Gene alised T ansla ion o Indi ec U ili y Func ions
DENIS CONNIFFE
Na ional Uni e si y o I eland Maynoo h
Abs ac
This pape conside s he de i a ion o new demand sys ems om exis ing ones h ough eplacing an
indi ec u ili y unc ion by , whe e p is a ec o o p ices and y
is income. This is a gene alisa ion o Go man ansla ion
),( yU p})/(,{ j
ypyyU jj


p
,( yU )
jj p



p and will be shown o be
e ec i e in e ms o p oducing new demand sys ems wi h bo h good egula i y and lexibili y
p ope ies.
JEL Classi ica ion: D 11
Keywo ds: T ansla ion, indi ec u ili y unc ions, demand equa ions.
Add ess o co espondence: Denis Conni e, NIRSA, Economics Depa men , Na ional
Uni e si y o I eland, Maynoo h, Co. Kilda e, I eland. Tel: 353 1 7086299; Fax: 353 1 7083934;
e-mail: [email p o ec ed]
2
I INTRODUCTION
Go man (1975) in oduced he “ ansla ion” de ice o inco po a e ex a pa ame e s in o u ili y
unc ions and demand equa ions. I is he o iginal indi ec u ili y unc ion, whe e p is a ),( yU p
ec o o p ices and y is income, he ansla ed u ili y unc ion is
),( jj pyU


p, (1)
whe e he j

a e he subsis ence quan i ies, y is assumed jj p


and summa ion is o e n
commodi ies. Go man showed ha i we e he o iginal demand equa ions, he ansla ed
),( yqip
equa ions a e
ijji pyq





),(p. (2)
Fo example, i is he simple homo he ic u ili y unc ion y/P, whe e P is a weigh ed
),( yU p
geome ic mean o p ices, )log(log jj pP

 , he demand equa ions iii pyq /


a e ansla ed o he amous S one-Gea y linea expendi u e sys em (LES)
)( jj
i
i
ii py
p
q



 .
The idea o his pape is o eplace (1) by he mo e gene al ansla ion o o
),( yU p


















j
y
p
yyU j
j


,p. (3)
This educes o (1) i all 1
j

. The condi ion jj py


 is eplaced by
. (4)

j
jypy jj



1
which will hold o posi i e j

i y is no oo small. I a i

is nega i e, i

mus also be nega i e.
This pape is pa icula ly conce ned wi h how gene alised ansla ion can p oduce demand sys ems
wi h bo h good egula i y and lexibili y p ope ies. Regula i y means ha , gi en app op ia e anges
o he pa ame e s, he indi ec u ili y unc ion complies wi h he cons ain s implied by a ional
economic beha iou .1 Ideally, his should be possible o all p ices and incomes (global egula i y),
bu
1 Tha is, a con aximises di ec u ili y unde a budge cons ain . This implies he indi e
u ili y unc ion ),( yU p should be homogeneous o deg ee ze o in income y and p ices p, non-
sume m c
3
should a leas hold o all alues o hese a iables ele an o he si ua ion unde s udy. Flexibili y is
also equi ed in ha he co esponding demand sys em, while sa is ying egula i y, should be able o
model a easonably comp ehensi e spec um o consume beha iou - so he possible alues o income
and p ice elas ici ies, which a e unc ions o pa ame e alues, should no be se iously es ic ed2. The
in luen ial ‘ lexible unc ional o ms’ app oach, employing Taylo se ies app oxima ions o gene al
u ili i y (o cos ) unc ions, sough sys ems embodying lexibili y and hoped o egula i y, bu i seems
ha o en hei lexibili y depends on hei pa ame e s being allowed o ake alues ha con adic
egula i y3. Such models gene ally canno es i obse ed consump ion pa e ns do o do no acco d
wi h economic heo y. The app oach in his pape will be o s a om globally egula sys ems and o
imp o e hei lexibili y by gene alised ansla ion4.
P ope ies o he gene al ansla ed u ili y (3) a e examined in sec ion 2 and he co esponding
demand sys em de i ed. Income and p ice elas ici ies a e ob ained and p esen ed in e ms o he
elas ici ies o he pa en sys em and he pa ame e s o he gene alised ansla ion. Sec ion 3 illus a es
hese esul s by conside ing a pa icula case, ai ly pa simonious in pa ame e s, ha is in e es ing in
i s own igh . Sec ion 4 applies gene alised ansla ion o mo e pa ame e ich, hough s ill globally
egula , u ili y unc ions. In sec ion 5 he possible applica ion o gene alised ansla ion o non-
globally egula u ili ies is discussed. Finally, in sec ion 6, connec ions be ween gene alised ansla ion
and Hou hakke ’s (1960) indi ec addilog sys em a e explo ed5.
II. TRANSLATED DEMAND EQUATIONS AND ELASTICITIES
Le
dec easing in y, non-inc easing in p, and con ex o quasi-con ex in p. These cons ain s imply
co esponding condi ions (agg ega ion, homogenei y, Slu sky symme y and nega i i y) on he demand
equa ions.
2 Mo e o mal de ini ions o lexibili y exis , di e ing in de ail. See, o example, Diewe (1974).
3 Ca es and Ch is iansen (1980), Ba ne and Lee (1985) and Coope and McLa en (1992) ha e
discussed he di icul y o econciling lexibili y and egula i y o such sys ems.
4 O cou se, he e a e o he app oaches o imp o ing egula i y p ope ies. Ba ne (1983), Ba ne and
Lee (1985) and Chal an (1986) epo ed wide egula i y egions esul ing om app oxima ions based
on Lau en , Mun z-Sa z and Fou ie expansions a he han Taylo se ies. Lewbel (1987) and Coope
and McLa en (1996) ha e also p oposed sys ems.
5 Lewbel (1985) has al eady used he e m gene alised ansla ion in a pape on inco po a ing
demog aphic e ec s in o demand equa ions, which was a heme o conside able in e es o Go man.
Bu he gene alised ansla ion o his pape is so ob iously a gene alisa ion o Go man’s ha i seems
inapp op ia e o call i any hing else.
4
j
y
p
y


, yz j
j








ha he ansla ed u ili y is I s homogeneous o deg ee ze o in income and
as i ough o be i a alid u ili y unc ion, i is clea ha is also.
),( zU p. ),( yU piso
p ices, ),( zU p
y
z
z
zU
y
zU






),(),( pp



 











yyz
j
jj
j
j

1 (5) 












jj pp
zU

),(p
ssuming he o iginal u ili y is non-dec easing in income, he de i a i e o wi h espec o z
non nega i e. Taking he e ms wi hin he chain b acke s, equa ion (4) implies he i s and second
ms combined a e posi i e while he hi d is also posi i e because, as p e iously men ioned,
),( zU pA
is
i

e and
i

a e p esum o hed o ha e he same sign. S e ansla ed u ili y is non-dec easing in income.
y
z
z
U
p
U
pi
zU
i







),(p


















p
UU i
ii

(6) 



1
i
yzpi

u ili ies.
show ha abo e so s will be
gula . This co esponds o he si ua ion wi h Go man ansla ion. Fo example, he simple
omo he ic u ili y unc ion y/P,

So assuming he o iginal u ili y is non-inc easing in p ices, he ansla ed u ili y is also. P oo s o he
con exi y o ansla ed u ili ies a e mo e complica ed and depend on he o ms o he o iginal
They a e p o ided in Appendix 1 and me income le el he ansla ed sys em
e
)log(log jj pP



h , is globally con ex in p ices, while
ypy jj /)(

 , which gi es he LES, is con ex i
. (7)
s
nds o inc ease wi h ime. The e m ‘e ec i e global egula i y’ has been applied o egula i y
222 2)( ppy


iijji
So al hough Go man ansla ion inc eases lexibili y by in oducing ex a pa ame e s, i may in alida e
egula i y a e y low incomes. This has no been seen as a di icul y, because, a leas o analyse
wi h ime se ies da a, in e es ocuses on in e ences alid o ecen o cu en ime pe iods and income
e
5
e e ywhe e excep a low incomes6. Gene alisa ions o (7) o gene alised ansla ion can be deduced
a e sec ions.
F om Roy’s lemma he demand equa ions a e
om he esul s o Appendix 1 and will appea in l



y
jij )1(1 




















j
i
p
y
p
z
zU
p
zU
yq
j
i
ii
i
i




),(
/
),(
),(
1
pp
p,
e deno es he demand equa ion om he ansla ed u ili y. Also by Roy’s lemma,
i
q
whe























j
y
p
yyqzq
z
zU
p
zU j
jioio
i


,),(
),(
/
),( pp
pp ,
whe e eno es he demand unc ion de i ed om he o iginal u ili y unc ion. So
io
qd









































j
i
j
y
p
y
p
y
p
yyq
yq
j
jij
i
ii
j
jio
i





)1(1
,
),(
1
p
p ,
o , mo e idily,


















1
,
1i
y
p
zi
ii


p,
o , in e ms o budge sha es


),( q
V
yq ioi p




















i
y
p
y
z
zw
V
yw i
iiioi


,
1
),( pp
, (8)
whe e
j
y
Vjj




)1(1 pj







c oss-p i
ms o and ,
he income and p ice elas ici ies o he o iginal sys em. They a e:
and ),( ywi pis he budge sha e o he ansla ed sys em as a unc ion o p and y, and ),( zwio pis
he budge sha e o he o iginal sys em as a unc ion o p and z. The income, own-p ice and ce
))( zp,
io
E),( zeiko p
elas ici ies o he ansla ed sys em a e de i ed in Appendix 2, in e
6 The e m seems due o Coope and McLa en (1996).

6





  jjoi
i
io
jj iio
i
io
i zw
yw
zw
yV
z
ywzE
yw
zw
E

),()1(
),(
),(
1),(),(
),(
),( p
p
p
pp
p
p









 yV
z
yw
zw
zEzwzeywzee
i
io
ioioiioi iiioii ),(
),(
1),(),(),(1),()1(),( p
p
ppppp














 ywzee k kikoik ),()1(),( pp

yV
z
yw
zw
yw
yw
zEzwze
k
ko
i
k
ioioiko ),(
),(
1
),(
),(
),(),(),( p
p
p
p
ppp
he
xample o he nex sec ion illus a es his.
D TRAN
E en i he elas ici ies o he o iginal sys em a e e y es ic ed, hese elas ici ies a e much less so. T
E
III GENERALISE SLATION OF THE CONSTANT BUDGET SHARE MODEL
Re u ning o PyU /, whe e )log(log jj pP



, he o iginal demand equa ions, in
budge sha e o m, a e
iio
w

, o cons an budge sha es. Income and own p ice elas ici y a e uni y
and c oss-p ice elas ici ies a e ze o. F om (8) he ansla ed demand equa ions a e








































j
j
y
p
y
p
y
p
w
j
jj
i
i
ii
j
ji
i





11
1
, idily,.
o



















i

i
iiii y
p
y
z
V
w

1 (9)
is shown in Appendix 1 ha his sys em7 will sa is y e ec i e global egula i y i
I
22
222
1
22)1(










ii pp ii

(10)









 y
pz
y
pz iiiiiiii

,
which holds i 

iiji




,1, and .1



I could e en hold o an 0
i

i
 10 i

, al hough ha may be pu ely academic. Ob iously, (10) educes o (7) i all j

= 1.
I (9) is o ha e much p ac ical alue, i should be mo e lexible han he LES, which i becomes
7 This sys em has been examined in g ea e de ail in Conni e (2002a), bu was no hen unde s ood as
a case o gene alised ansla ion.
7
when all j

= 1. The limi a ions o he LES include he linea i y o all i s Engel cu es8, i s inabili y
e inc
pa a io o income elas ici ies
ansla ed income elas ici y, d opping, o con enience,
ip s indica ing ansla ed and o iginal,
o ca e o in e io goods and he ying o p ice e c s o ome aspec s o p ice changes. In
icula , he a io o he c oss-p ice elas ici ies ik
e and jk
e equals he
i
Eand j
E, and he possibili y o complemen a y goods is excluded.
F om he o mula o he p e ious sec ion o
he subsc




i
jjjji
i
iwyVw
 
  iii z
wE )1(
1



.
y
ne anges d
i
Eis no gene ally a simple mono onic unc ion o income. A commodi y could be a luxu , o a
cessi y a di e en o income, and could e en be an in e io goo ( o a posi i e i

and small
i

). Howe e , as y he budge sha es (9) end o cons ancy and i
E o uni y, which is no
un easonable and is a p ope y o many o he demand sys ems. Tha Engel cu es can ake a la ge
he demand equa ion and he income elas ici y, al hough as
F om he o mulae o he p e ious sec ion, he own-p ice and c oss-p ice elas ici ies
a ie y o shapes is clea om y
ey will app oach linea i y. h
a e








 wwe iiii

)1(1 yV
z
wi
i
ii


1
and











 Kk
i
kik yV
w
w
e

1,

 iz

s o c oss-p ice elas ici ies need no equal a ios o income elas ici ies. The compensa ed p ice
so a io
e ec
y
q
q
p
qi
k
k
i





is
8xample, Lau, 1986) is
Eons an p ices, a e no
One o he ew gene ally ag eed indings om empi ical s udies (see, o e ha
ngel cu es, he ela ionships be ween expendi u e and income a c n-linea o a
leas some goods.
8
)}1()1({)}1(1{ i
ikki ppppyVyVy
ikkikiki qqy
zz
qq

kjjjjki w







and his can gene ally be nega i e o posi i e, allowing o complemen s as well as subs i u es. So he
ex a n pa ame e s con ained in (9), ela i e o he LES, do subs an ially inc ease lexibili y9. O
ou se, any demand sys em wi h jus 3n-1 pa ame e s, will all sho o ull lexibili y i he e a e mo e
lly i da a
e sca ce, o obse ed p ice anges do no embody su icien independen a ia ion o es ima e a e y
e ailed model o in e ac ing p ice e ec s.
NIOUS UTILITIES
igina . Ini ially, a u il
o so10, pa s
om wo u ili on ame e s, a e
onca e in p ices. Then (Conni e, 2002b) he sum, p oduc and he ecip ocal o ecip ocals
(ha monic mean) a e also globally egula u ili y unc ions. Fo example, wi h and
c
han a ew commodi ies. Some imes, howe e , pa simony o pa ame e s is desi able, especia
a
d
IV GENERALISED TRANSLATION OF LESS PARSIMO
ea e lexibili y o p ice e ec s can be modelled by gene alised ansla ion o a less pa simonious, G
bu s ill globally egula , o l u ili y i y o he o m Py/will be conside ed, whe e
is a unc ion o mo e han n pa ame e s, bu is conca e in p ices. P
I is easy o gene a e a globally egula homo he ic u ili y unc ion wi h 2n, ame e
y unc i s /y and 2
/Py , whe e 1
P and 2
P, bo h unc ions o n pa
1
P
c
j
j
pP

=
1
jj pP

=
2, wi h i

and i

posi i e, he ha monic mean o he u ili y unc ions is



jjj pp
y
Uj


,
which is globally egula wi h 2n – 1 pa ame e s. I gi es he demand sys em
jjj
iiji
ij
j
wp+p
p+p
=





,
which s ill, o cou se, gi es a uni a y income elas ici y, bu p ice elas ici ies o
ollak (1972) desc ibed a class o demand equa ions o he o m ),,/( Wyp q iii

9 P whe e W is a
homogenous unc ion o all p ices and income, as exhibi ing “gene all alised sepa abili y”. Because
p ices, excep , ake e ec h ough W, he e a e implied symme ies in how commodi y demands
a e a ec ed by p ices o o he goods. The LES and he Indi ec Addilog Sys em (IAD) o Hou hakke
(1960) a e o his class. Bu (9) is no , p o ided he
i
p
j

a e non ze o.
10 Cons ain s such as 0

j

educe pa ame e s by one.
9
)( jjji
iiiii ppw j



)1( j
we


pj


and
j
pj


)( jjji
kiikik ppw j



we

 .
sys em wi h 4n pa ame e s, gi en by equa ion (8), wi h
elas ici ies gi en by subs i u ing he abo e in o he o mulae o sec ion II. Many o he sys ems a e
o p c ion
Combining simple u ili y unc ions is no he only way ex a pa ame e s could ha e been ob ained.
iginal homo he ic u ili y unc ions wi h mo e han 2n pa ame e s a e qui e possible. Fo example,
Gene alised ansla ion hen p oduces a -1
ob ainable by o he combina ions ai s o u ili y un s.
O


2
1
2
1
kjjk pp
y
U

,
could ha e been aken whe e he jk

a e posi i e and jk

= kj

11. This has n(n+1)/2 pa ame e s and
gi es he demand sys em
2
1
2
1
2
1
2
1
2kjjk
jiji
i
pp
pp
w




,
wi h p ice elas ici ies
2
2
1
2
1
)(4
2
1
kjjki
iii
iii
ppw
p
we




and
2
2
1
2
1
2
1
2
1
)(4 kjjki
ikki
kik
ppw
pp
we



 .
spi e o a ela i e p o usion o pa ame e s his sys em s ill embodies es ic ions on p ice
ec s as
In
e
y
q
p
i
k
i


=
qq
k

2
2
1
2
1
ki
22
)(4 kj
ik
pp
pp



si i e i
11 
jk
is po ik

is, so uling ou complimen a i y. So gene alised ansla ion, which gi es a
11 This is ac ually he homo he ic case o he gene alised Leon ie u ili y unc ion.
16
iii p
P
Pp
P
P
pz
U
z
U
Pz
U













log11
- and 0,
1
2
2
2
2
.
u he mo e F










































ji
22
2
i
2
2
2
2
p
Plog
p
Plog
P
z
log
p
Plog
P
z
log
jiji
ii
pp
P
P
z
pp
U
and
p
P
P
z
p
U
o (A1) and (A2) become
and s















































2
2
ii
2
i
2
2
p
z
p
Plog
2
p
Plog
log1
ii p
z
z
p
P
z
P
and


P


























ijjiji
2
p
z
p
Plog
p
z
p
Plog
p
Plog
p
Plog
log1 z
pp
P
z
ji
.
be ew i en as These can









































2
2
2
2
2
211loglog1
i
i
ii
ip
z
z
p
z
p
z
z
p
P
z
p
P
z
P
and

































 jijjiiji ppzp
z
pp
z
p
z
pp
z




 
 
 
 
zzzP
z
zPP
P
11log1loglog .
a i
he Hessian ma ix o wi h espec o p ices, which is posi i e
e ini e17, plus which, is posi i e semi-de ini e, whe e G is he ec o wi h i h e m
2
1
o he Hessian m x (wi h espec o p ices ) o he ansla ed u ili y is (apa om he S
PlogP/1 mul iplie ) z by minus
'GG , d
ii p
z
z
p
P
z



1log ,
lus a diagonal ma ix wi h i h e m
p
1
2
i

2
)1( 











y
p
p
p
zi
i
iii
i

minus , whe e H is he ec o wi h i h e m
zHH /'
17 I
P
is conca e, is conca e. Plog

17
1










i
y
p
p
zi
ii
i


.
While 'H
H
 is nega i e semi-de ini e, he addi ion o i o he diagonal ma ix wi h i h e m
2
22
2

 i
p
z

22
 i
gi es a nonnega i e de ini e ma ix (diagonals posi i e and p incipal mino s o highe o de ze o).
ed u ili y is con ex wi h espec o p ices i he n
ul iplied by z minus he diagonal ma ix wi h elemen s
2









yp ii
i

So he ansla ega i e o he Hessian o log P
m
z
y
p
p
yiiiii 


pii
i
i
i
22
22
1
2)1(

















(A3)
a i e de ini e.
on exi y o he cons an budge sha e model
is nonneg
C
)log(log jj pP


Now le . The Hessian o log P wi h espec o p ices is So U is
on ex wi h espec o p ices i
diagonal.
c
z
y
p
p
y
p
p
z
ii
i
ii
i
i
iii
i
i
22
22
1
2
2)1(























is posi i e, o
22
2222 2)1( 






ii p
p
p
zi
iii
i
iiii

, (A4)
1











y
z
y
pi

hich is equa ion (10) o sec ion 3. I all h
ej

a e posi i e, he le hand side is o o de , while
e igh hand side e ms a e o o de and , so since
2
y
w
i
y

2i
y

22
i

his posi i e he condi ion holds
o ided y (and hence z) is no small. The i s igh hand side e m is nega i e i 0 <
i

p < 1, so ha a
e o i

migh e en be compa ible wi h con exi y in ha si ua ion. Fo a leas one o he j

znega i e,
be he smalles (mos nega i e). The g ea es powe o y in z is hen wi h coe icien
s
y

1
le s

18
s
ss s
p


(

is nega i e i s

is). So he le hand side de , while he igh hand
ms a e o o de and and i is clea he c i ical e m is i= s. Compa ing coe icien s
, he condi ion is
s
y

22
is o o
si
y

2i
y

22
e

22g ea e han . This will be ue i 0 >
2
s

s

s
yss


 > -1, e en i s

ois
small (o e en ze o). So e ec i e global egula i y holds18 i 



iiji




,1, and  .1

ain con ex wi h espec o p ices i he nega i e o he Hessian o log P
ul iplied by z minus (A3) is nonnega i e de ini e. Le he smalles eigen alue o minus he Hessian
a ix o log P, ha is, he ma ix wi h i,k h e m
Con exi y wi h Pconca e in p ices
The ansla ed u ili y is ag
m
m
ki pp
P


log
2

. I log P is s ic ly conca e minus he Hessian is posi i e de ini e and can be w i en equal o be
Q+

I, wh de e I is he iden i y ma ix, Q is posi i e semi-de ini e an

is posi i e. Then he condi ion
o egula i y is
22
222 )1 













p
z
p
zi
ii
iii



.
1
2
(

ii
yypi
i



(A5)
ms he same as (A4) wi h eplacing i

/2
i
p

This see , bu i is ac ually a much s ic e condi ion. I s
equi alen , whe e s

deno es he smalles i

would ha e been
2
2
22
1
22)1(











ip
pz
p
pz ii


2









i
yy iiiiiiis


,
all i and no jus i=s. So (A5) p obably exagge a es he income equi ed o e ec i e global
gula i y, al hough as he le hand side is o o de (assuming
o
2
y)

i

e , while he igh hand side
ms a e o o de and , he e is no doub i will hold as y inc eases. Less demanding
ed o sp i ic ases o P. Again, i
i
y

2i
y

22
e

condi ions could be de i ec c =0 o some obse a ions (log P no
18 The condi ions on he j

and j

co espond o hose o he egula i y o he u ili y unc ion o he
indi ec addilog sys em. The alidi y condi ions o he IAD ha e been deba ed in he li e a u e mo e
han once, as he exchanges be ween Gamela os (1973, 1974) and Some maye (1974), and be ween
Akin and S ewa (1979) and Mu y (1982) es i y.
19
i

s ic ly conca e) e ec i e global egula i y could s ill hold i 0 < < 1, al hough he case is
e

p obably no o p ac ical in e es . Fo nega i he same a gumen as be o e gi es he condi ion
nd again his will be ue i 0 >
ss
p


2>2
s

as

> -1.
ice index.
Hessian A(3) o
he He
Con exi y wi h Pa anslog p
As be o e, con exi y equi es he nega i e o he o log P mul iplied by z minus
be nonnega i e de ini e. The nega i e o ssian o Log P is


 LL
,
i
p/1 ,

is he ma ix o coe icien s jk

whe e L is he diagonal ma ix wi h i h elemen and
is he diagonal ma ix wi h i h elemen I , ollowing es ima ion o he

./ 2
ii pS jk

,  is ound o
be conca e he con exi y equi emen becomes
22
22
1
22)1(











ii
z
p
pzS i


.
2






y
p
p
y
i
iiiiiiii

his gi es he condi ion o sec ion V. The condi ion is he same as (A4) wi h eplacing
i
S i

T and
e same a gumen s apply as ega ds he equi ed anges o
i

and i

. Bu while he i

hwe e
ons an s, he a e unc ion o p ices and he alidi y o he a gumen depends on hei being
exi y wi h a Gene alised Leon ie u ili y
e u n o (A1) and (A2) o ake accoun o e ms ha no longe anish. Howe e ,
uch o he p e ious app oach will s ill apply. The ( ansla ed) u ili y unc ion is
i
Sc
osi i e. p
Con
I is now necessa y o
m
1
2
1
2
1
2
1
2


















U
















z
p
z
p
z
pj
i
ij
j
j

nd could be w i en in he o m , whe e
*/ Pz
a
2
1
2
1
2
1
2
1
2* jiijiij pppzP 

,
20
al hough is now a unc ion o income, as well as p ices. The con exi y o he o iginal u ili y wi h
espec o p ices mus imply ha ha is conca e in p ices and he e o e is also.
*P
*P *logP
2
i
*
*2
2
*2
2
p
logz
*log 














P
Pp
P
P
z
p
U
ii
nd
a

iii pz
PzPPU 














*logloglogz 2**2
P
P
p
P
P
pz 
 
 




 pz
*log1
*
i
2
*
*
i
i
C
p
P
P



*log1
*
, say. (A6)
Al hough he i s e m o (A6), which is easily shown o be
2
1

i

2* )(2 






i
p
z
P
z/1 beis o o de , he e m , which may be shown o
i
C
2
1
2
1
2
jiji pp
pz 





2
1
3*
1
)(
jiij
i
pP
. This will be impo an la e . Re u ning o (A1), wo o i s e ms
z/1
is o o de
ii
ip
z
pz
U
p
U






2
2
22
now gi e
i
i
i
ippzP
PpP 








*
*2* i
z
C
z
p
z
z
P
z















 2
11
p
log1
*22
i
*
.
he es o (A1) is
Pz log
2
T
2
2
2
2
2
i
ip
z
z
U
p
z
z
U
















. (A7)
a ly, e ms om (A2)
Simil
ijjiji p
z
pz
U
p
z
pz
U
pp
U











222
gi e
21
jijiji p
z
p
z
zPp
z
z
P
z
p
z
z
P
z
Pz 




log1
*log
** P
pp
P






















 *
11
p
log1
p
**2
us
 ji
pl
i
ji p
C
p
C
j
zz




nd, om he emaining e m o (A2), a
ji p
zzU 
2
2
. (A8)
p
z

ome e ms a e almos iden ical o hose occu ing in he case, wi h
Pz/*
P
Sins ead o , and,
llowing om m o a posi i de ini e ma ix (minu
y he Hessian o ) plus wo posi i e semi-de ini e ma ices minus wo diagonal ma ices
i h diagonal e ms
P
as be o e, can be exp essed as o he su e s *
/Pz
*
logPb
w
2
 2
*
2






i
p
z
zP and 2
i
p
z
z
U




.
espec i ely o o de
i

22/1

and i


2/1
I is wo h no ing ha hese e ms a e in income.
, hose in and can be seen as ollowing om
i
Cj
C
Tu ning o he ex a e ms in ol ed
'
*
*1
'' 















 pp
zz
zCPLL , (A9) 
zP
C
whe e Lis he ec o wi h elemen s
i
ip
z
CzP zP 


*1, *
n s . The i s ma ix in (A9) is posi i e semi-de ini e. The second
l ma ix wi h diagonal e ms
e diagonal ma ix wi h diagonal e ms
i
C
and Cis he ec o wi h eleme
is nega i e semi-de ini e, bu becomes posi i e semi-de ini e i he diagona
2*
2i
zCP is added o i . The same applies o he hi d e m i h
2
*
2









i
p
z
zP
is added. These diagonal ma ices ha e o be sub ac ed again, o cou se.
The i s e m in (A7) and he (A8) e m a ise om he ma ix

22
'
2
2






















pp
zz
z
U
and since








2
1
2
1
2
1
2
1
2
2
3
jiijjj pp
z
p
P
p
U


(A10)
is nega i e he ma ix is nega i e semi-de ini e. Adding he diagonal ma ix wi h diagonal e ms




3
*
2
jj
z
2
2
2





zU
2
 
i
p
z
p oduces a posi i e semi-de ini e ma ix. No e ha (A10) is o o de 2
3

z.
o he inal sub ac ed diagonal ma ix is
So he i h elemen
2
2
2
*
4



p
z















i
z
U
zP
2*
2
22i
i
zCP
p
z
z
U




 , (A11)
whe e, o cou se,
1










i
y
p
p
zi
ii
i


and
1
2
2)1( 











i
y
p
p
p
zi
i
iii
i


.
ei

F om ea lie commen s i is clea he e ms o (A11), assuming h posi i e, a e espec i ely o
de o
i
y

2
1
2, i
y


1
2
1

y.
2 and
As al eady said, minus he Hessian ma ix o , mul iplied by , is posi i e de ini e in
ices and i is easily seen o be o o de
*
logP *
/Pz
z o y. p So he same a gumen s as p e iously will
o income and own p ice elas ici ies o ansla ed sys ems
n enience o n , he dependence o quan i ies on p ices and income will no be indica ed
xplici ly excep o z, he ansla ed income.
The income elas ici y is
emons a e e ec i e global egula i y. d
Appendix 2: De i a ion
Fo co o a ion
e
23
=yq
y
i 





























1
1i
y
p
zq
V
i
iiio


.
y
q
q
y
Ei
i
i 

= V
1
























1
)1( i
y
p
yy
z
z
zq iiiiio


y
V
V
qi


.
y
qi


ha
No ing
and
j
y
p
yy
Vj
jjj












)1(
1
V
y
z


his is

ii
y
p
yV
q
y
p
yVz
zq

i
ijji
iiiiio
























)1(
1
)1( 1
.
o
S

i
E

ii
pq iiiiio



 1
)1( 1
y
p
VyVqz
z
q
yi
ijj
i i


















)1( .
Using

)((
1
zw
y
z
VwzqVq
y
p
y
p
y
p
y
i
ii 





p
ioi ioi
ii
ii
i
ii











 


,
e he and e he budge sha es, and ha
i
w)(zwio awhe
z
zq
zq
z
zEio )(
io
io 
)(
)(
becomes
i
E





  jjoi
i
io
jj iio
i
io zw
yw
zw
yV
z
wzE
w
zw

)()1(
)(
)(
1)(
)( .
The o icwn-p e elas ici y is
=
ii
i
pq
p

















i
i
i
i
ii p
q
q
p


e













1
1i
y
p
zq
V
i
iiio


.
V
1


























2
)1()( i
y
p
yp
z
z
zq
p
zq iiii
i
io
i
io


i
i
p
V
V
q


.
i
i
p
q

=
No ing ha
1
)1(
1










i
i
iii
iy
p
yp
V


,
1









pi
i
y
p
zi
ii


and
24
his is

yV
zq
qq
V
zq
q
pV
zq
q
z
zq
p
zq
V
io
i ii
io
i
i
iio
i
io
i
io }
)(
){1(
}
)(
{
)1(
}
)(
{
)(
1















.
o equals

ii
e
S














ze
zqio )(
)(
V
zq
q
y
p
q
zE
zq
zqp
Vq
io
i
ii
i
i
io
i
ioi
iio
i
)()1()1(
)(
)(

,
o








 yV
z
w
zw
zEzwzewzee
i
io
ioioiioi iiioii
)(
1)()()()1)(1()(

.
Th
e c oss-p ice elas ici y is















k
io
k
io
i
k
p
z
z
zq
p
zq
Vq
p)(
k
k
p
V
V
p


 



k
i
i
k
ik p
q
q
p
e












 zE
zq
ze
Vq io
i
iko
i
)()( Vq
zq
y
qpzqqpzq
k
kok k
k
iok kio )(
1)1(
)()(

=
o







  yVww
zEzwzewzee
k
ko
i
k
ioioikok kikoik 1)()()()1()(

.
 z
zww )(


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R