.

For transitivity, suppose x %C y and y %C z. Need to show x %C z.

x %C y means there exist a menu Axy with x ; y 2 Axy and such that x 2 C(Axy ). y %C z means there also exists a menu Ayz with y ; z 2 Ayz and such that y 2 C(Ayz ). Since C(fx ; y ; zg) is nonempty, there are three cases: Case 1: x 2 C(fx ; y ; z g). Then we are done as x %C z . Case 2: y 2 C(fx ; y ; z g). Observe x ; y 2 fx ; y ; z g \ A xy , x 2 C(A xy ), and y 2 C(fx ; y ; z g). By WARP, we must have x 2 C(fx ; y ; z g) and we are done as x %C z . Case 3: z 2 C(fx ; y ; z g). Observe y ; z 2 A yz \ fx ; y ; z g, y 2 C(A yz ) and z 2 C(fx ; y ; z g). Then, WARP implies y 2 C(fx ; y ; z g). Now apply Case 2.

We have x %C z in all cases, thus %C is transitive.

WARP Implies Rationalizable II Step 2: show that if WARP holds then %C rationalizes C. Proof. We need to prove that C(A) = C%C (A) = fx 2 A : x %C y for all y 2 Ag First, show that C(A)

C%C (A):

Suppose x 2 C(A). Then for any y 2 A, x %C y , since x ; y 2 A. So C(A)

Now show that C%C (A)

C%C (A)

C(A).

Suppose x 2 C%C (A): for any y 2 A, there exists some Bxy such that x 2 C(Bxy ). Since C is nonempty, …x some z 2 C(A). WARP applied to x ; z 2 Bxz \ A, x 2 C(Bxz ), and z 2 C(A) delivers x 2 C(A). So C%C (A) C(A).

Since we have C(A) C(A) = C%C (A).

C%C (A) and C%C (A)

C(A); we conclude that

Rationalizable Implies Sen’s Axioms Proof. Since rationalizable means C = C% , we can assume that %=%C by last class result. First, show that Sen’s

holds.

Let x 2 B A and x 2 C(A), we need to prove that x 2 C(B ). For all y 2 B , x ; y 2 A and x 2 C(A), i.e. x %C y . Thus x 2 C%C (B ) = C(B ) and therefore Sen’s holds.

Now, show that Sen’s

holds.

Let x ; y 2 C(A), A B , and y 2 C(B ); we need to prove that x 2 C (B ) For all z 2 B , y %C z. Since y 2 C(A) A and x 2 C(A), x %C y . By transitivity, for all z 2 B , x %C z. Thus x 2 C%C (B ) = C (B ), proving Sen’s holds.

This concludes the proof of the Theorem: C satis…es Sen’s

and

, C satis…es WARP , C is rationalizable.

WARP and Classic Demand Theory This is the study of consumption bundles that maximize a consumer’s utility function subject to her budget constraint. In the next few weeks, we will do this using calculus, but some conclusions can be obtained by observing a consumer’s choices.

n goods: consumption x 2 X = Rn+ , prices p 2 Rn++ , and income w 2 R+ De…nition A Walrasian demand function maps price-wage pairs to consumption bundles: x : Rn++

R+ ! Rn+ such that p x (p; w )

w

x (p; w ) 2 Bp;w = fx 2 Rn : p x w and xi 0g: The de…nition assumes a unique choice ¤rom a given budget set (why?). Thus

Choice Over a Budget Set Since Bp;w represents the menus of consumption bundles a consumer can a¤ord; classic demand theory says x (p; w ) = C (Bp;w )

Properties of Demand Functions De…nition A Walrasian demand function is homogeneous of degree zero if x ( p; w ) = x (p; w ) for all

> 0:

In words: nominal price changes have no e¤ect on optimal consumption choices. De…nition A Walrasian demand function satis…es Full Expenditure if p x (p; w ) = w : The consumer spends all of her income (this is sometimes also called Walras’ Law for individuals).

Weak Axiom on Budget Sets De…nition A Walrasian demand function satis…es the weak axiom of revealed preference if p 0 x (p; w ) w 0 and ) p x (p 0 ; w 0 ) > w 0 0 x (p; w ) 6= x (p ; w ) for all pairs (p; w ) and (p 0 ; w 0 ).

Equivalently (recall that here the choice rule is single valued): C(Bp;w ) 2 Bp 0 ;w 0 and C(Bp 0 ;w 0 ) 6= C(Bp;w ) ) C(Bp 0 ;w 0 ) 62 Bp;w Suppose prices and income change from (p; w ) to (p 0 ; w 0 ). The old consumption bundle x (p; w ) is still a¤ordable since p 0 x (p; w ) w 0 ... ... yet the consumer changes her choice (x (p; w ) 6= x (p 0 ; w 0 )).

Then the new choice could not have been a¤ordable in the old situation, because otherwise she would have chosen x (p 0 ; w 0 ) while facing (p; w ).

Exercise Verify that the original WARP axiom, imposed on the limited choice data included in x , is equivalent to the stated weak axiom of revealed preference.

Weak Axiom and Compensated Demand De…nition The pairs (p; w ) and (p 0 ; w 0 ) are a compensated price change if p 0 x (p; w ) = w 0 : A compensated price change gives the consumer enough income so that at the new prices she can still purchase the bundle she chose before. Question 8, Problem Set 1. Suppose x is homogeneous of degree zero and satis…es Full Expenditure. Prove that if x satis…es WARP for all compensated price changes, i.e. p 0 x (p; w ) = w 0 and ) p x (p 0 ; w 0 ) > w 0 0 x (p; w ) 6= x (p ; w ) then it satis…es WARP for all price changes (even the uncompensated ones).

If the weak axiom of revealed preference holds for these special price changes, then it holds for all price changes.

Law of Compensated Demand Proposition (Law of Compensated Demand) Suppose x (p; w ) is homogeneous of degree zero and satis…es Full Expenditure. Then the weak axiom of revealed preferences is satis…ed if and only if, for any compensated price change (p; w ) and (p 0 ; w 0 ) with w 0 = p 0 x (p; w ), (p 0

p) [x (p 0 ; w 0 ) 0

x (p; w )]

0;

0

with strict inequality if x (p; w ) 6= x (p ; w ). Roughly, price and compensated demand move in opposite directions: if price goes up, demand goes down. Remarks This is true only for compensated demand. We will, in a few lectures, prove a di¤erential version of this result (negative semide…niteness of the Slutsky matrix) using calculus. The point here is to show that the law of compensated demand results from homogeneity of degree zero, full expenditure, and rationality (as de…ned by the weak axiom of revealed preference). The calclulus stu¤ is extraneous.

WARP implies the Law of Compensated Demand Proof. Suppose the weak axiom of revealed preference is satis…ed. The result is immediate if the demands are equal, so without loss of generality assume x (p; w ) 6= x (p 0 ; w 0 ). Since p 0 x (p; w ) = w 0 and w 0 = p 0 x (p 0 ; w 0 ) by Full Expenditure, we have p 0 [x (p 0 ; w 0 ) Since p

0

x (p; w )

x (p; w )] = 0:

w , the weak axiom implies p x (p 0 ; w 0 ) > w .

Thus: p [x (p 0 ; w 0 )

x (p; w )] > 0:

So:

(p 0

p) [x (p 0 ; w 0 )

p 0 [x (p 0 ; w 0 ) | {z

=0

x (p; w )]

=

p [x (p 0 ; w 0 ) | {z >0

< 0:

x (p; w )] }

x(p; w )] }

Law of Compensated Demand implies WARP Proof. By Question 8 in PS1, it su¢ ces to verify the weak axiom of revealed preference for compensated price changes. So, suppose p 0 x (p; w ) = w 0 and x (p; w ) 6= x (p 0 ; w 0 );

We need to show that p x (p 0 ; w 0 ) > w .

Then, the law of compensated demand states: 0 > (p 0

p) [x (p 0 ; w 0 )

= p 0 x (p 0 ; w 0 ) | {z } =w 0

=w

x (p; w )]

p 0 x (p; w ) p x (p 0 ; w 0 ) + p x (p; w ) | {z } | {z }

p x (p 0 ; w 0 ):

=w 0

=w

Social vs. Individual Choices Social choices may not seem rational even if all individuals in a society are rational. Individuals f1; 2; 3g have complete and transitive preferences over fa; b; cg as follows: a 1b 1c b 2a 2c : c 3a 3b Now suppose we let C(A) = fx 2 A : jfi : x 2 C%i (A)gj

jfi : y 2 C%i (A)j for all y 2 Ag:

In words, x is chosen by society if there is no alternative that would be chosen by strictly more individuals than x .

Then C(fa; b; cg) = fa; b; cg, but C(fa; bg) = fag.

So b is chosen from the larger menu, but not from the smaller, violating Sen’s . The outcome of a three-way choices may be di¤erent than the outcome of a two-way choice, but this violates IIA.

Econ 2100

Fall 2015

Problem Set 1 Due 9 September, Wednesday, at the beginning of class 1. Prove the following: if a binary relation R is asymmetric and negatively transitive, then it is transitive. 2. Let R be an equivalence relation on X. Prove the following: (a) 8x 2 X; x 2 [x];

(b) Given x; y 2 X, either [x] = [y] or [x] \ [y] = ;. 3. Prove that if % is a preference relation (i.e. it is complete and transitive), then: (a) - is a preference relation; (b)

is asymmetric and transitive;

(c)

is an equivalence relation;

(d) x % y and y

z imply x % z; and

(e) x % y and y

z imply x

z.

4. Prove that if % is a preference relation, then C% (A) 6= ; whenever A is …nite. 5. Prove that % is complete and acyclic (but not necessarily transitive) if and only if C% (A) 6= ; whenever A is …nite. 6. Prove that if C is rationalizable, then it satis…es WARP. 7. Suppose a choice correspondence satis…es the weak axiom of revealed preferences. Consider the following two possible revealed preferred relations. C and C de…ned as follows: x x (a) Show that (b) Is

C

C C C

y , there is some A such that x; y 2 A, x 2 C(A), and y 2 = C(A) y , x %C y but not y %C x

and

C

are the same (that is, show that x

C

y,x

C

y).

transitive? If not provide a counterexample. If yes, prove it.

8. De…ne a choice rule C as resolute if, for all A X, jC(A)j 1, i.e. the decision maker chooses a unique option from every menu. Assume C is nonempty. Prove or provide counterexamples to the following statements: (a) If C satis…es Sen’s

(b) If C satis…es Sen’s

and C is resolute, then C is rationalizable.

and its revealed relation %C is antisymmetric, then C is rationalizable.

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9. Suppose x is homogeneous of degree zero and satis…es Full Expenditure. Prove that if x satis…es the weak axiom of revealed preference for all compensated price changes (i.e. p0 x (p; w) = w0 and x (p; w) 6= x (p0 ; w0 ) ) p x (p0 ; w0 ) > w ), then it satis…es the weak axiom of revealed preference for all price changes (even the uncompensated ones). 10. You are given the following partial information about a consumer’s purchases in two di¤erent years. She consumes only two goods. Year 1 Year 2 Quantity Price Quantity Price Good 1 100 100 120 100 Good 2 100 100 ? 80 Over what range of quantities of good 2 consumed in year 2 would you conclude: (a) That her behavior is in contradiction with the weak axiom? (b) That the consumer’s consumption bundle in year 1 is revealed preferred to that in year 2? (c) That the consumer’s consumption bundle in year 2 is revealed preferred to that in year 1? (d) That there is insu¢ cient information to justify any of the previous answers?

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