Cha p t e r 1 T h eory of C h oi ce

In th ese no tes, I w i l l s um m a rize th e b asi c id eas i n c ho ice t h e or y , w h ic h y ou m u st b e fa m iliar w ith f rom 1 4.12 1. I w ill d esc rib e th ree w a y s o f m o d e l in g i nd iv idu a l c ho ice, n a m e ly c h oice fu n ctio n , p refe ren ce, a n d u tilit y m a x im iza t ion . I w il l p r esen t t h e con - di ti ons under w h i c h o ne c a n u s e e a c h mo de l . O n e c a n al w a ys us e c hoi c e f unct i o ns i n m o delin g a d ecision m a k e r’s c ho ice a t a giv e n s itu a ti o n . I n o rd er to rep r esen t a c h o i ce fu n ction b y a c o m p l ete a nd tra n sitiv e prefer en ce rela tion , o ne m u st h a v e a n on -em p t y c h o i ce fu n c tio n th at sa tis fi es the w ea k a xio m of r e v e aled pr eferen ce. F i na l l y , a c om plete an d t ran s fe rab l e p re fere nc e r elat io n c a n b e re pre sen te d b y a utilit y f un ctio n, a s lo ng a s it co n t in uo u s .

1. 1 A l t ernat i v e s

C o nsid er a set X o f a l tern ativ es. A l tern ativ es ar e m utu a l l y e x c lusiv e in th e s ense tha t on e c ann o t c h o ose t w o d i stinct al tern ativ es at th e s am e t im e. T a k e also th e s et of feasib le altern ativ es exh a u s tiv e so th at a d ecision m a k e r’s c ho i c es w ill alw a ys b e d e fi ned. 1

1 N o te that this is a m at ter o f m o d eling. F o r i nstance, if w e ha v e opt i ons C o ff ee and T e a , w e d e fi ne alternativ es as C = Co ff ee but no T e a, T = T e a but no Co ff ee, CT = Co ff ee and T ea, a nd NT = no Co ff ee and n o T ea.

3

1. 2 C hoi c e

W h ile X con s ists o f all p o ssible a ltern a ti v es, som e of these a l t ern a tiv e s m a y n o t b e feasib le for t he decision m a k e r. H e is co nstrain e d t o c ho o s e f rom a set A ⊂ X . A c h o i c e fu n ction d escrib e s w ha t a d ecisio n m a k e r w o u ld h a v e c h o sen u n d e r v ario us h y p o theti c al co nstrain t s.

De fi ni t i on 1 A ch o i c e f u n c t i o n i s a m appi ng c : 2 X \ { ∅ } → 2 X \ { ∅ } su ch that c ( A ) ⊆

A fo r a l l A ⊆ X .

He re , c ( A ) i s mea n t t o b e t he s e t o f a l l al t e r na ti v e s t hat t he de ci s i on ma k e r may ch o o s e f r o m A . H is act u a l c h o i ce w ill b e a s in gle a lter na tiv e w i th in c ( A ) . N o t e t h a t c ( A ) i s no n- e m pt y b y d e fi ni t i on. I n c anoni c al mo de l s , i t i s a l s o a s s u med t hat t he c h oi ce fu n ction sa tis fi es t h e f ol l o wi ng as s u m p ti on.

A x i o m 1 ( W eak A xi om of R e v e al ed P r ef erenc e s) Fo r a n y A, B x, y ∈ A ∩ B , i f x ∈ c ( A ) an d y ∈ c ( B ) , t h e n x ∈ c ( B ) .

⊆ X and a ny

T h e W ea k A xiom o f R e v e aled P r eferences s ta tes t h a t i f x i s c h osen in the p resen ce o f y ( s o t h a t i t i s r e v e a l e d t h a t x is at lea s t a s g o o d a s y ), th en w h enev er y is c h o sen i n th e p r esen ce o f x , x cou l d h a v e b een c h osen, t o o . T his a xio m em b o d i es t w o a ssu m p tion s. F i rst, the c h o ice i s a resu lt of b i na ry com p a r ison . S eco nd , t he u n d erlyi n g p referen c e u sed i n t h e c o m p a r i s o n i s n o t a ff ected b y t h e set A of a v ailab l e a lternativ e s. ( F or ex am p l e, th e d ecisio n m a k e r d o e s n ot lea r n f ro m t h e a v aila ble c h o ice s .)

1. 3 P ref e rence

A re l a t i o n (on X ) i s a sub set o f X × X . A r e l a t i o n º is said to b e co m p l e t e if a n d o n l y if, g iv en a n y x, y ∈ X , e i t h e r x º y or y º x . A r e l a t i o n º is sa id to b e tr a n sitive if a n d on ly if, g iv en an y x, y , z ∈ X ,

[ x º y an d y º z ] ⇒ x º z .

De fi ni t i on 2 A r e l a t i o n i s a pr eferen ce relatio n i f and o nl y i f i t i s c om pl et e a n d t r an - sitive .

G i v e n a n y p r eferen ce relatio n º , t h e strict pr efer en c e  is d e fi ned b y

x  y ⇐⇒ [ x º y and y º / x ] ,

an d t h e in d i ff er en c e ∼ is d e fi ned b y

x ∼ y ⇐⇒ [ x º y and y º x ] .

He re , x º y m e ans t ha t t he de ci s i on mak e r fi nds x at least a s g o o d as y ; x  y mea n s t ha t t he de ci s i on mak e r fi nds x strictly b e tter t h a n y , a n d x ∼ y mea n s t ha t t he d ecision m a k e r i s i nd i ff eren t b et w e en x an d y .

N o w , co nsid er a d ecision m a k er w h o c ho o ses a b est a l tern a ti v e a cco rdin g t o a pref - erence rel a ti on º wi t h i n e a c h se t A ⊆ X of a v aila ble a lter na tiv e s . H i s c h o ice f u n ction c º is giv e n b y

c º ( A ) = { x ∈ A | x º y ∀ y ∈ A } ¡ ∀ A ∈ 2 X \ { ∅ } ¢ .

A n im p o rtan t q u estion i s w hic h c h oice f u nction s c an b e th ou gh t o f c om i n g f rom s uc h a

de c i s i o n m a k e r. Thi s i s f o rm ul a t ed i n t h e f ol l o wi ng de fi ni ti on.

De fi ni t i on 3 A c h o ic e f u n c tio n c i s rep r esen ted b y º i ff c = c º .

R e p r esen t atio n b y a p r eference rela tion º m e an s t h a t d e c ision m a k er ’s c h oic e s a re m a d e a s if h e tries t o c ho o s e a b est a v ailab l e e lem e n t acc o rd ing t o º . I t t ur ns out t hat th e w eak a xiom of rev e aled p r eferen ces i s e quiv a l en t t o s uc h a rep r esen ta tion .

The o rem 1 A ssu m e that X is fi n i te. A ch o i c e fu n c tio n c is r e p r e s en te d b y s o m e p r e f - er e n c e r e la tio n º if a n d o n l y i f c satis fi e s we ak axi o m o f r ev e a l e d p r e f e r e nc e s .

It is a u s e fu l e x e r c ise t o s h o w t h a t i f c is rep r esen ted b y so m e prefer en ce relation º , th en it sa tis fi es A x i o m 1 . F or the c on v e rse, de fi ne º c by x º c y ⇐⇒ x ∈ c ( { x, y } ) . U n d e r A xio m 1 , it is a n o t h e r u sefu l exer cise to sho w tha t c = c º c .

1. 4 U t i l i t y

A r e l a t i o n º ca n b e re p r e s e n t e d b y a u tilit y f u n c t ion U : X → R in th e f o l l o w i n g sen se:

x º y ⇐⇒ U ( x ) ≥ U ( y ) ∀ x, y ∈ X. (O R )

T h e f o l lo w i n g theorem s tates t ha t a relatio n n eed s t o b e a pr eferen ce rela tion in o r d e r to b e re pre sen te d b y a utilit y f u n c t io n .

The o rem 2 A ssu m e tha t X is fi n i t e (or c ou nt abl e ). A r el ati o n c an b e pr esente d b y a u t ility f u n c tio n i n t h e s e n s e o f ( O R ) i f a n d o n ly if it is c o m p le te a n d t r a n s itive. M o r e o v er ,

if U : X → R r e pr esen ts º , a n d i f f : R → R is a s trictly i n c r e a s in g f u n c t io n , th e n

f ◦ U al s o r e pr es e n t s º .

B y the l ast state m e n t , w e c all s u c h u tilit y r ep rese n t at io ns or dinal . T o p r o v e t h i s result f o r fi nite X , d e fi ne U ( x ) = # { y ∈ X | x º y } an d c h e c k that U repr esen ts º whe n º is com p le te an d t ran s itiv e. W e ar e m a i n l y i n t ere s ted i n d e c ision u n d e r u nc erta in t y . I n th at case, the n a t ura l set of alter n a t iv es (e.g. t h e set o f a l l p o ssible l o tteries) is in fi ni t e . Wh e n X is in fi nite, o ne also n e eds t o i m p o s e a co n t in uit y assu m p tion .

De fi ni t i on 4 A p r e fer e n c e r ela t io n º is sa id to b e co n t i n uo us i f and o nl y i f t he upp e r- an d l ow er-c ontour set s { y | y º x } and { y | x º y } ar e c l o se d f or every x ∈ X .

x

T h at is, t h e w e a k p r eferen ce is a l w a y s p r ese rv e d in t h e lim it. T h e m a in re sult in th is l e c t u r e i s t h a t c o n t i n u o u s p r e f e r e n c e r e l a t i o ns can b e r epresen ted b y ( co n t in u o u s ) u tili t y fu n ction s:

The o rem 3 A ssu m e that X i s a c omp a ct, c onvex s ubs e t o f a s e p a r a bl e m et ri c s p a c e . A pr efer en c e r e l a tion º c a n b e r ep r e sen t e d by a c o n tin u o u s u t ility fu n c tio n U : X → R in t h e s ense of (OR ) i f and o nl y i f º is c o n tin u o u s .

T h is resu l t is a g en era lizatio n o f w ell-kno w n r esu lts b y W old ( 1 9 4 3 ), D e br eu (1 95 4), an d A rro w - H a h n (1971). O n e can e asily (i.e. y ou sh ou ld) c h e c k th at i f º is rep r esen ted by a c o n t i nu o u s u t i l i t y f u n c t i o n U : X → R i n th e sen se of (O R ) , t hen º i s c o n t i n uo us . Y o u m u s t h a v e seen th e p ro of o f th e c on v e rse f or th e s p e cial ca se co nsid ered b y D e b r eu.

A s a n exercise, sho w tha t lexicog r ap hic p referen c e r elatio n c an no t b e r ep resen t ed b y an y u t i lit y f u n c tio n (if y ou do n’t r em em b e r f rom 1 4.12 1). F ind a lso a d isc o n t in u o u s

p r eferen ce rel a tion tha t i s rep r esen ted b y a d i sco n t in uo u s u t ili t y fu nction . H en ce, c on ti - n u it y o f p references is not s up er fl u o u s for o rdin al repr esen ta tion , b u t it is n o t n ecessa r y , eith er. 2

T w o p r o p e rties o f c on ti n u o u s p refer e nces w i ll b e u seful in th e sequ e l. F o r a n y x ∈ X ,

de fi ne t h e i ndi ff eren ce set b y

I ( x ) = { y | x ∼ y } .

The fi rst p rop e rt y i s t hat I ( x ) is a c l o sed set (b eca use I ( x ) = { y | y º x } ∩ { y | x º y } ).

T h e s econd p rop e rt y i s t hat I ( x ) in tersects a n y con t i n uo us pa th th at con n ects a s u p er i o r altern ativ e t o a n i n f erior o ne:

Lem m a 1 Ta k e a n y x 0 , x 00 ∈ X wi th x 0  x  x 00 a n d a n y c o n tin u o u s m a p p in g φ : [0 , 1] → X w ith φ (1) = x 0 and φ (0) = x 00 . T h e n , t h e r e e x i s t s t ∈ [0 , 1] such t h at φ ( t ) ∈ I ( x ) .

T h is im m e d i a t ely f o llo w s fro m T h eor e m 3 an d t h e in ter m ed iate v a lue t he ore m . ( B y The o r e m 3 , U ◦ φ i s con t in uous, a nd U ( φ (1)) = U ( x 0 ) > U ( x ) > U ( x 00 ) = U ( φ (0)) .) N o rm ally , t h i s f a c t i s p ro v e d a s a m a in step to w a r d s p ro vin g T h eorem 3 , a s y o u m a y rem e m b er fro m th e e arlier c l a sses.

2 Some form of coun tabili t y /con tin u it y i s n ecessary for r epresen t abilit y . X m u st b e separable w ith resp ect to the o rder top o logy of º , i.e., it m u st con t ain a coun table s ubset that is dense w ith r esp e ct to the o rder top o logy . ( See T heorem 3.5 i n K reps, 1988.)

M IT OpenCourseWare http://ocw.mit.edu

1 4.123 Microeconomic Theory III

Spring 2010

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