Cha p t e r 5

St o c hast i c D o m i nance

In th is lectu r e, I w i l l i n t ro d u ce n o tion s o f s to c h astic d om ina n ce th at a l lo w o ne to d e - ter m in e t h e pr efe r en ce o f a n ex p ect ed u t ilit y m ax im iz er b e t w een som e lot t eries w ith m i n i m a l k no w l ed ge of the d e c ision m a k er ’s u t ilit y f un ctio n.

A s in th e p reviou s l ectu r e, tak e X = R as t h e s e t of w e al t h l e v e l a nd l e t u be t h e d ecision m a k e r’s u tili t y fun c tio n . A ssu m e tha t u is w e a k ly inc r ea sing . T he lotte ries ar e d istrib uted b y th eir c u m ula t i v e d istrib u t ion f u n c tion s. D esign ate F and G generic di st ri but i o n f u nc t i ons . I w i l l a s s u me t h ro ug ho ut that F and G ha v e a b ounded s u pp ort [ a, b ] wi t h F ( a ) = G ( a ) = 0 and F ( b ) = G ( b ) = 1 .

I w ill de fi ne t w o n ot i o ns o f s t o c has t i c do m i nance :

1. F i rst-or der s t o chast i c dom i nanc e: whe n a l ot t e ry F do m i nat e s G in t h e s e n s e o f fi rst-o r der s to c h a s tic d om in an ce, t he d ecisio n m a k e r p refer s F to G re gardl e s s of wha t u i s , a s l o n g a s i t i s w e a k l y i n c r e a s i n g .

2. S e c o nd-or d er st o c hast i c dom i n anc e : whe n a l o t te r y F dom i nates G in th e sen se o f secon d - ord e r sto c h astic d om ina n ce, th e d ecision m a k e r p refers F to G as long as h e is risk a v erse a n d u is w e ak ly in crea sing .

5 . 1 F ir st -o r d e r S t o c h a s t ic D o m i n a n c e

De fi ni ti on 12 F o r a n y lo tte ries F and G , F fi rst-or der s to c h a s tically d o m i n a tes G if an d o nly i f

F ( x ) ≤ G ( x ) ( ∀ x ) .

43

44 CHAP TER 5 . S TOCHAS TI C D OM I NANCE

N o te th at fi rs t- o r de r s to c h a s t i c d o m i n a n c e c a n b e o bt ai ne d b y t rans f e rri n g p roba - b i lit y w eig h ts u p w a rd s. F o r a n illu s tra t ion , a ssu m e th at F and G are c on t i n u ous a nd strictl y i n creasin g o n [ a, b ] . S upp o s e t ha t l ot t e ry x is distrib u t ed w i th G . F or ev ery rea lizat io n x , l et us g i v e th e d eci s i o n m ak er in stea d

y ( x ) = F − 1 ( G ( x )) .

Wh e n F fi rst-ord e r s to c h a s tica l l y d o m in ates G (i.e. F ( x ) ≤ G ( x ) ) w e w ou ld b e givin g hi m m ore t han x a t ev ery r ealizati o n :

y ( x ) = F − 1 ( G ( x )) ≥ x.

H e nc e , unde r t he ne w s c h e m e , he w o ul d b e g e t t i ng e x t r a w e a l t h F − 1 ( G ( x )) − x ≥ 0 at ev ery r e a liza t ion . B u t t he new l o t ter y y i s di s t ri but e d b y F : f or an y y ¯ ,

Pr ( y ( x ) ≤ y ¯ ) = P r ¡ x ≤ y − 1 ( y ¯ ) ¢

= G ¡ y − 1 ( y ¯ ) ¢

= G ¡ G − 1 F ( y ¯ ) ¢

= F ( y ¯ ) ,

whe r e t he fi r s t e q u a lit y i s t h e fa ct th at y is increa si n g , t he secon d equa li t y is b y th e fa ct th a t x is distrib u t e d b y G , a nd t h e t hi rd e q ua l i t y i s b y d e fi nition of y ( y − 1 ( y ¯ ) = G − 1 F ( y ¯ ) ).

A s lon g as th e d ecisi o n m ak er prefers h a v ing m ore w ealth t o l ess (i. e . u is in cre a sin g ), he w o ul d p ref e r t o h a v e t he l a t t e r s c h eme y ( x ) , w hi c h i s di s t ri but e d b y F , r a t h e r t h a n x , w hi c h i s di s t ri but e d b y G . T h e n e xt result states this form ally .

The o rem 7 F o r a n y lo tter ies F and G , F fi rs t - or de r s t o c h as t i c a l l y d omi n at e s G if an d

on l y if t h e d e c i s ion m aker we akl y pr ef ers F to G u n d e r e v e r y w e a k ly in cr e a sin g u t ility fu n c tio n u , i.e., R u ( x ) dF ≥ R u ( x ) dG .

Pr o o f . ( ⇐ ) S u p p o s e F do e s no t fi rst-o r d e r s to c h a s tica ll y d om ina t e G . T he n, there exists x suc h tha t F ( x ) > G ( x ) . D e fi ne u ≡ 1 { x>x ∗ } by u ( x ) = 1 if x > x an d 0 otherw ise. C l ea rl y ,

Z u ( x ) dF = 1 − F ( x ∗ ) < 1 − G ( x ∗ ) = Z

u ( x ) dG.

5 . 2 . S E COND- O RDE R S T OCHAS T I C DOMI NANCE 45

( ⇒ ) I w i l l pro v e t hi s p a r t under t h e a s s u mpt i o n that F an d G are c on t i n u ous a nd strictl y in creasin g o n [ a, b ] . I n t ha t c a se, as w e h a v e seen ab o v e,

Z u ( y ( x )) dF ( y ( x )) = Z

u ( y ( x )) dG ( x ) ≥ Z

u ( x ) dG ( x ) ,

w h ere t h e equ a lit y b y y ( x ) = F − 1 ( G ( x )) and t he i n e q ual i t y i s b y t h e f a c t t hat u ( y ( x )) ≥

u ( x ) for e v e ry x , w h i c h is tru e b eca use y ( x ) ≥ x an d u is w e a k ly inc r eas i n g .

5. 2 S econd- order S to c h as ti c D om i n anc e

N o w a ssu m e th at F an d G ha v e the s ame m ean, s o that one d o e s n ot domi nat e t h e other i n t he sense o f fi rs t - orde r s to c h as t i c d om i n ance . C an w e s t i l l s a y t hat a ri s k - a v e rs e d ecision m a k e r p refers F to G wi t h out k no wi ng hi s u t i l i t y f unc t i o n u ? I n t u i tiv e ly th is w o ul d b e t he ca s e a s l o ng a s F in v o lv e s less risk tha n G . I w ill n e x t fo rm alize t h i s i de a, an d t h i s w i l l l ead t o t h e notion of secon d-ord e r s to c h astic d om inan ce.

De fi ni ti on 13 F o r a n y lo tte ries F and G , F s e c o nd- o r de r s to c h a s t i ca l l y do m i nat e s G i f and o nl y i f t he de ci s i on mak e r we ak l y pr e f ers F to G und er every w e a k l y in cr e a sin g c o n c a v e u tility fu n c tio n u .

Thi s d e fi n i tion is dire ctly g iv e n in ter m s o f t h e fi na l g oa l. I w ill n e x t g iv e a no th er equ ilv alen t d e fi ni ti on, w hi c h f o rm a l i z e s the i de a t hat G is risk ier t ha n F .

De fi ni ti on 14 F o r a n y lo tter i es F and G , G is a m e an-preserving s pread of F if a n d on l y i f

y = x + ε

fo r s o m e x ∼ F , y ∼ G an d ε such t h at E ( ε | x ) = 0 for a l l x .

Im a g in e t h a t f o r ev ery r e a liza t ion x , w e a d d a n o i s e ε a n d g iv e d ecisio n m a k e r y = x + ε . S i n c e E ( ε | x ) = 0 , t hi s o nl y m ak es t h e c o n s u mpt i o n ri s k i e r w i t ho ut i m pro v i n g i t s exp ectation . In oth e r w ord s , w e a r e spr e ad ing t h e pr ob ab il ities w ith o ut c h an gin g th e m e an. I f t he deci sion m a k e r i s r isk a v e rse, he w o uld n ot lik e t o h a v e t his s c h em e. H e w o ul d r a t he r c ons u m e x . I n d eed , t his w ill b e th e c a se. B e fo re sta t ing t h i s f or m a l l y , it is i n stru ctiv e t o c o m p a r e this sc hem e to th e o n e in th e fi rst-or der s to c h asti c d o m ina n ce.

46 CHAP TER 5 . S TOCHAS TI C D OM I NANCE

In tha t case, w e w ere g i v i n g h im a n extra a m o un t c o n su m p tion at ev ery r ealiza t ion x . W h ile th i s cou l d i ncrea s e t he v a rian ce o f th e c on su m p tion , t h e decision m a k e r k n e w th at he w a s g etti n g if an yth i ng m o re. H e lik ed tha t sc h e m e . H ere, w e are i ncrea s i n g t h e v a ri ance wi t h out i ncre as i n g t he e x p e c t at i o n. H e c a n g ai n o r l os s b y t he c h ange . B e i ng risk a v erse, h e d o es n o t lik e t he c h a n g e .

The o rem 8 A ssu m e th at R xdF = R xd G . T he f o l l o w i n g a r e e q ui v a l e nt .

1. u ( x ) dF ( x ) ≥ u ( x ) dG ( x ) fo r e ver y w e a k l y i n c r e a s in g c o n c a v e u t ility fu n c tio n

u .

2. G i s a m e a n- pr es ervi ng s p r e ad of F .

3 . F o r every t ≥ 0 , R t G ( x ) dx ≥ R t F ( x ) dx .

Pr o o f . I fi rs t s ho w t ha t 2 i m pl i e s 1 . U nder 2 , w e c a n w ri t e

Z	u ( y ) dG ( y ) =	Z	E [ u ( x + ε ) | x ] dF ( x )
		Z
	≤		u ( E [ x + ε | x ]) dF ( x )
	=	Z	u ( x ) dF ( x ) ,

ob ta inin g 1 . H er e, th e fi rst e qua l i t y i s b y t he l a w o f i tera ted e xp ecta tion s a n d b y th e assum p tio n th at y = x + ε , t he in eq u a lit y i s b y J en sen ’s in eq u a lit y ( a s u is con c a v e), an d t h e last equa li t y b y th e a ssum p ti o n tha t E [ ε | x ] = 0 .

T o sh o w th at 1 i s e qu iv alen t t o 3 , d e fi ne m a pping I : R → R by I ( t ) = t [ F ( x ) − G ( x )] dx .

Cl e a r l y , I ( a ) = 0 . S i n c e F and G ha v e th e s a m e m ea n , i t i s also tru e th at I ( b ) = 0 (see th e t extb o o k ) . A pp lying i n tegr a tio n b y p a rts t w i ce, o n e th en ob tain s t h a t

Z u ( x ) d ( F ( x ) − G ( x )) = Z

u 00 ( x ) I ( x ) dx.

C o nd ition 1 is tru e i ff t h e l e f t h a n d s i d e i s n o n n e g a t i v e f o r a l l u wi th u 00 ( x ) ≤ 0 ev ery - w h ere. B y th e e qua l i t y , th e l a tter h o l ds if an d o n l y i f I ( x ) ≤ 0 ev ery w h e r e , i .e ., C o nd ition 3.

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

1 4.123 Microeconomic Theory III

Spring 2010

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