Preference Relations and Revealed Preference BINARY RELATIONS

For a given set X , let X x X denote the usud Cartesian p d a c t of all ordered pairs ( z , y ) , where both z and y are from X .

A binary relation B on the set X is formdly defined as a aubat of X x X - write B E X x X, and (t, y) E B if the o r d d pair ( z , y ) is in the relation B . Another, quicker way to m-te (z , y) E B is zBy , which can be read as ' z Bees y " or " z stands in the relation B to y." If (z,y) 4 B , I'll write "not zByn or tBy.

(8) Let X = {1,2,3) and B = {(1,1), (1921, (1,319 (2,319 (3,111. (b) Let X = all people in the world and I d B be the relation "&am at least one given name with.* (c) Let X = R (the red line, ranember) and let B be the nldian "greater or equal ton; that is, B =?. (d) Let X = R and let B be the relation: zBy if jz - yl> 1. (e) Let X = R and let B be the relation zBy if t - y is an integer multiple of 2.

There is a long list of properties that a given b i i d a t h might or might not have. The properties that will be importsat in this book are the following. A binary relation B on a set X is:

rt@xitx if zBz for all z E X; if z ~ z f o r d z E X; symmetric if zBy implies yBz; asymmetric if zBy implies y j z ; antisymmetric if zBy and yBz imply z = y ; tmnsitiue i f zBy and yBa imply zBz;

Refnence Rehations and Rrornlai P@ma 9

ncgaiidy imnsitm if zBY and yBz imply rib; complrtc or canccted if for all z, y E X, zBy or yBz (or both;

u n or s are never exclusive in this book unless s p d c d y mentioned); ureaklyumncctedifforall s,y E X , z = y or zBy or yBz; acyclic if zl Bz2, zzBzs,. . . z,-~ Bz, imply zl # z, .

Example (a) (above) is weakly connected, but nothing else. Exrunple (b) is reflexive and symmetric. Example (c) is dexive, antisymmet- ric, transitive, negatively transitive, complete, and w d y connected. Example (d) is idex ive , symmetric. Example (e) is reflexive, sym- metric, transitive.

PREFERENCE RELATIONS

In this section, we take up the following simple story. There is a set of items X , and Totrep ia willing to express his preferences among these items by making paired comparisons of the fonn: "I strictly prefer z to yn which is written z >- y. "Strict preferencen is a binary relation on X . Consider the following properties that this binary relation might possess:

(a) Asymndty - if z is strictly preferred to y, then y is not strictly preferred to z. (What do you think of this? Reasonable normatively? HOW about descriptively? Think of these questions for each of the following.)

(b) ~mifmrfmrfy - if z is strictly p r e f d to y and y is strictly pre- ferred to z , thexi z is strictly preferred to z .

(c) r+ti& - no z is strictly prefenwi to itself. (d) Nqatioe ttnnsitivity - if z is not strictly preferred to y and y is not strictly preferred to z, then z is not strictly preferred to z.

Negative transitivity ia a hard property to deal with intuitively in the form given, so let me develop an dtemative statement that is completely equivalent.

(2.1). A b i relation B is negstively transitive iff (if and only if) zBz implies that, for all y E X , zBy or yBz.

Pnwf. (Very pedantic.) The statement [M implies N] is the mne aa the statement [N or not M], thus [M implies N] is the uane as [not N implies not Mj. (The second equivalence is called contrapmition.) Thus [(zBz) implies (+By or yBz for aII y E X)] is the same as [{not(zBy or yBz for all y E X)) implies {not zBz)J which is [{there exists y E Z with ~8~ and y ~ t ) implies {zi)z)], whicb is negative transitivity.

Now bad; to F . Is negative transitivity reasonable? b it r e d to say that if z is strictly preferred to z, then for dl y either t irr h'ctly preferred to y or y is strictly preferred to z? As anormati* propaty, I think it is (barely) masonable. But as a descriptive property, 1 don't think it is reasonable. Suppose X = (0,oo) x (0, oo), whac t = (zl, t 2 ) E X is interpreted as the commodity b d e z l bottles of beer and 2 2 bottles of wine. Tot- (if his tastes are like my own) would certainly say that (10,lO) + (9,9). But -dm (15,6). Totrep might not be willing to say either that (10,lO) + (15,6) ap (15,6) + (9,9) - he might plead that comparisons d l e d for an too difficult for him to make.

Despite these difficulties with negative transitivity, it is standard to proceed assuming that + is asymmetric and negatively transitive.

Definition (2.2). A binary relation + on a set X is called a pnfnnrce relafion if it is asymmetric and negatively transitive.

Proposifia (23). If + is a preference relation, then >- is idexive, transitive, md acyclic.

Proof. (a) Asymmetry directly implies irdexivity. (b) Suppose z >- y and y + z. By negative transitivity (and La~ls (2.1)), z + y implies that either z >- y or t >- z . But z + y is impossible because y F z is assumed and + is asymmetric Thus z + z, which is transitivity; (c) If 21 >- t 2 , z ~ + 25,. . . , zn-1 + 2,. then by t-tivity + x,. Since >- is imflexive, this implies z l # z,. Thus + is acyclic.

When we are given a binary relation + that arprescs strict pnt- erence, we use it to define two other binary relations:

where w

e are using as shorthsnd for >= or for %

ot + ." The relation is d

ed

racllk pnfmne~, although it redly expresses the absence of

strict ptefetence The relation - is called intiifereme - it expr-

the absence of strict pteference in either direction, which is perhaps

not quite the same thing as active indifference.

Wf

Ml

(2.4). If + is a preference relation, then: (a) For d

2 and y, exactly one of z

+ y,y + z or z - y holds. (b) 2 is com

plete and tdtiv

e.

(c) - is reflexive, symm

etric, and transitive. (d) w

+z,z-y,y+

zimply

w+

y and r+

z.

(e) 2)-y ifTz+

yor z-y. (f) z

ky

and ytzim

ply

z-y.

Proof. (a) foII0w

s hm the definition of - and the fact that + is

asymm

etric. (b) By the asym

metry of +, either z

3 y or y 3 z (or both) for dl

z and y, thus 2 is complete. For transitivity of t

, note that this follow

s immediately from the negative transitivity of +.

(c) N is reflexive because + is irreilexive. - is sym

metric because the

definition of - is symm

etric. For transitivity, suppafe t

- y - z. Then z + y 3 z and z 3 y 3 t

.

By negative transitivity of +, z)(z+

z,or Z

-z. (d) If w

+ z - y, then by part (a) one of w + y or y - w

or y + w.

But y >- w

is impossible, since then trsnsitivity of + w

ould imply

y + 2. A

nd y - w is impossible, since then transitivity of - w

ould im

ply z - w

, cantradicting w + t. Thus w + y m

ust hold. The other part is pim

ilarIy done. (e) z 2 y iff y 3 z iff z + y or z - gt (by part (a)). (f) This is'im

mediate from the definitions of & and -.

Note w

ell the plot: Totrep expresses strict preferences, from

w

hich we define weak preferences and indifference. It is strict pref-

erence that is basic - Totrep is not being called upon to express any judgm

ents concerning weak preference or indifference, and he m

ight disagree w

ith our use of those terms to describe the negation of strict prdennce.

Another possible plot w

ould be to ask Totrep to ex

pm

weak

preferencar or preferepce or indifference. That is, the basic relation is

Prtfemuc Relations and Retm

lui Pqkencc 11

I 2

. This is a plot that is followed in many developm

ents of &dm the-

ory, ad

in the standard treatment it leads to the sam

e mathem

atical results:

Proposition 05). Given a binary relation 2' on a set X

, d&

taro new

binary relations hJ and -' from

2' by

Then if k' is complete and transitive, +' will be a preference relation.

Moreover, if w

e start with a binary relation k', define +' and -' aa

above from k', and then define k and - from

+I by

then 2' and w

ill agree, as will

and -. The proof is I&

as an exercise. So it doesn't matter w

hetha we ntart with a strict preference relation that is asym

metxic and negatively

transitive or with a weak preference relation that is com

plete and transitive - w

e end up in the same place. For reawns of intapretation

I prefer to take strict preference as being basic But it is a matter'of

personal taste, and most authors do it the other way.

REVEALED PREFERENCE THEORY

In the previous section, t6e story was that Totrep waa maU

q paired com

parisons between item

s in X. B

ut especially ftom a dt .

scriptive point of view, we would like to start w

ith an even more basic

concept - that of choices made rather than p

ref~~

llcc~

arpnsssd. That is, from

a descn'ptin point of view what we stx is an individ- ual's choice behavior - we have to connect that behavior as best we can w

ith his preferences which are neva directly e

xp

d.

The indi- vidual's choice behavior reveals his preferences, hence the nam

e of this subject: revealed preference thw

ry. This subject also hs, som nor-

mative justifications - taking preferences as given, how should choices

be made? B

ut this subject is of greatest interest from the d

dp

tive view

point.

To keep matters sim

ple, throughout this section I'll assume that

the choice set X is finite. E

epeddy if the application you are think- ing of is dem

and for consumption bundles or for any item

that is in- finitely divisible, this is not a very nice sim

plification. For nonempty

subsets of X, I'll

use notation such as A, B, etc.

The set of all nonem

pty subsets of X will be denoted P(X

).

DeFnition (2.6). A

choice fundia for a (finite) set X is a function

c : P(X) +

P(X

) such that for all A E X

, c(A) E A

.

The interpretation is: If Totrep is offered his choice of anything in the set A

, he says that any mem

ber of c(A) will do just h

e.

If Totrep's preferenas are given by the bi

i relation + (and by

the corresponding >- and -), it is natural to suppose that he &cm

s according to the d

e that from a set A

, anything that is undominufad

will be okay. In symbols, define a function c(-, +) : P(X

) by

It is dear that for any + , c(A, +) E A

, but it isn't dear whether

c(A, +

) # 0. Thus it isn't clear that c(-, +) is a choice function.

That will be som

ething to be investigated The otha questions to be looked at are:

(a) Fiom the norm

ative point of view: G

inn a relation >- (not nec- essarily a preference relation), w

hen is o(., +) a choice function? If + is a preference relation, w

hat propeaties does c(-, +) have? (b) R

om the descriptive point of view:

Givm

a choice function c, w

hen is there, a bi

i relation + such that c(-) =

c(-, +)? When is this b

ii

relation a pnference nlation? (NB

., this last question is the &

tit$ one, aa w

e're going to be building models w

here individuals are ausum

ed to be maximizing their prdermces according to some

preference relation.)

PIopwitia (2.7). It a binary relation + is acyclic, then c(., +) is a

choice function.

Proof. We need to show

that for A E P(X

), the set

is nonempty. Suppose it was em

pty - then for esch z E A then exids

a y E A such that y + z. Pick zl E A (A

is nonempty), and let

zl be zl 's "y ". Let z3 be zl's y ", and so on.

In 0th- words,

zl, 23, zj, . . . is a sequence of elemeats of A

where

... 2, + 2,-1 + ... + z1 +z1. Because A

is a finite set, there must exist some rn and n such that

z, =

z, and m

> n. But thie m

uld be a cyde, and + ia assum&

to be acyclic. The neassary contradiction is establishad.

Note the follow

ing instant oxolhy: If + is a prdercn~b A-

then <-,+) is a choice function. Also, we can strengthen (27) M

folhs.

Propwitia (2.8). For a binary &

ion +, c(., >-) is a choice function

ifT + is acyclic. Proving this is left ss an exercise.

Next we survey som

e properties of choice functions. Th dsb

sic axiomatic property of choice is H

outhakker's &om

of naacd

pnfereLlce.

Houfhakkds &om

(29). If z and y are both in A and B and if

z E c(A

) and y E dB), then z E c(B

).

In words, if z

is sometim

es chosen (&om A) w

hen y is availab15 then w

henever y is chosen and z is available, z is also chosen. H

outhskker's axiom is broken into two pieces by Sen:

Sen3 prop@ a (2.10). If z E B

G A and t E c(A

) , then z E c(B). Sen's paraphrase of this is: If the w

orld champion in som

e game C a

Pakistani, then he must slso be the cham

pion of Pakistan.

Sen's propnty @ (23U. If z, y E c(A), A

G B and y E c(B

), then z E c(B

).

Sen's paraphrase: If the world cham

pion in some game is s

Pakidmi,

then all champions (in this gam

e) of Pakistan are also world champi-

ons.

P

F R D- Y B ,.

- 8 .

CL 5' VJ' u * rn-

90 9 Y V

rn R ix 8

Pnfcnurc Relntions and Rmtnlcd Pnfam

u 17

A binary nIntion + is acydic if and only if

thncdtsachuicefundion c=c(.,+) for + w

hich implies that, but is not im

plied by e satisfics Sm

's a, and none of the previous three im

ply or are implied by

c safisfit5 Sm's p.

This is just an introduction to revealed prefmee theory - there

is a very large literature on the subject. Two im

portant questions, both related to the applications of these ideas to classical dem

and theory in m

icroemnom

ica, that we haven't discussed are:

What if X

is an infinite et? (Where did w

t use the finiteness of X in the developm

ent above?) For one approach to an answer to this question, see problem

8 of the next chapter.

Suppose we can't observe Totrep's choices from d

subsets of X . That

is, suppcwe c is defined only for a subset of P(X

). You can set? how this w

ould provide problems, especially if sets of the form

(2, y) are

not in the domain of c. A

nd you can see why this is a natural question

- espccidly if we have in m

ind a descriptive theory of choice. What

can be said in such caw?

PROBLEMS

(1) For eseh of the five examplea on page 7, show

that the b'

i

relation has precisely the list of properties that are &bed to it on

page 8 (froal the list of properties on pages 7 and 8).

(2) A binary &tion

E that b refiexive, symm

etric and transitive is called an q

uid

- relation. (For exam

ple, if >- is a preference relation, then - ia an equivalence relation; d

Pmposition (2.4)(c).)

Hen is an essy proof that if E

is symm

etric and transitive, then it is autom

aticdly dexive (thus reflexive could be deleted from

the list of properties): Fix = E

X and take some y such that zE

y. Then y

Et by symmetry, and hence xEz by transitivity.

Unfortuuately,

this easy proof is &m

. W

hy?

(3) Prove Proposition (2.5).

(4) Show that the properties (for a b

ii

relation) of asymm

etry ad

negative transitivity are independent. d

(5) Prow Proposition (2.8): For a binary relation + (on a finite set

X), c(., +

) is a choice function if and only if + is acyclic. In what sense is it im

portant here that X is finite?

(6) Prove that for any binary relation +, 4, +) dc&d

as in the display on page 12 satisfies Sen's a. (Is it im

portant here that X is finite? In w

hat sense is it important that + is acyclic, even though

you weren't told to assuxne acycliaty?)

(7) Give an exam

ple of a finite set X and an acyclic binary nlation

where c(-, +) does not satisfy Sen's property 8.

(8) In and around Propositions (2.13) and (2.14), I seem to get very

confused about whetha negative transitivity is dl I need to prow

H

outhskker's axiom, and thus asymmetry.

Unw

&

me. H

ave I im

plicitly used asymm

etry in the proof of (2.13). and if so, when? D

eal with the proof. I know (and so do you, if you did problem

(4)) that negative transitivity dasn't im

ply asymm

etry, and I don't want

an example of that - I w

ant to know what is going on in the pm

of.

(9) Give an exam

ple of a finite set X and a choice function q on

P(X

) that satisfies Sen's a but such that there is no binary &ion

+ such that c(., +) = c.