Compa ibili y o Expec ed U ili y and App oaches o Risk o a Class o σμ /
Non Loca ion-Scale Dis ibu ions
By Ge y Boyle and Denis Conni e
Abs ac
P oo s o compa ibili y o he expec ed u ili y and app oaches o inco po a ing unce ain y in
decision making exis o a leas some u ili y unc ions and loca ion-scale dis ibu ions. Bu he e a e
se e e cons ain s and i is desi able o in es iga e compa ibili y mo e widely. We do so o he class o
dis ibu ions ha a e ans o mable o loca ion-scale o m by conca e ans o ma ion and whe e he
u ili y unc ions emain conca e unde ans o ma ion. The class is impo an , con aining dis ibu ions
such as he logno mal and Pa e o, usually conside ed mo e app op ia e o modelling income o weal h
han hose in he loca ion-scale amily.
σμ /
1. INTRODUCTION
As is well known, wo equen ly employed app oaches o inco po a ing unce ain y abou some key
a iable – o en income o weal h - in analyses o he compa a i e s a ics o op imal decision-making
a e analysis and maximisa ion o expec ed u ili y. The o me app oach akes unce ain y as
ep esen able by he s anda d de ia ion o he a iable. The decision make is assumed o p oceed by
cons ained maximisa ion o some unc ion o he mean and s anda d de ia ion, which is inc easing in
he mean, dec easing in he s anda d de ia ion ( he mono onici y condi ions) and quasiconca e in
σμ /

and

. The expec ed u ili y app oach commences om a u ili y unc ion which is mono onically
inc easing and conca e in he a iable. Unce ain y is embodied in a p obabili y dis ibu ion o he
a iable and he decision make e alua es ou comes in e ms o expec ed u ili y. Elemen a y ex s
o en ema k ha he app oaches a e equi alen o ei he a quad a ic u ili y unc ion o no mali y o
dis ibu ion, al hough he la e case is eally g ea ly cons ained by being condi ional on he exis ence
o he expec ed u ili y. Fo o he si ua ions, he consis ency o he wo app oaches has been deba ed a
conside able leng h in he li e a u e, which is so ex ensi e ha only publica ions di ec ly ele an o he
heme o his pape will be men ioned.
Meye (1987) and Sinn (1989) showed ha equi alence o expec ed u ili y maximisa ion and
analysis could be ex ended om he no mal dis ibu ion o he loca ion-scale amily o dis ibu ions
al hough again his is ac ually condi ional on he exis ence o expec ed u ili ies. This is one o he key
issues in discussing equi alence. Some o he mos equen ly ad oca ed u ili y unc ions u(x) do no
ha e expec a ions unde no mali y, o some o he loca ion-scale dis ibu ions, bu do unde plausible
dis ibu ions ha a e no in he loca ion-scale amily. Fo example,
σμ /
2
xxu log)(

o , wi h

xxu )( 10



,
do no ha e expec a ions unde no mali y. No ha e hey expec a ions unde o he loca ion-scale
dis ibu ions whe e x can be , such as he Gumbel. The e a e loca ion-scale dis ibu ions o which
x mus be posi i e, such as he wo pa ame e uni o m and he wo pa ame e exponen ial, bu hese a e
su ely implausible as models o income o weal h.
0
Howe e , he expec a ions do exis o se e al wo pa ame e dis ibu ions ha a e no o loca ion-
scale o m and ha a e plausible as models o income o weal h. The Pa e o dis ibu ion was one o
he i s ad oca ed ( o example, A nold, 1983) o ha pu pose1, bu i is no in he loca ion-scale
amily. The log no mal has been equen ly employed o income modelling, jus i ied by bo h
heo e ical ( o example, Ai chison and B own, 1957) and empi ical indings and, o cou se, i is no o
loca ion-scale ype. O he non loca ion-scale wo pa ame e dis ibu ions and a ious mul ipa ame e
dis ibu ions ha e also been p oposed ( o example, Bandou ian, McDonald and Ju ley, 2003).
Meye ’s insigh ha i an o iginally mul ipa ame e non loca ion-scale dis ibu ion is employed in
ci cums ances whe e all pa ame e s excep he loca ion and scale pa ame e s a e held cons an i
e ec i ely becomes a loca ion-scale dis ibu ion, does es ablish equi alence o some applied
p oblems. Bu his si ua ion, e med he ‘loca ion scale condi ion’, mus be o limi ed occu ence.
So i is impo an o in es iga e i he equi alence be ween he expec ed u ili y unc ion and
app oaches holds o he dis ibu ions o which he expec a ions exis , bu ha a e no in he
loca ion-scale amily. The app oach in his pape is o in es iga e wo pa ame e dis ibu ions o x ha
a e no loca ion-scale, bu whe e he dis ibu ion o y = h(x) is loca ion scale, whe e h(x) is a
mono onically inc easing and conca e unc ion o x. Fo example, i x is log no mal, y = log x is
no mal and i x is Pa e o, y = log x ollows a wo pa ame e exponen ial and his, like he no mal, is a
loca ion-scale dis ibu ion. I we also equi e ha ou u ili y unc ions a e conca e when exp essed as
unc ions o y, we can show ha equi alence will hold in he sense ha
σμ /
),())((


VxuE sa is ies
he mono onici y condi ions h oughou ),(


space and is quasiconca e in

and

in a egion o
ha space. In ou p oo s we will exploi Meye ’s (1987) esul s o he loca ion-scale case, al hough
we will need o b oaden his p oo somewha . The loca ion-scale amily is de ined by he densi y
1 Nowadays i is ecognised ha he Pa e o is unsui able as a gene al income dis ibu ion, because i
does no i well a low incomes, al hough i does i well o a popula ion o highe income g oups. Bu
ha migh be he ele an popula ion in con ex s such as in es men .
3









x
1,
whe e

is he loca ion and

he scale pa ame e . The mean and s anda d de ia ion a e
1
k



 and 2
k



espec i ely, whe e and a e cons an s. Meye assumed
1
k2
k

and

equal o

and

, which would be ue o a no mal, bu no o some o he loca ion-scale
dis ibu ions, and he lowe and uppe a iable bounds independen o pa ame e s. These assump ions
a e no necessa y and would, o example, p e en us om employing he wo pa ame e exponen ial,
whe e he lowe bound depends on he loca ion pa ame e . Ou somewha mo e gene al p oo is gi en
in Appendix A.
2. CONCAVE TRANSFORMATIONS TO LOCATION-SCALE DENSITIES
As al eady men ioned, y = h(x) is mono onically inc easing and conca e and y has a loca ion-scale
dis ibu ion. The case y = x co esponds o x loca ion-scale. Suppose he in e se unc ion is
. Clea ly, any numbe o dis ibu ions o x wi h he equi ed p ope y can be
ound by ob aining he dis ibu ions o wi h h(x) any app op ia e unc ion and y
ollowing any o he loca ion-scale dis ibu ions. Bu whe he hese dis ibu ions o x a e good i s o
income o weal h dis ibu ions is ano he ma e . Some migh be, e en i hey ha e no appea ed in he
li e a u e, bu p obably many would no be. So, al hough much o he pape main ains he gene ali y o
y = h(x), ou examples will use y = log x, whe e we know app op ia e dis ibu ions exis .
)()(
1ygyhx  
)()(
1ygyhx  
Suppose he u ili y unc ion u(x) =u(g(y)) is also conca e in y. Examples implying amilia u ili y
unc ions a e:
bybyyyuxxbxxu 2/1,)(,1,)(loglog)( 22 
10,)(,1,)(log)( 


yyuxxxu
0,1)(,0,)( log  


yx eyuxeAxu
yyuxxxu log)(,1,loglog)(

 .
4
Since y = h(x) has a loca ion-scale dis ibu ion, we know ha he expec a ion o u(y), ,
has
),( **

V
*
2 he equi ed p ope ies o mono onici y and conca i y wi h espec o he mean and s anda d
de ia ion o he a iable y. O cou se, he expec a ion o u(x) wi h espec o he dis ibu ion o x
mus also be , since i amoun s o jus a change o a iable in in eg a ion. Howe e , we
a e in e es ed in i s p ope ies wi h espec o

*

V),( **


and

and no and . We will in es iga e hese
in he nex wo sec ions.
*

*

Since he e a e e iden ly conca e u ili y unc ions o x ha will no be conca e in e ms o y, we a e
es ic ing he class o u ili y unc ions somewha . Some may be o ally excluded, while o he s may
equi e es ic ion o he ange o he a iable o pa ame e s. Fo example,
,0,)( log  xexxu x

,10



is conca e in x, bu wi h y = log x,
y
eyu

)(
is no conca e in y as i s second de i a i e is posi i e. Howe e ,
, x > 0.
x
exu


1)(
wi h y = log x would gi e
.
y
e
eyu


1)(
and hen

yey eee
y
uy




1
2
2
.
which is nega i e i he b acke ed e m is. So conca i y equi es 1

, o y > - log

.
3. THE MONOTONICITY CONDITIONS
Since sa is ies mono onici y condi ions we know ha
),( **

V
*



V and *


V
2 Assuming V exis s, o cou se. )(log)log(log yExE

will no exis i y is no mal, ha is, i x is
log no mal, bu i will i y is wo pa ame e exponen ial, ha is, i x is Pa e o.
5
a e posi i e and nega i e espec i ely. Now


















*
*
*
*
VVV
and
















*
*
*
*
VVV .
So i



*
and



*
(1)
a e posi i e, as is plausible om y = h(x) being a mono onically inc easing unc ion o x, and



*
and



*
(2)
a e nega i e, as is plausible om y = h(x) being conca e in x, he i s de i a i es o V wi h espec o

and

will be posi i e and nega i e espec i ely. The de ailed p oo is gi en in Appendix B. O
cou se, (1) and (2) imply he slope o an indi e ence cu e






 VV
d
d
S/
is posi i e.
Fo a i s example, we ake he case o x log no mally dis ibu ed, so ha y = log x is no mally
dis ibu ed and he u ili y unc ion as u(x) = log x. This migh seem a a he i ial u ili y unc ion o
choose, since hen u(y) = y, which is on he limi o conca i y, while and he slope
o he indi e ence cu e in space is ze o. Howe e , he case is impo an , bo h his o ically
and because many ex books on in es men o po olio heo y ( o example, El on and G ube , 1995,
p.234 ) s a e ha expec ed u ili y and analysis a e equi alen o a log u ili y unc ion gi en log
no mali y. The claim is based on esul s in El on and G ube (1974), bu he case had p e iously been
analysed by Felds ein (1969) and commen ed on by o he s.
*** ),(

V
),( **

μ σ/
S a is ical ex books w i e he pa ame e s o he log no mal as he mean and a iance o he log o
he a iable, ha is, o and . The mean and a iance o he logno mal a e
*

*


6
2**
2
1



e
and


1
2*2**
22  


ee .
Exp essing in e ms o
*


and

gi es







 2
2
*1log
2
1
log



(3)
and







 2
2
2* 1log



. (4)
Since is also
*

),(


V, di e en ia ion o (3), as pe o med by El on and G ube (1974 ), gi es
)(
2
22
22







V and 22








V,
which a e posi i e and nega i e espec i ely. U ili y is inc eased by inc easing

o ixed

o
dec easing

o ixed

. So i he se o possible alues in ),(


space is bounded by a conca e
on ie he expec ed u ili y maximum will lie upon ha on ie . The issue o p ecisely whe e on he
on ie depends on how he slope o he indi e ence cu e
22 2






 d
d
S (5)
changes along he cu e and his will be e u ned o in he nex Sec ion. O cou se, he signs o he
de i a i es o V wi h espec o

and

we e al eady gua an eed by he gene al p oo in Appendix B.
In ac , o e e y u ili y unc ion u(x) = u(g(y)), conca e in y as well as x, wi h y = h(x) ollowing a
loca ion-scale dis ibu ion, expec ed u ili y is compa ible wi h analysis in his sense o he
σμ /
maximum occu ing on he on ie .
The e a e ob iously e y many possibili ies, bu keeping o logno mal x, one in e es ing case is
.
yx eAeAxAxu


  log
)(
whe e 1

. In e ms o y, his is he e y equen ly employed cons an absolu e isk a e sion
u ili y unc ion. F om he well known momen gene a ing unc ion o he no mal dis ibu ion
),( **

V2*2*
2
1



 eA .
7
I may be wo h emembe ing ha , which is no a unc ion o in
**** /

 ddS *

acco dance wi h cons an absolu e isk a e sion. Subs i u ing (3) and (4) in o gi es
),( **

V
),(


V=)/1log()1(
2
log 22





 eA , (6)
wi h



22
22 )2(









VA
V,

22
)1(









VA
V
and
22 )2(
)1(






S. (7)
So S is a unc ion o

and since

22
22
)2(
)2(







S
S. (8)
This will be nega i e, showing dec easing absolu e isk a e sion, which is o en conside ed plausible, i
22 )2(

 . I could be posi i e o low mean income (o high

and a iance), bu as will be
seen in he nex sec ion, quasiconca i y equi emen s a e hen in inged. Sinn (1989, p.152 ) p esen s a
u ili y unc ion o he o m (6), al hough his de i a ion ollowed a qui e di e en pa h.
Assuming log no mali y is no essen ial, o cou se. Suppose x has a Pa e o dis ibu ion
 


x
x
1
1.
The mean and a iance o he dis ibu ion a e
1




and )2()1( 2
2
2





The log o a Pa e o has he (loca ion-scale) wo pa ame e exponen ial densi y
,,
1
)(
)(


yey
y




whe e


/1 and


log. The mean and a iance o he exponen ial a e
and .


*22*


8
Taking u(x) = log x gi e as in he log no mal case, bu exp essing in e ms o
s *** ),(

V*


and gi es ins ead o (3)
2


























 1111log),( 2
2
2
2
2
2
2
22
*











V .
nd no ing ha
Again, aking
yx eAeAxAxu


  log
)(
a













edye yy
1
)(
1
,
we ge

































 2
2
2
22
22
22 11
/11
/11
),( AV
s ead o (6). Many o he ),(


V
incan be ob ained co esponding o a ious u(x) and dis ibu ions o
n he ac abili y o he in eg als in ol ed, ma hema ical exp essions can
me imes be di icul .
4. THE QUASICONCAVITY CONDITIONS
x, al hough, depending o
so
We also wan quasiconca i y o V wi h espec o

and

, so ha he indi e ence cu es a e
con ex. The quasiconca i y condi ion is non-nega i i y o
3
2
2
2
2/2
222


































































VVVVVVVV . (9)
While his will hold o low alues o

o all u ili y unc ions and dis ibu ions, Appendix B shows i
will no emain so o e he whole o ),(


space o he class o u ili ies u(x) =u(g(y)) de ined in
Sec ion 2. The quasiconca i y egion depends on bo ies o ma ion o loca ion-
scale and he beha iou o he indi e ence cu e o ),( in ),( space. Seeking a single
o mula co e ing all ans o ma ions and u ili y unc ions leads o ex emely unwieldy algeb aic e ms.
h he p ope o he ans
x log no mal x and any u(x) = u(g(y)), conca e in y as well as x, non nega i i y o (9)
equi es ha
**

V**

Fo he case o
9























































**
*
21
2
)( *
2*
2
*
*
22
*
*
3
*
*
22
22









S
d
dSSS
S, (10)
wi h
*
*
*
*
*
*
*






S
S
S
d
dS ,
be non-nega i e. This shows ha quasiconca i y ce ainly holds in he in e al
2
1
*
*2











S, (11)
when he elas ici y o he indi e ence cu e in space
),( **

*
*
log
log

d
Sd
is equal o uni y. When he elas ici y is g ea e han uni y he in e al expands, since

can become
somewha la ge , and when he elas ici y is less han uni y he in e al con ac s co espondingly. Bu
i is di icul o make mo e clea -cu s a emen s wi hou conside ing speci ic u ili y unc ions. Simila
condi ions o (10) can be ob ained o o he dis ibu ions ans o mable o loca ion-scale o m, bu in
he case o any pa icula u ili y unc ion, i is usually much easie o p oceed by ob aining S as in he
p e ious Sec ion and hen examining he posi i i y o
.














S
S
S
d
dSS
d
dS
This is wha Felds ein (1969) did o he case o u(x) = log x wi h x log no mal. Simple di e en ia ion
o (5) shows
322
2222
)2(
)2)((






d
dS
so ha con exi y o he indi e ence cu e equi es 2/

. This has al eady been ob ained
mo e ediously in Appendix B and would also ollow om (11). As men ioned in he p e ious sec ion,
El on and G ube (1974) also examined his case, bu unlike Felds ein, did no hink he lack o
16


dww kwu
k
V)()(...)('
1
*1
2

and since is dec easing in x, nega i e alues o
)(' xu 1
kw

a e being mul iplied by la ge alues
han a e posi i e alues. So, in iew o (A1), he de i a i e o V wi h espec o *

is nega i e.
Then he slope o an indi e ence cu e
*
/
**
*
*






 VV
d
d
S
is posi i e.



dww u
V)((...)''
*2
2

is nega i e since is nega i e and
)('' xu


dww kwu
k
V)()(...)(''
1
*
2
1
2
2
2
2

is nega i e o he same eason. The sign o


dww kwu
k
V)()(...)(''
1
** 1
2
2

is unclea . I is posi i e i is posi i e and nega i e i is nega i e, bu in ei he
)(''' xu )(''' xu
si ua ion he Cauchy-Schwa z inequali y implies
2
2
2
2
2
2
**
** 
















VVV
is non-nega i e. So V is a conca e unc ion o and , indi e ence cu es a e con ex, and he
*

*

equi alence o analysis o he expec ed u ili y app oach is e iden . While conca i y is no
σμ /
essen ial o con exi y o an indi e ence cu e and ha quasiconca i y will su ice, conca i y implies
quasiconca i y. Also, he main ocus o his pape is o u ilise hese esul s o ex end he equi alence
o dis ibu ions ha a e no loca ion-scale, bu a e ans o mable o ha amily.
APPENDIX B
EQUIVALENCE FOR TRANSFORMABLE DISTRIBUTIONS AND RESTRICTED U(X)
P o ing Mono onici y
To show ),(


Vsa is ies mono onici y.we ha e o p o e ha he e ms o (1)

17



*
and



*
a e posi i e and ha he e ms o (2)



*
and



*
a e nega i e. The dis ibu ion o x is no loca ion-scale, bu ha o y=h(x) is. Now h(x) is assumed
inc easing and conca e. The in e se ans o ma ion x=g(y) mus be such ha x=g(h(x)). Then
.1 x
h
y
g





So g(y) is inc easing in y. Also
2
2
2
2
2
0x
h
y
g
x
h
y
g
















and since h(x) is conca e, i s second de i a i e is nega i e and so he second de i a i e o g(y) is
posi i e. Tha is, g(y) is inc easing and con ex. Clea ly
dy
y
yg
u
l
c
c











 



)(
1
o , wi h he same subs i u ion as in Appendix A,




u
l
c
c
dww
k
kw
g)())(**(
2
1

.
Taking he limi s o in eg a ion as unde s ood and, o con enience, omi ing a gumen s o g



,)('
*dww g


which is posi i e since g(y) is inc easing in y.


dww kwg
k)()('
1
*1
2


and since is inc easing in y, nega i e alues o
)(' yg 1
kw

a e being mul iplied by smalle alues
han a e posi i e alues. So, in iew o (A1), he de i a i e wi h espec o *

is posi i e. Also
18

dy
y
yg
u
l
c
c











 




2
2)(
1
o









u
l
c
c
dww
k
kw
g)())(**(
2
2
1
2

.
Then
.)(')(2
*
2
*
2dww gg












No ing ha , by de ini ion,
(B1)
dww g )()(0


and emembe ing is inc easing, nega i e alues o
'g


g
a e being mul iplied by smalle alues
han a e posi i e alues and so (B1) implies he de i a i e o

wi h espec o is posi i e. Again,
*

.)()(')(
2
*
2
*1
2
2dww kwgg
k











When and
1
kw 


g he sign inside he in eg al is posi i e. When and
1
kw 

g i
is posi i e and when and
1
kw 

g he sign inside he in eg al is also posi i e8. Then he ac
ha is inc easing ensu es he de i a i e o
'g

wi h espec o is posi i e. So he e ms in he
*

ma ix
M
=




















**
**








a e all posi i e. Now













d
d




















**
**


















*
*


d
d
and o cou se
8 Since

is he mean o a posi i e inc easing con ex unc ion o y, i will be much la ge han ,
which is ze o o a no mal o uni o m, .5572 o a Gumbel and 1 o a wo pa ame e exponen ial.
1
k
19











*
*


d
d




























**
**
.












d
d
So




























**
**
=1
M
and e iden ly he diagonal elemen s a e posi i e and he o diagonal elemen s nega i e. So



*
and



*
a e posi i e and



*
and



*
a e nega i e and ),(


Vsa is ies mono onici y.
In es iga ing Quasiconca i y
We need ),(


Vquasiconca e in

and

o ensu e he indi e ence cu e is con ex. The
condi ion o his is ha
3
2
2
2
2/2
222


































































VVVVVVVV
(B2)
be posi i e. Now







































*
*
*
*2
2VVV
o , wo king ou e ms
2
*2
*2
*2
*
2
*
2*
2**
**
2
2
*
2*
22

























































VVVVV .
Simila ly
2
2


V
is
20
2
*2
*2
*2
*
2
*
2*
2**
**
2
2
*
2*
22

























































VVVVV
and


V
2
is
.
2
*2
*
*2
*
**
2*
2****
**
2**
2*
2




































































































VV
VVV
Subs i u ing hese in o he nume a o o (B2) many e ms cancel and he emaining e ms gi e




















































2
*2
*
****
2
*2
*
222
2




VVVVVVV
2
****

























2
*
*
*
*2
*2
*2
*2
*










































VVVV
2
*
*
*
*2
*2
*2
*2
*










































VVVV




































































*
*
*
*
*
*
*
*
*2
*
*2
*
2VVVVVV .
The i s e m o his exp ession is he p oduc o a squa ed e m and he condi ion ha is
),( **

V
quasiconca e. Bu ha condi ion mus be ue because Appendix A p o ed conca e in
),( **

V
*

and . So he i s e m is non-nega i e. Howe e , i is unclea ha he sum o all e ms is non-
*

nega i e o all (

,

) space. In he ex eme si ua ion o u(x) = h(x) = y, and all
*** ),(

V
second de i a i es o V wi h espec o and a e ze o and he i s e m o he exp ession is ze o.
*

*

As migh be expec ed he emaining e ms educe o

























































2
*
2
****
2
2
*222
2
*














. (B3)
So we equi e , he expec a ion o he ans o ma ion, o be quasiconca e in
*


and
21

. Now i we ake x log no mal and y = h(x) = log x, so ha y is no mal, (B3) can be shown by a he
edious e alua ion o e ms o be
2222
22
)(
)2(




and his can be nega i e unless

2. Remaining wi h x logno mal, bu any u(x) ha is a
conca e unc ion o y = log x, e y labo ious manipula ion enables (B2) o be w i en























































**
*
21
2
)( *
2*
2
*
*
22
*
*
3
*
*
22
22









S
d
dSSS
S, (B4)
whe e
*
*
*
*
*
*
*






S
S
S
d
dS .
Ano he o mula ob ained du ing he de i a ion o (B4) is



































































**
*
)2(
21
2
1
*
2*
2
*
*
*
22
2
*
*
2
*
*
2
*
*
22












SS
SSS
S
S, (B5)
which is employed in Sec ion 5.
Re u ning o (B4), he second e m wi hin chain b acke s in (B4) could be posi i e, nega i e o
ze o depending on whe he he elas ici y o he slope along he indi e ence cu e associa ed wi h
),( **

V is g ea e han, equal o, o less han, uni y. The i s e m could be nega i e oo, because
*** ),(

V implies and all e ms wi hin he chain b acke s anish excep , 0
*S22 2


implying, as be o e,

2 o posi i i y. So he egion o quasiconca i y in ),(


space
depends on bo h he quasiconca i y o he ans o ma ion expec a ion and he elas ici y o he
indi e ence cu e in .
),( **

Co esponding condi ions o (B4) can be ob ained o o he dis ibu ions ans o mable o loca ion-
scale o m, bu a emp ing o ob ain a single o mula applicable o all he ele an dis ibu ions and
u ili y unc ions seems o esul in almos in ac able algeb aic exp essions. In any e en , he

22
quasiconca i y egion o pa icula cases is mo e easily ob ained by di ec examina ion o he
con exi y o he indi e ence cu e in ),(


space ia he posi i i y o






S
S
S
d
dS
wi hou explici conside a ion o o i s indi e ence cu e. The examples in sec ion 4
),( **

V
exempli y his.
I may be wo h emembe ing ha o su icien ly small

, he expec a ion o any u ili y unc ion
can be w i en

)(''
2
1
)(),()( 2

uuVxuE  ,
so ha
)(
)('
)(''




A
R
u
u
S ,
whe e is he A ow-P a coe icien o absolu e isk a e sion. Then
A
R



















)(
1)( 2A
A
R
R
S
S
S
d
dS
and his is posi i e, e en i dec easing absolu e isk a e sion holds, i

is small enough. So e e y
indi e ence cu e commences wi h a con ex egion, he ex en o which depends on he p ope ies o
he dis ibu ion and he u ili y unc ion.
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Uni e si y P ess.
A nold, B. C. (1983) Pa e o Dis ibu ions, Fai land, Ma yland: In e na ional Co-ope a i e Publishing
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de Unce ain y 2nd Ed., New Yo k: Sp inge -Ve lag (Is
Ed. 1983, Ams e dam: No h-Holland).
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Sinn, H.-W. (1989) Economic Decisions un