GENERATING GLOBALLY REGULAR INDIRECT UTILITY FUNCTIONS
Denis Conni e
Economics Depa men , Na ional Uni e si y o I eland Maynoo h
Abs ac
Despi e hei sca ci y in he li e a u e, an abundance o globally egula indi ec u ili y unc ions,
in ol ing as many pa ame e s as desi ed, exis s. They a e easily cons uc ed as a unc ion o
simple homo he ic componen u ili ies.
JEL Classi ica ion: D 11
Keywo ds: Global egula i y, indi ec u ili y unc ions.
Add ess o co espondence: Denis Conni e, 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: Denis.Conni [email protected]
2
1. INTRODUCTION
Regula i y means ha an indi ec u ili y unc ion complies wi h he cons ain s implied by a ional
economic beha iou 1. Regula i y is global i i holds o all (posi i e) p ices and incomes, gi en
app op ia e anges o he alues o pa ame e s occu ing in he unc ion. This sho pape is conce ned
wi h he gene a ion o such globally egula unc ions.
2. COMBINING HOMOTHETIC COMPONRNTS
Conside a se o homo he ic u ili y unc ions, each o he o m
(1)
,/),( kk PyyU k
p
whe e k
is posi i e and is inc easing in p ices, homogeneous o deg ee
k
Pk
in p ices and is a
conca e unc ion o p ices wi h nega i e de ini e o semi-de ini e Hessian. Then he ecip ocal o is
k
P
con ex in p ices, because
2
2
2
2
32
12 12
i
k
k
i
k
ki
k
p
P
P
p
P
Pp
P
and
ji
k
k
j
k
i
k
k
ji
k
pp
P
P
p
P
p
P
P
pp
P
2
23
12 12 ,
so he Hessian o is
1
k
P
'
3
2
pp
kk
k
PP
P,
which is posi i e semi-de ini e, minus he Hessian o , which also gi es a nonnega i e de ini e
k
P
ma ix. So is con ex in p, non-dec easing in y, non-inc easing in p and as well as ),( yUkp
homogeneous o deg ee ze o in income y and p ices. So i is globally egula . A ew p ope ies o
k
Ua e wo h examining. As is al eady ob ious, is a conca e unc ion o p ices. Fo posi i e
1
k
U
1
, , being an inc easing conca e unc ion o a conca e unc ion, is conca e in
kk UU 1
1 Tha is, a consume maximises di ec u ili y unde a budge cons ain . This implies he indi ec
u ili y unc ion should be homogeneous o deg ee ze o in income y and p ices p, non-
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.
),( yU p
3
p ices. Fo 1
, is an inc easing con ex unc ion o a con ex unc ion and so is con ex in
k
U
p ices. Fo 1
, i is simple o e i y ha
k
Phas a nega i e de ini e o semi-de ini e Hessian
and he e o e i s ecip ocal is con ex in p ices and so is
kk UU
k
Py k/. So is
k
U
con ex in p ices o all posi i e
. Also, is con ex in p ices as i equals
k
Ulog Py
kloglog
and since is an inc easing conca e unc ion o a conca e unc ion, i is conca e and minus i is
Plog
con ex.
Now conside he unc ion
1
kkUU , (2)
whe e he k
a e non-nega i e and sum o uni y2. Commence wi h posi i e 1
. is conca e
k
U
in p ices and since sums o conca e unc ions a e conca e,
kkU is conca e. Then
1
kkU
is a dec easing unc ion o a quasi-conca e unc ion and so is quasi-con ex. Since
kkkkk UU
yy
U1
1
1
and
i
k
k
kkkk
ip
P
P
UU
p
U
1
1
1
Uis inc easing in income and dec easing in p ices. I is ob iously homogeneous o deg ee ze o in
income and p ices since each is. So U is globally egula o
k
U1
.
Now ake
nega i e and pu
. Then (2) becomes
1
kkUU .
As al eady shown is con ex in p ices, so
k
U
kkUis con ex and since an inc easing unc ion o
a con ex unc ion is quasi-con ex, Uis quasi-con ex in p ices. I is clea ha is again inc easing U
in income, dec easing in p ices and homogeneous o deg ee ze o in income and p ices. Finally, o
2 The unc ion (2) is o he o m o he ecip ocal o a CES (cons an elas ici y o subs i u ion) p ice
index, al hough he e i is an index o componen u ili ies no p ices.
4
0
, he amilia limi ing a gumen as 0
( o example, Diewe , 1993) gi es
o
k
k
UU
kk UU loglog
.
Since each is con ex in p ices, he sum is, and since an ilog is an inc easing con ex unc ion,
k
Ulog
Uis. So (2) gi es a globally egula u ili y unc ion o all 1
.
Fo u ili ies o he o m , which a e special cases o (1), he sums, p oduc s and ha monic
k
Py/
means, which a e special cases o (2), ha e been examined in Conni e (2002). Howe e , hey a e o
limi ed in e es as he esul ing u ili ies mus all be homo he ic.
3. EXAMPLES
Conside
jj p
y
U
1 and
jiij pp
y
U
2.
Bo h a e easily shown o be con ex in p ices, he o me s ic ly so, p o ided he j
and ij
a e
posi i e and ij
= ji
. Take 3/1,3/2 2
iand 1
, so ha (2) is a weigh ed ha monic
mean o and . I is
1
U2
U
1
2
1
2
1
2
1
23
y
p
y
p
y
pj
i
ij
j
j
, (3)
Diewe ’s (1974) gene alised Leon ie u ili y unc ion. Diewe p o ed i s global egula i y, bu ha is
immedia ely e iden om he de i a ion he e. Mo e impo an ly, (3) is jus one o many possibili ies,
all globally egula . Taking 1
, wi h he same
alues as be o e, gi es a weigh ed a i hme ic
mean o and
1
U2
U
1
2
1
2
1
1
2
1
3
1
3
2
y
p
y
p
y
pj
i
ij
j
j
. (4)
The demand sys ems ollowing om (3) and (4) a e no he same and show di e ences ha a e
possibly impo an in p ac ice. Fo example, he income elas ici ies o he gene alised Leon ie
demand sys em a e
5
2
1
2
1
2
1
2
1
2
1
2
1
2
2
1
y
p
y
p
y
p
y
p
y
p
y
p
w
E
j
i
ij
j
j
j
i
ij
i
i
i
i
,
which mus be g ea e han a hal . The income elas ici ies co esponding o (4) a e
2
1
2
1
2
2
1
2
1
2
1
2
1
2
2
1
2
1
/2
/
2
3
jiijjj
j
j
jiijjj
i
i
i
i
ppp
y
p
ppp
y
p
w
E
’
which mus be less han 3/2, which migh be mo e accep able han cons aining hem o exceed a hal .
Regula i y need no imply he capabili y o model a wide ange o consume beha iou . The
gene alised Leon ie ’s de iciencies in his ega d ha e been men ioned by Ca es and Ch is ensen
(1980) and Diewe and Wales (1987) and (4) has i s own in lexibili ies. Hoewe e ,
, o a
,
could be ea ed as an unknown pa ame e o gain mo e lexibili y. The case o 0
, gi ing
p oduc s o u ili ies is pa icula ily in lexible since he esul ing unc ion is s ill homo he ic.
Many o he choices o and a e ob iously possible; o example, could be aken o be
1
U2
U1
U
j
j
p
y
1,
whe e 1
j. Using (2) o combine his wi h he p e ious wi h
2
U
=1 and 2/1
21
gi es
1
2
1
2
1
2
y
p
y
p
y
pj
i
ij
j
j
,
and he co esponding demand equa ions a e
2
1
2
1
1
2
1
2
1
y
p
y
p
y
p
y
p
y
p
y
p
w
j
i
ij
j
j
ij
i
j
i
i
j
j
.
6
Clea ly, h ee o mo e componen u ili y unc ions could be employed in (2) o p oduce e en mo e
hea ily pa ame e ised, bu s ill globally egula , u ili y unc ions. While he lexibili y o he
esul ing demand sys ems would be inc eased, especially i
, o he
we e also pa ame e ised,
he da a equi emen s o he es ima ion o so many pa ame e s could be a se ious p ac ical di icul y.
4. DISCUSSION
This pape has shown how o gene a e highly pa ame e ised globally egula u ili y unc ions om (2)
using he globally egula componen s (1). Familia and mo e pa simonious u ili y unc ions could also
be seen as ollowing om (2). Hou hakke ’s (1960) indi ec addilog unc ion
j
j
jp
y
,
whe e i
and i
ha e he same sign and i
>-1, is globally egula and is ob iously (2) applied o
j
py/ o he powe o j
wi h 1
and jj
. The componen s employed in sec ion 3 could
be conside ed as esul ing om applica ion o (2) o s ill simple componen s. Fo example, le
kk pyU /, a u ili y unc ion co esponding o expendi u e o he whole budge on good k.
Using (2) wi h 1
and jj
gi es he o sec ion 3.
1
Ukiik ppyU /, a u ili y unc ion
co esponding o expendi u e o hal he budge on good i and hal on good k, is (2) applied o and
i
U
k
Uwi h 0
and 2/1 ki
. Then applying (2) o wi h
ik
U1
and
o
gi es
2
Uo sec ion 3.
O cou se, many u ili y unc ions ha e appea ed in he li e a u e ha a e no o he class gene a ed
by (2) om componen s o he o m (1). Bu hey a e no globally egula .
REFERENCES
Ca es, D. and L. R. Ch is ensen (1980): “Global p ope ies o lexible unc ional o ms”, Ame ican
Economic Re iew, 70, 422-432.
Conni e, D. (2002): “Sums and P oduc s o U ili y Func ions”, Economic and Social Re iew, 33,
285-295.
Diewe , W. E. (1974): “Applica ions o duali y heo y”, in F on ie s o Quan i a i e Economics Vol.
II, ed. By M. D In iliga o , and D. A.Kend ick, Ams e dam: No h-Holland, 106-171.
Diewe , W. E.and T. J. Wales (1987): “Flexible unc ional o ms and global cu a u e
Condi ions”, Econome ica, 55, 43-68.
Hou hakke , H. S. (1960), “Addi i e P e e ences”, Econome ica, 28, 244-256.
