Decision Types, Personal Characteristics and Risk Aversion

A C T A U N I V t R S I 1 A T I S L O D Z I E N S I S _____________________FOLIA OECONOMICA B5. 1988

X I . R E S E A R C H O N C O N S U M E R B E H A V I O U R

Franz Böck er*

DECISION TYPES,-PERSONAL CHARACTERISTICS AND R ISK AVERSION

1. In t r o d u c t io n

S in c e D a n ie l B e r n o u l l i 's c o n s lu s io n s o f the S t . P e te rs b u rg game r e s u l t s i t i s e v id e n t th a t in d i v i d u a l s do not fo llo w the p r i n c i p l e o f s t a t i s t i c a l e x p e c ta tio n when d e c id in g under r i s k . Two c la s s e s o f models have been develop ed to d e s c r ib e and e x p la in the outcomes of d e c is io n s under r i s k : " c l a s s i c a l " d e c is io n models f o llo w in g the p - 6 - r u le e t c . and d e c is io n models based on the von Neumann-Morgenstern u t i l i t y c o n s t r u c t . M a n a g e ria l d e c is io n s are m o stly based on p r i n c i p l e s s i m i l i a r to those o f t h e u - 6 - r u le , t h e o r e t ic a l p apers a re based p re d o m in a n tly on the von Neumann-Mor- g e n s te rn u t i l i t y (vNMu) co n c e p t.

Two reason s a re d e fa v o ra b le to any m a n a g e ria l use of the vNMu: f i r s t th e re i s a la c k o f o p e r a t io n a l models o f the vNMu and, s e ­ cond, th e re i s no c l e a r e v id e n c e on the r i s k a v e r s io n l e v e l in d i f f e r e n t r i s k s i t u a t i o n s .

In t h i s paper we a re aim ing f i r s t to d e velo p o p e r a tio n a l r i s k models based on th e vNMu axiom s, second , to p a ra m e te riz e them and, f i n a l l y , to t e s t h yp o th eses on the r i s k a v e r s io n r a t e l e v e l s depending on c h a r a c t e r i s t i c s o f the d e c is io n maker as w e ll as the d e c is io n s e t t i n g . Thus, we hope to d e velo p a more o p e r a t io n a l as w e ll as th eo ry-b ased d e c is io n th e o ry under r i s k .

* P r o f e s s o r , D r, Regensburg U n i v e r s i t y (F e d e r a l R e p u b lic of Germany) .

2. The vNMu Theory o f R i s k A ve rsio n

2 .1 . B e r n o u l l i 's P r i n c ip le and the vNMu Concept

A cco rd in g to th e o ry i n d iv id u a ls d e c id e under r i s k not a c c o rd ­ in g to the m ath em atical e x p e c ta tio n o f the d e c is io n outcom es, but in com p lian ce w ith the m ath em atical e x p e c ta tio n o f tran sform ed d e c is io n o u tp u ts. The th e o ry i s w e ll founded on s e t s of axiems [3 2 ,2 3 ,1 1 ,3 ]. The vNMu th e o ry has been c r i t i z i s e d [1 ,6 ,1 8 ] and - more r e c e n t ly - s p e c i f ie d in d e t a i l s [ 5 , 4 ] . The f o llo w in g p o in ts a re m o stly a c c e p te d .

1. The vNMu i s a co m p o sitio n o f r i s k u t i l i t y as w e ll as income u t i l i t y . I t i s assumed th a t the income u t i l i t y fu n c tio n is con­ cave f o r p o s i t i v e outcomes and convex f o r n e g a tiv e ones. T h e re fo re , even i f a d e c is io n maker i s > r i s k n e u t r a l h is vNMu fu n c tio n should be convex.

2. Based on the u n c o n t r o v e r s ia l com plete o rd e r axion we d e fin e w ith X the random d e c is io n o u tp u t, w as the w e lfa r e i n d i c a t o r , and L (X , p ( x) ) as the l o t t e r y w ith the outcomes X and the p r o b a b il­ i t y p ( X ) :

{ r i s k a v e rs io n }* - * { u ( E (X ♦ w )) > U (w ) + u ( L ( X , p ( X ) ) } (1 ) [ r i s k p r o n e n e s s }* * ju (E (X + w )) < u (w ) + u (L ( X , p ( X ) ) j

The l e v e l of r i s k a v e r s io n or r i s k proneness i s depending not o n ly on d e c is io n m a k e rs 's c h a r a c t e r i s t i c s but on the type o f l o ­ t t e r y , too.

2 .2 . Arrow P r a t t R a t e of R is k Av e r s io n and Hypotheses on the R is k A v e rs io n

For monotone in c r e a s in g vNMu fu n c tio n s w ith e x is t in g second o rd e r d i f f e r e n t i a l s P r a t t [26] and A r r o w [2] d e fin e d the r i s k r a t e r x f o r any d e c is io n outcome x:

(

2

)

By c o n v e n tio n :

| r ( x ) > o } » - » r i s k a v e r s i o n } { г ( x ) < o ] « — » r i s k p r o n e n e s s }

The r i s k a v e rs io n r a t e r ( x ) i s not depending of the i n d i v i d u a l 's s t a t e of w e lf a r e .

There are numerous h yp o th eses on the g r a p h ic a l form of the vNMu fu n c tio n [3 .1 2 ,1 3 ] so«,e o f them a re d e p ic te d in F ig . 1 and 2

F ig . 1. Concave ( I ) , convex ( I I ) and l i n e a r ( I I I ) von Neumann-Morgenstern u t i l i t y cu rve

F o r x > 0 ( I ) i s supposed to be the most r e le v a n t c u rv e when d e a lin g w ith u su a l m a n a g e ria l d e c is io n s . ( I I ) i s s a id to be t y p i ­ c a l fo r a l l games of h a z a rd , ( I I I ) should be a p p lic a b le fo r re p e ­ t i t i v e d e c is io n s o f minor im p o rtan ce. ( I ) to ( I I I ) d e s c r ib e a r i s k b e h a v io r which i s c o n s ta n t as re g a rd s type o f b e h a v io r as d e fin e d in [ j ] whereas ( I V ) and (V ) a re d e s c r ib in g a r i s k b e h a v io r which i s - depending on x - p a r t l y r i s k prone and p a r t l y r i s k a v e rs e . N orm ally the tu r n in g - p o in t in ( I V ) i s not t h e o r e t i c a l l y d e te rm in e d , but supposed to bo e m p i r ic a ll y found w ith o u t c o n s id e rin g the e n o r­ mous s p e c i f i c a t i o n , i d e n t i f i c a t i o n as w e ll as v a l i d a t i o n problem s a r i s in g from t h i s method of tu rn in g p o in t f i x i n g . A cco rd in g to F i s h b u r n and K o c h e n b e r g e r [12] the tu rn in g p o in t i s the d e c is io n m akers' i n d iv id u a l t a r g e t p o in t (r e f e r e n c e p o i n t ) . The vNMu fu n c tio n a c c o rd in g to F r i e d m a n and S a- V a g e [13] (V ) has - to our knowledge - n e v e r been e m p i r ic a ll y t e s t e d ; i t i s supposed to be a p e r s o n - s p e c ific g e n e ra l fu n c tio n which i s a p p lic a b le f o r a l l d e c is io n s of an i n d i v i d u a l , in con ­ t r a s t the vNMu c u rv e s ( I ) to ( I V ) a re d e c is io n type s p e c i f i c .

won N eum ann-M orgenatornutility ( u )

F ig . 2, Convex-concave von Neumann-Morgenstern u t i l i t y c u rv e ( I V ) and u t i l i t y c u rv e as to Friedm an and Savage (V )

3. S p e c if y in g vNMu F u n c tio n s

3 .1 . B a s ic Assum ptions

The B e r n o u l li p r i n c i p l e can be fo rm u la te d as in ( 4 ) :

u(X- + w) » w ♦ E ( u ( X ) ( 4 )

S in c e X equal to a c o n s ta n t (L

={x,

p (x ) » 1 .0 ; p(xO - o j i s a s p e c i a l case o f X as a random v a r i a b l e we co n clu d e t h a t the u t i l i ­ t y s c o re s o f th e extrem e outcomes o f a l o t t e r y should be n um eri­ c a l l y . i d e n t i c a l to the u t i l i t y s c o re s o f the extrem e p o in t s o f the r e le v a n t l o t t e r y . By a tra n s fo r m a tio n as g iv e n in ( 5 ) t h is p r e r e q u is it e i s met:

max {x / L } -*1.0 and u (m a x {x / L f)- * l.0 ^

min {x/l }- » 0 and u (m in |x /l })- * 0

As a consequence the vN M u-functions have to be i d e n t i f i e d spe­ c i f i c a l l y f o r each maker and each d e c is io n ty p e . We suppose t h a t a d e c is io n maker may be c o m p a r a tiv e ly r i s k a v e rs e i f he has to tak e a v e ry r is k y d e c is io n and c o m p a r a tiv e ly r i s k prone or r i s k ne­ u t r a l i f he has to tak e a d e c is io n w ith low r i s k .

3 .2 . S p e c i f i c a t i o n s o f vNHu F u n c tio n s

W ith in c a p i t a l th e o ry [3 2 ,7 ,1 6 ,2 5 ] g e n e r a lly concave vNMu c u rv e s a re ćssumed. A m a th em a tica l model t h a t i s p a rsim o n io u s and based on t h is th e o ry i s the f o llo w in g one.

tŁ--.q u (x ) * (a + bx) b

w ith г ( x ) * 1

(

6

)

a ♦ bx

a ,b > 0 in ( 6 ) in d ic a t e s r is k a ve rs e b e h a v io r. (6 ) s ta n d a rd iz e d to u (x min * 0 0 and u ( “ max * l 0 ) * 1 0 9iv e s ( 7 ) :

u (x ) * X w ith с * b - 1 _{с > 0} w ith r ( x ) = l - ~. 5. с -i- X bx (7 ) Fo r с < 1.0 ( 7 ) shows a r is k a v e rs e b e h a v io r and fo r с > 1.0 a r is k prone b e h a v io r.

C o n stan t r is k a v e rs io n is g ive n assuming the fo llo w in g vNMu f u n c t io n : u (x )

ť

" ’

fo r fo r b > 0 b * 0 (0> w ith r ( X ) « r * b (6 ) s ta n d a rd iz e d :

X

1-e x -b (1-e -bx) for fo r b $ 0 u (x ) w ith r ( x ) a г - h

(8 ) and (9 ) in c lu d e any r is k n e u tr a l and the maximal r is k a ve rs e b e h a v io r (W ald r u le ) as s p e c ia l c a s e s . The u t i l i t y fu n c tio n (9 ) i s w e ll founded on axioms [4_{] .}

The vNMu fu n c tio n s g iv e n a re th e o ry- b a sed , but they are o n ly s t r i c t l y concave or s t r i c t l y convex. Concave and convex-concave vNMu fu n c tio n s can be e stim a te d by the subsequent fu n c tio n w ith s ta n d a rd iz e d ran g es:

a u (x ) * (1 ♦ b)

xa ♦ b a , b > 0

w ith r ( x ) * M a D -- y aí iĽ j L _n (x 3 ♦ b)x

and {a £ 1 ^ jc o n c a v e vNMu f u n c t io n }

|a > 1 jco n vex -co n cave vNMu f u n c t io n } tu rn in g p o in t fo r a > 1 and a <

J

f H ?

Г П • "

The r is k r a t e in (1 0 ) is a complex fu n c tio n o f two e x p l i c i t

para-(

1 0

)

m eters whereby one param eter is d om in atin g. A much s im p le r u t i l i ­ ty fu n c tio n i s the f o llo w in g one:

u (x ) * ebe ^ a, b > 0

w ith r ( x ) * 2-<- ~ -ft

xZ (1 1 )

and { b ž 2 •<=> {convex vNMu fu n c tio n

{ b < 2 <=*{convex-concave vNMu fu n c tio n tru n in g p o in t f o r b < 2 : x * y

(1 0 ) and (1 1 ) are v e ry f l e x i b l e fu n c tio n s ; they g iv e the o p p o rtu ­ n it y to p a ra m e te riz e a l l r e le v a n t vNMu fu n c tio n s ( a l l but vNMu fu n ­ c t io n as to Friedm an and S a v a g e ).

3 .3 . I d e n t i f i c a t i o n of vNMu F u n c tio n s

F i s h b u r n and K o c h e n b e r g e r [1 2 ] have summa­ r iz e d th e r e s u l t s on r i s k f u n c t io n s ' p a r a m e te r iz a t io n s :

( a ) Very o fte n th e re a re tu r n in g p o in ts in u t i l i t y fu n c tio n s w ith th e t a r g e t p o in t as a tu r n in g p o in t [2 2 ,1 9 ]. P a r t l y t h i s t a r g e t p o in t i s e q u iv a le n t to the z e ro - p o in t [3 0 ], p a r t l y the t a r g e t p o in t has a p o s i t i v e [1 5 ] or a n e g a tiv e outcome [1 4 ].

( b ) The vNHu f u n c tio n i s m o stly convex below the t a r g e t p o in t and concave above the t a r g e t p o i n t .

( c ) The u t i l i t y fu n c tio n i s s te e p e r below the t a r g e t p o in t as above th e t a r g e t p o in t .

F is h b u rn and Kochenberger I d e n t i f i e d the f o llo w in g u t i l i t y fun­ c t i o n s :

u ( x ) * ax a > 0 w ith r ( x ) « 0 (1 2 )

u ( x ) = axb a ,b > 0 w ith r ( x ) * ^ (1 3 )

u ( x ) = a ( l - e ' bx) a ,b > 0 w it h t ( x ) * r * b (1 4 )

The u t i l i t y fu n c tio n s have been e s tim a te d p a r t w is e (a b o ve and be­ low the t a r g e t p o i n t s ) by m in im iz in g the sum o f squared d e v ia t io n s . In t o t a l , t h i r t y u t i l i t y fu n c tio n s taken from f i v e a r t i c l e s have been re v ie w e d . The r e s u l t s as re g a rd s ( a ) to ( c ) in 3.3 a re g iv e n in Tab. 1.

T a b l e 1 T e sts as reg a rd s the h ypotheses ( a ) to ( c ) a c c o rd in g to-

F is h b u rn and Kochenberger Target p o in t

(f r e q u e n c y ) Shape of u t i l i t y fu n c tio n (fre q u e n c y )

0 > 0 < 0 1 2 .con-i • con­ • convex- concav- non _{1 2}

cav vex -concav -convex g ive n

23 3 4 30 3 5 13 7 2 30

Model w ith h ig h e s t in t e r n a l v a l i d i t y (fre q u e n c y ) _{F u n c tio n be­} be (1 3 ) .ow tt (1 4 ) ie ta i t ie d •get po non g ive n in t z ; abot (1 3 ) /e the (1 4 ) ta rg t ie d e t poin non g iv e n t z : low tr g et pc steepe above quency (1 2 ) e tar- in t is r as ( f re- ) (1 4 ) 12 13 3 2 30 15 15 - - 30 29 24

The r e s u l t s c o n firm the h yp o th eses ( a ) and ( b ) . The c o n c lu ­ s io n [19] th a t in d i v i d u a l s a re r i s k prone below the t a r g e t p o in t and r i6 k a v e rs e above the t a r g e t p o in t i s however not a n e c e s s a ry one because the vNMu fu n c tio n i s j u s t a co m p o sitio n of the income u t i l i t y fu n c tio n which i s b a s i c a l l y re p re s e n te d by a convex-con- ca ve c u rv e o f the r i s k u t i l i t y fu n c tio n .

A. H ypotheses on the Shape of thie vNMu F u n c tio n fo r D i f f e r e n t L o t t e r i e s and D i f f e r e n t I n d i v id u a ls

There i s an u n a n im ity th a t in d i v i d u a l s may show d i f f e r e n t rates o f r i s k a v e r s io n ; the one th a t the same in d i v i d u a l s show d i f f e ­ re n t r i s k a v e rs io n r a t e s f o r d i f f e r e n t l o t t e r i e s cannot be excluded. We fo rm u la te the f o llo w in g h y p o th e se s , t h e r e f o r e :

H , : fo r x > 0: r > 0. H^: fo r x < 0: r < 0.

H^: the g r e a t e r the span o f a l o t t e r y the g r e a t e r | r | . H^: men a re more r i s k prone than women.

Hrj: the o ld e r the i n d iv id u a l the b ig g e r | r | . Hg: the h ig h e r th e e d u c a tio n the lo w er | r j .

5. The E m p ir ic a l Study

5.1 Data and S p e c i f i c d_j/NMu Fu n c t ions

The e m p ir ic a l stu d y i s based on f i v e l o t t e r i e s . Fo r each l o t ­ t e r y s e v e r a l p r o b a b i l i t y d i s t r i b u t i o n s have b een.used to e s tim a te the u t i l i t y s c o re s (s e e Tab. 2 ).

4 T a b 1 e 2

L o t t e r i e s used in the e m p ir ic a l study L o t t e r y Outcomes of the l o t t e r y

( * ) . P r o b a b i l i t y d i s t r i b u t i o n s used f o r e s tim a tio n p u r­ poses A 0 + 1 000 В + 1 000 -1 000 _{. •} 0 .2 / 0 .8; 0 .4 / 0 .6; 0 .5 / 0 .5; С 0 -1 000 0 .6 / 0 .4; 0 . 7 /0 .3 ; 0 .B / 0 .2 0 ♦ 500 - 500 E ♦ 2 000 -2 000

The u t i l i t y fu n c tio n s f o r p a ra m e te r iz a tio n have been s e le c t e d on the b a s is o f t h e o r e t ic a l re a s o n in g . Zero outcome has been chosen as the t a r g e t p o in t o f the u t i l i t y f u n c tio n . The fu n c tio n s , l o t t e r i e s and d ata p o in ts u n d e rly in g the p a ra m e te r iz a tio n a re g i ­ ven in Tab. 3. ( 7 ) , ( 9 ) , (1 0 ) and (1 1 ) a llo w to e s tim a te a l l d i s ­ cussed u t i l i t y f u n c tio n s , e s p e c ia lly those which have been a n a ly ­ zed by Fis h b u rn and K ochenbeťger.

A l l u t i l i t y fu n c tio n s - h a v e been e stim a te d on the in d iv id u a ls ' le v e l whereby f o r some o f the p a rtw is e e s tim a tio n runs th e r e s id u ­ a l d egrees o f freedoms a re v e ry low . The in d iv id u a ls . have been s e le c t e d by random, . they a re d e s c rib e d by the subsequent socio d e- m ographic v a r ia b le s (T a b . 4 ).

Data used to e s t in a t e the vNMu fu n c tio n s T a b l e 3

L o t t e r ie s

Data used s ta n d a r d iz a tio n of outcomes x(x* trans­

formed outcom es)

P r o b a b ilit y d is t r ib u t io n s A - E _{xmin ± ®} xmax 1 1 a l l B .D .E x = 10—* x' = 0 1 X = 1 max 0 .2 / 0 .8; 0 .4 / 0 .6 B .D .E _{X«nin é 0} min 0 .6 / 0 .4; 0 . 7 /0 .3 ; x = 0 —- к ' = 1 í 0 min , xmax 4 ť o\e/o.2 U J 1 < a l l B .D .E x = 0—* x' = 0 xmax ' * 0 .2 / 0 .8; 0 .4 / 0 .6 B .O .E _{xmin i ®} 0 .6 / 0 .4: 0 . 7 / 0 .3 ; X II 0

1

x II _{*— 0 .8 / 0 .2} A - E xmin ‘ ® xmax é * a l l A - E xmin _i ° — x = 1 max a l l U t i l i t y fu n c tio n u (x ) = x (7 ) u (x ) = — i--- r- ( l - e _b x ) 1 - e ‘ b (9 ) a u (x ) = (1 ♦ b ) ---x + b

(

1 0

)

(

11

)

R is k a v e rs io n r i s k a v e rs e or r i s k prone r i s k a v e rs e or r i s k prone r i s k a v e rs e or r i s k prone r i s k a v e rs e o r r i s k prone r i s k a v e rs e or r i s k prone r i s k a ve rs e or r i s k prone r i s k a v e rs e or r i s k prone- r i s k a v e rs e r i s k prone o r r i s k prone- r i s k a v e rs e D ec is io n T yp es , Pe rs o n al C h a ra c te ri s ti c s

T a b l e 4 P e rs o n a l c h a r a c t e r i s t i c s of the f o r t y resp ond ents in c lu d e d in the study and h y p o th e s is on the c h a r a c t e r is t ic s ' i n f l u ­

ence on r is k a v e rs io n Sociodem ogrflphic

v a r ia b le s

Outcomes H y p o th e s is :

r is k a v e r s io n . . .

Age <30; 31-40; 41-50; 51-60; in c re a s e s w ith age

>60

Sex men, women i s h ig h e r fo r womer

than fo r men

Form al e d u ca tio n p rim a ry , secortad ary, c o l l e ­ d e cre a ses w ith f o r ­ ge, u n iv e r s it y le v e l mal e d u c a tio n

5 .2 . The R e s u lts

Based on th e f o r t y respondents, in t e r v ie w s , the u t i l i t y fu n c­ t io n s ( 7 ) , ( 9 ) , (1 0 ) , and (1 1 ) have been e s tim a te d .

5 .2 .1 . The P a ra m e te rs of the U t i l i t y F u n c tio n s

The r e s u l t s "of the 1760 e s tim a tio n runs (40 p e rso n s, 4 mo­ d e ls , 5 l o t t e r i e s w ith 1/3 d ata s e t s ) a re g iv e n in the subsequent t a b le (T a b . 5) whereby a s t a t i s t i c a l and not an a lg e b r a ic parame­ t e r iz a t io n p roced ures have been used C u rrim / S a rin .

T ab le 5 g iv e s the fre q u e n c ie s th a t s p e c i f i c models have the h ig h e s t in t e r n a l v a l i d i t y ' f o r the f o r t y in d iv id u a ls s e le c t e d ; mo­ d e l ( 7 ) e .g . shows 15 tim es the h ig h e s t in t e r n a l v a l i d i t y . In con­ t r a s t to models ( 7 ) , ( 9 ) and (1 1 ) modul (1 0 ) has two e x p l i c i t pa­ ra m e te rs ; a h ig h e r in t e r n a l v a l i d i t y o f (1 0 ) has to be exp e cte d , t h e r e f o r e . The in t e r n a l v a l i d i t i e s of the model (1 0 ) a g a in s t those of the models ( 7 ) , (9 ) and (1 1 ) h a v e ' been te s te d s t a t i s t i c a l l y . Taking in to c o n s id e r a t io n the d i f f e r e n t numbers of e x p l i c i t p a ra ­ m eters the maximum in t e r n a l v a l i d i t y fre q u e n c ie s which ace g iv e n in b r a c k e ts a re the r e le v a n t ones.

The main r e s u l t s o f the e s tim a tio n s can be summarized as sub­ seq u en t: w ith the zero outcome as t a r g e t ^point and p o s it iv e outcom es, l o t t e r i e s (A , B * , D+ and E +) the u t i l i t y fu n c tio n is s i g n i f i c a n t l y more o fte n concave than convex ( s i g n i f i c a n t l e v e l : < 0.003%) w ith n e g a tiv e d e c is io n outcom es, l o t t e r i e s (B ~ , D, 0"

T a b l e 5 R e s u lts of th e p a ra m e te r iz a tio n runs f o r a l l fo u r models and f o r t y persons

L o t- i t e r y Oata used fo r para-Model w ith v a l t d i tyh ig h e s t in t e r n a l (f r e q u e n c y ) . Person- s p e c i f i c Shape of u t i l i t y J u n c t io n (f r e q u e n c y ) m etn riz

a-t io n (7 ) (9 ) (1 0 ) (1 1 ) m inim al stand- ared d e v ia t io n con­ cave con­ cave concave--convex 1 2 A a l l x-u- - p a ir s 15 (1 8 ) 0 (1 6 ) 17 (6 ) 0 (0 ) 40 0.013-0.044 33 (3 7 ) 3 ( 3 ) 4 (0 ) 40 В a l l x-u- - p a ir s x - u - p a irs below t a r g e t 0 В (0 ) 0 (0 ) 32 1 ( 1 ) 39 (3 9 ) 40 40 0.013-0.107 0.0004-0.105 0 (0 ) 0 0 (0 ) 40 40 (40) 40 40 x - u - p a irs above t a r g e t 20 20 - - 40 0.004-0.063 39 1 - 40 С a l l x-u- - p a ir s В (9 ) A ( 5 ) 4 (0 ) 24 (2 6 ) 40 0.022-0.104 0 (0 ) 12 (14) 28 (26) 40 D a l l x-u- - p a ir s x - u - p a irs below target 1 10 (1 ) 2 (5 ) 30 6 (1 ) 31 (3 3 ) 40 40 0.009-0.093 0.000-0.077 1 (2 ) 1 2 (4) 39 37 (33) 40 40 x-u-p’a i r s above target 15 25 - _ 60 0.012-0.156 40 0 • 40 E a l l x-u- - p a ir s x - u - p a irs below target 0 0 (0 ) 0 (0 ) 40 1 (0 ) 39 (4 0 ) 40 40 0.021-0.094 0.002-0.117 0 (0 ) 0 0 (0 ) 40 40 (A0) 40 40 x-u-pai rs above target 16 24 “ - 40 0.003-0.066 40 0 • 40 - . 1 93 (9 7 ) 185 (1 9 7 ) 29 (8 ) 133 (1 3 B ) 440 - ft - - 440 D ec is io n T yp es , Pe rs o n al C h a ra c te ri s ti c s

and С ) the u t i l i t y fu n c tio n is s i g n i f i c a n t l y more o fte n convex/

t Q

/convex-concave than concave ( s i g n i f i c a n t l e v e l : < 10 ) . The con­ vex-concave u t i l i t y fu n c tio n s a re dominated by i t s convex p a r t .

The г е з и Н з c o n firm the h y p o th e s is d e s c rib e d in c h a p te r 3 of t h is pap er.

5 .2 .2 . The A n a ly s is of the U t i l i t y F u n c tio n Param eters

The hypotheses g iv e n in c h a p te r 4 of t h is paper have been ana­ ly z e d v ia ANOVA. The r is k param eters in ( 7 ) , (9 ) and (1 1 ) and the more im p ortan t r is k p aram eter in (1 0 ) have been c l a s s i f i e d as the dependent v a r i a b l e . The r e s u l t s are g iv e n in the subsequent Tab les 6 to 9.

T a b l e 6 ANOVA runs on the r is k param eter e s tim a te s o f model (7 ) w ith the l o t t e r i e s A and С as a whole and the l o t t e r i e s B,

0 and E d iv id e d in two p a r t s Independent

v a r i a b l e s

Sum of sq uares Degrees of freedom F - v a lu e L e v e l o f s i g n i f i ­ cance L o t t e r y 2 882.395 ... .... r ■ 7 22.64 < 0. 001 Age 63.183 4 0.87 0.45 Sex 31.330 1 1.72 0.30 Form al e d u c a tio n 55.402 3 1.02 0.40 R e s id u a ls 5 530.240 304 - -Sum 8 562.550; 319 - -T a b l e 7 ANOVA run s on th e r i s k p aram eter e s tim a te s o f model (9 ) w ith the l o t t e r i e s A and С as a whole and the l o t t e r i e s B,

0 and E d iv id e d in two p a r ts

Independent v a r i a b l e s

Sum o f sq u ares D egrees of

freedom F - v a lu e L e v e l of s i g n i f i ­ cance L o t t e r y 1 452.689 7 153.14 0.001 Age 30.471 4 5.62 0.001 Sex 14.587 1 10.76 0.002 Form al e d u c a tio n 13.963 3 3.43 0.02 R e s id u a ls 411.964 304 - -Sum 1 923.674 319 -

-T a h 1 e 8 ANQVA runs on the r is k param eter e s tim a te s of model (1 0 )

w ith the l o t t e r i e s A to E as a whole Independent

v a r ia b le s Sum of squares Degrees of freedom F - va lu e L e v e l of s ig n i f i- cance L o t te r y 26.245 4 53.29 < Ü.0Ü1 Age 0.904 4 1.84 0.20 Sex 0.282 1 2.29 0.20 Form al e d u ca tio n 0.416 3 1.13 0.40 R e s id u a ls 21.300 173 - « Sum 49.147 185 T a ii 1 e 9 ANÜVA runs on the r is k param eter e s tim a te s of model (1 1 )

w ith the l o t t e r i e s A to E аз a whole Independent

v a r i a b l e s Sum of sq uares Degress of f reedom F - va lu o L e v e l of s ig n i f i- cance L o t t e r y 23.168 4 101.78 < 0.001 Age 0.568 4 2.50 0.04 Sex 0.355 1 6.24 0.02 Form al e d u c a tio n 0.467 3 2.74 0.05 R e s id u a ls 10.642 187 - • Sum 35.200 199 -

-Fo r a l l models the v a r i a b l e l o t t e r y is h ig h ly s i g n i f i c a n t , where as the socio d em o g rap h ic v a r i a b l e s a re o n l y , p a r t l y s i g n i f i c a n t i n ­ dependent v a r i a b l e s . ANOVA shows f o r model ( 9 ) - by f a r - the h i ­ g h est in t e r n a l v a l i d i t y .

The n u m e ric a l v a lu e s o f d i f f e r e n t models r i s k p aram eter a re g iv e n in Tab. 10.

When o p e r a t io n a liz in g the r i s k of a d e c is io n v i a the span of outcomes th e se d a ta c o n firm in f i v e o f s ix c a s e s h y p o th e s is с as g iv e n in c h a p te r 3.3 o f t h i s p a p e r. Fo r model (9 ) the f in d in g s can be summarized as f o llo w s :

T a b l e 10 Average r is k p aram eters fo r d if f e r e n t l o t t e r i e s and

models

Model Average risk coefficien t for lottery

para­ meter A В B+ В ' С D D+ 0" E E* E" (7 )/(b ) 2.63 - 2.90 -2.61 -2.92 - 3.11 -2.74 - 3.29 -3.66 (9 )/(b ) 2.30 - 2.23 -2.55 -0.88 - 1.59 -1.72 - 2.46 -3.51 (10)/(a) -0.47 -0.07 - - 0.66 -0.06 - - -0.12 - -( li ) / -( b ) -0.42 -0.09 - - 0.62 0.05 - - -0.05 - -T a b l e 11 Average r is k param eters f o r d if f e r e n t r is k y d e c is io n s

L o t t e r y /p a rt of lo t t e r y R is k l e v e l outcomes of (= span of the l o t t e r y ) Average r is k c o e f f i c i e n t 0 ł 0 500 1.59 А, В+ 0 1 000 2.30 Е* 0 2 000 / 2 .3 8 ; 2.23/ D” -500 0 _{-1. 72}2.46 В * , С -1 000 0 -1.72 Е ' -2 000 0 / 2 .5 5 ; -0.88/ _-3.51

From Tab. 11 we conclude* th a t the h ig h e r the r i s k i s the hig-her i s the ave rag e a b s o lu te v a lu e o f the r is k p a ra m e te r. T h is h y­ p o t h e s is is b a s i c a l l y co n firm e d , j u s t one minor r e s t r i c t i o n com­ p a rin g the l o t t e r i e s 0 and В " , С emerges. Fo r the range o f po­ s i t i v e outcomes the f in d in g s a re i n t u i t i v e l y a p p e a lin g , however f o r the range o f n e g a tiv e outcomes i t i s hard to e x p la in them.

6. C o n c lu s i ons

Base’d on e x p e rim e n ta l in t e r v ie w in g von Neumann-Morgenstern u- t i l i r y f u n c tio n s have been e s tim a te d . F u rth e rm o re , v a r i a t i o n s of the r i s k p a ram eters have been e x p la in e d by v a r i a b l e s d e s c r ib in g the in d iv id u a l d e c is io n makers as w e ll as the d e c is io n s e t t i n g s .

F o r d e c is io n s e t t in g s w ith p u re ly p o s i t i v e outcomes d e c is io n making can be d e s c rib e d v e ry e f f e c t i v e l y by a concave u t i l i t y

f u n c tio n , fo r d e c is io n . s e t t in g s w ith p u re ly n e g a tiv e outcomes b> a convex u t i l i t y f u n c tio n . The u t i l i t y fu n c tio n which seems to be the most r e le v a n t one is a u t i l i t y fu n c tio n w ith з co n s ta n t r is k a v e rs io n r a te ( 9 ) .

T h is m o d e l's param eter may be e x p la in e d by e x te r n a l v a r ia b le s v e ry e f f i c i e n t l y , to o . I t turned out th a t o ld e r , fem ale and le s s educated i n d u iu d a ls a re more r is k a v e rs e than younger, male and more educated in d iv id u a ls . F u rth e rm o re , the more r is k y a d e c is io n is - r is k o p e r a tio n a liz e d by the span o f outcomes - the h ig h e r are the r is k a v e rs io n r a t e in the p o s it iv e outcomes area and the r is k promeness r a te in the n e g a tiv e outcomes a re a . We are a d vo ca tin g th a t t h is phenomenon is not due to r i s k b e h a v io r but due to in ­ come u t i l i t y judgements which a re alw ays mixed in e m p ir ic a l s t u ­ d ie s .

The study has heen based on somehow a r t i f i c a l l o t t e r i e s , i t would be w o rth w h ile to d u p lic a te the study w ith more r e a l i s t i c d e c is io n s e t t in g s and ta k in g in to account p r o fe s s io n a l v a r ia b le s ( e . g . fin a n c e managers r is k a v e rs e , m arketing managers *--- - r is k p r o n e ).

jB 1 i o g raphy

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Franz böcker

RODZAJE O EC YZJI

CECHY OECYDENTA I NIECHĘĆ DO PODEJMOWANIA RYZYKA

W s y t u a c j i r o z p a try w a n ia jed n orod n ych a lt e r n a t y w d e c y z y jn y c h p o ję c ie u ż y te c z n o ś c i stosow ane przez von Neum ann-Moigensterna jest w ła ś c iw ą k o n s t r u k c ją , s łu ż ą c ą w y ja ś n ia n iu o raz p rzew id yw an iu e f e ­ któw in d yw id u a ln yc h d e c y z j i . Na p o d sta w ie k s z t a łt u f u n k c ji użytecz­ n o ś c i oraz w skaźnika n ie c h ę c i do podejmowania ry z y k a , zależn eg o od cech d ecyd en ta oraz ro d z a ju d e c y z j i , można sform ułow ać s ze re g h i ­ p o te z .

W b ad aniu eksperym entalnym resp o n d e n ci z o s t a l i poddani w ie lo ­ k r o t n ie d z i a ł a n iu przypadku ( l o t e r i a ) . O p ie r a ją c s i ę na d z ia ła n iu przypadku ( l o t e r i i ) o raz p rzy u w z g lę d n ie n iu s p e c y fic z n y c h cech r e ­ spondenta oszacowano s t a t y s t y c z n i e w ła ś c iw ą mu fu n k c ję u ż y te c z n o ­

ś c i . ‘ P rze te sto w a n o rów nież h ip o te z y d o tyczą ce n ie c h ę c i do p o d e j­ mowania ryzyka sform ułowane d la różnych grup społeczno-dem ograf icz- nych oraz d ]a różnych typów l o t e r i i .

Zgodnie z z a ło ż e n ia m i te o re tycz n y m i fu n k c ja u ż y te c z n o ś c i j e s t cz ęścio w o w k lę s ła a częściow o w yp ukła, co ja k s ię p rz y jm u je j e s t konsekw encją f u n k c ji u ż y te c z n o ś c i dochotfów. Okazuje s ię iż z a ró ­ wno typ l o t e r i i , ja k i n ie k tó r e zmienne sp ołeczn o-d em o g raficzn e wy­ j a ś n i a j ą dobrze wahania w skaźnika n ie c h ę c i do podejmowania ry z y k a .