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impac o global p ocesses on egional and spa ial de elopmen . I in ol es se e al hi d-
le el ins i u ions in I eland: NUI Maynoo h, Galway-Mayo Ins i u e o Technology, Ins i u e
o Technology, Sligo, Ma y Immacula e College - Uni e si y o Lime ick, Wa e o d Ins i u e
o Technology.
© 2002 by Denis Conni e. All igh s ese ed. Sho sec ions o he ex , no o exceed wo
pa ag aphs, may be quo ed wi hou he explici pe mission o he au ho p o ided ull c edi ,
including he © no ice, is gi en o he sou ce.
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While NIRSA is happy o acili a e he dissemina ion o wo king pape s p oduced by
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ABSTRACT
The e a e ela i ely ew known demand sys ems ha a e heo e ically sa is ac o y and
p ac ically implemen able. This pape in es iga es building mo e complex demand sys ems
om simple known ones by conside ing sums and p oduc s o basic u ili y unc ions, an
app oach ha does no seem o ha e been exploi ed p e iously in he li e a u e. Some o he
sys ems ha esul a e in e es ing and use ully ex end he ange o a ailable unc ions. E en
he simple sys ems ha a e no su icien ly lexible o he analysis o eal wo ld
consump ion da a may s ill be use ul o applied gene al equilib ium s udies and o
heo e ical explica ion. Al hough some sys ems, ins ead o being new, u n ou o be
edisco e ies o al eady known ones, he way in which hey a ise as combina ions o simple
componen s is o in e es in i sel in showing hem as sub se s o wide classes.
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I INTRODUCTION
An indi ec u ili y unc ion , whe e p is a ec o o p ices and y is income, and he
demand equa ions de i ed om i h ough Roy's iden i y
),( yU p
y
U
p
U
q
i
i∂
∂
∂
∂
−= /, (1)
sa is y demand heo y, o u ili y maximisa ion, p o ided mee s s ingen c i e ia.
These a e ha U be homogeneous o deg ee ze o in income and p ices (p), non-dec easing in
y, non-inc easing in p, and con ex o quasi-con ex in p. Then he demand equa ions sa is y
he equi ed cons ain s o agg ega ion, homogenei y, Slu sky symme y and nega i i y
u ili ies. These c i e ia o he alidi y o indi ec u ili y unc ions a e e y es ic i e on he
choice o unc ional o ms, e en wi h es ic ions placed on he pa ame e s occu ing in he
o ms. The e a e ela i ely ew known unc ions U ha sa is y alidi y condi ions o all, o
e en o all plausible alues o p ices and income and some o hem a e e y basic. This
pape in es iga es building mo e complex demand sys ems om simple known ones by
conside ing sums and p oduc s o basic u ili y unc ions.
),( yU p
The basic combina ion de ices, which will be desc ibed in sec ion II, a e qui e simple, bu a
leas as a as his au ho knows, hey ha e no been exploi ed p e iously in he li e a u e in
o de o expand he ange o alid demand sys ems. Some o he simple sys ems ha esul
and ha will be desc ibed in sec ion III, may no be as lexible as migh be desi ed o he
analysis o eal wo ld su ey o ime se ies da a on consume expendi u es on commodi ies.
Howe e , hey may s ill be use ul o applied gene al equilib ium s udies and o heo e ical
explica ion. Some mo e complex sys ems, o be de i ed in sec ion IV, a e mo e lexible and
pe haps use ully ex end he ange o a ailable unc ions. As migh be expec ed, some
sys ems, ins ead o being new, u n ou o be edisco e ies o al eady known ones. Howe e ,
e en he way in which hey a ise as combina ions o simple componen s is o in e es in i sel
in showing hem as sub se s o wide classes.
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II DEMAND EQUATIONS FROM SUMS AND PRODUCTS OF UTILITIES
Suppose we ha e wo (indi ec ) u ili y unc ions and sa is ying all alidi y c i e ia.
Then he c i e ia ob iously apply o
1
U2
U
21 UU
+
( he sum o wo quasi-con ex unc ions is
quasi-con ex o con ex) and indeed o 21
)1( UU
λ
λ
+
−
, whe e
λ
is a posi i e cons an , and
co esponding demand sys ems can be de i ed. Le ),(
11 yww ii p
=
and be
he se s o demand equa ions, in budge sha e o m, esul ing om applica ion o Roy's
iden i y o and espec i ely. Then by applying (1) o
),(
22 yww ii p=
1
U2
U21
)1( UU
λ
λ
+
−
and
simpli ying, he demand equa ions co esponding o his sum o u ili ies u n ou o be
y
U
y
U
y
U
w
y
U
y
U
y
U
ww iisi
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
+
∂
∂
−
∂
∂
−
=
21
2
2
21
1
1)1()1(
)1(
λλ
λ
λλ
λ
, (2)
o he o iginal indi idual demand o mulae weigh ed by (apa om cons an s) he de i a i es
o u ili ies wi h espec o income. The sub-sc ip s deno es he u ili ies we e summed.
Fo he special case o u ili y unc ions o he o m
1
1P
y
U= and
2
2P
y
U=, (3)
whe e and a e p ice indices, he alidi y o u ili y unc ions educes o he alidi y o
he p ice indices and i is hen e iden ha he u ili y unc ion
1
P2
P
21
)1( PP
y
U
λλ
+−
=
is also alid. So we will be in e es ed in he p ope ies o he weigh ed sum o u ili y
unc ions
2
21
2
1
21
1
)1()1(
)1( U
PP
P
U
PP
P
U
λλ
λ
λλ
λ
+−
+
+−
−
=.
Applying Roy's iden i y o his gi es
21
2
2
21
1
1)1()1(
)1(
PP
P
w
PP
P
ww iiwsi
λλ
λ
λλ
λ
+−
+
+−
−
=, (4)
he indi idual demand o mulae weigh ed (apa om cons an s) by he p ice indices, o he
ecip ocals o he de i a i es o u ili ies wi h espec o income.
4
Fo unc ions o he o m (3), he p oduc o u ili ies is a alid u ili y unc ion
λλ
2
1
1UU −1.
Applying (1) o his gi es
y
U
y
U
y
U
w
y
U
y
U
y
U
ww iimi
log
log
log
log
)1(
log
log
log
log
log
log
)1(
log
log
)1(
21
2
2
21
1
1
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
+
∂
∂
−
∂
∂
−
=
λλ
λ
λλ
λ
, (5)
he indi idual demand o mulae weigh ed by he elas ici ies o u ili ies wi h espec o
income. The sub-sc ip m deno es he u ili ies we e mul iplied.
1 I he logs o u ili y unc ions we e u ili y unc ions, hen he ac ha he sum o u ili ies gi es a alid
u ili y would su ice o he p oduc o u ili ies. Con exi y is he c ucial p ope y. Fo unc ions o he
o m (3), . Since P is a alid p ice index, i is conca e in p ices. The log
unc ion is conca e and inc easing, so is conca e and he e o e is con ex.
PyU logloglog −=
Plog Ulog
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III SIMPLE HOMOTHETIC COMPONENT UTILITY FUNCTIONS
Th ee simple u ili y unc ions can be gene a ed by di iding income by p ice indices
co esponding o (weigh ed) a i hme ic, geome ic and ha monic means. They a e
∑
=
jj
ap
y
U
γ
, (6)
∏
=j
j
gp
y
U
α
(7)
and
∑
=
j
j
hp
yU
δ
(8)
espec i ely. All sa is y alidi y condi ions p o ided he ss ','
α
γ
and s'
δ
a e posi i e wi h
.1=Σ j
α
The co esponding demand equa ion sys ems a e ob ained by applying (1) and o
(6) his gi es
jj
ii
ai p
p
w
γ
γ
Σ
= o
jj
i
ai p
y
q
γ
γ
Σ
=, (9)
whe e q deno es quan i y. These a e Leon ie demands in ha he a ios o quan i ies o
commodi ies a e always in ixed p opo ions, i espec i e o p ices o income. Fo he i h
commodi y he own-p ice elas ici y is ai
w
−
and he c oss-p ice elas ici y wi h espec o
p ice k is . No e he e is o e -pa ame e isa ion in (9) as any one
ak
w−
γ
could be elimina ed
by di iding i in o nume a o and denomina o . Bu he con en ion 1=Σ j
γ
is mo e
compa ible wi h he a e age p ice in e p e a ion. As is well known, applica ion o (1) o (7)
leads o he Be gson, o cons an budge sha e, demands, igi
w
α
=
, wi h own-p ice elas ici y
equalling minus one and c oss-p ice elas ici y ze o. Applying (1) o (8) gi es
∑
=
j
j
i
i
hi
p
p
w
δ
δ
o
∑
=
j
j
i
i
hi
p
p
y
q
δ
δ
2, (10)
wi h own-p ice elas ici y –2 + and c oss-p ice elas ici y . As o (9), he e is pa ame e
edundancy in (10) and a
hi
whk
w
1=Σ j
δ
con en ion ma ches wi h a ha monic mean p ice index.
O he simple u ili y unc ions a e easily w i en down, o example,

6
∑
=2
jj
p
y
U
φ
, (11)
which is alid i he s'
φ
a e posi i e and which gi es he demand equa ions
2
2
jj
ii
i p
p
w
φ
φ
Σ
= o 2
jj
ii
i p
yp
q
φ
φ
Σ
=, (12)
wi h own-p ice elas ici y 1-2 and c oss-p ice elas ici y
i
w k
w
−
. Again a edundan
pa ame e can be accoun ed o by imposing 1
=
Σ
j
φ
, which also pe mi s in e p e a ion o
he denomina o o (12) as a p ice index.
E en wi h he ou u ili y unc ions (6), (7), (8) and (11), he e a e qui e a ew po en ial
demand sys ems. Taking he u ili y unc ions wo a a ime, he e a e six possibili ies and he
h ee combina ion me hods ia (2), (4) and (5) makes eigh een demand sys ems. Bu how
much mo e lexibili y do hey gi e? Wi h income appea ing as simply as i does in he ou
s a ing poin u ili y unc ions, i is e iden ha no only ha e hei demand sys ems uni a y
income elas ici ies2, bu so will he combina ion sys ems because he weigh s in (2), (4) and
(5) a e unc ions o p ices and no income. So we a e only conside ing g ea e lexibili y in
esponse o p ice changes. Taking as a i s example he combina ion o (6) and (7) by (4),
gi es he demand sys em
∏
∏
Σ+−
+−
=
jjj
iiji
wsi pp
pp
wj
j
γλλ
λγαλ
α
α
)1(
)1( , (13)
which can be w i en as
ag
iigi
wsi pp
pp
w
λλ
λ
γ
α
λ
+−
+
−
=)1(
)1( (14)
whe e he sp'deno e means o p ices (which a e unc ions o pa ame e s) wi h he subsc ip
deno ing he ype o mean. By di iding nume a o and denomina o o (13) by
λ
−1and
w i ing
ii
γγ
λ
λ
=
−1
i is possible o w i e (13) as
∏
∏
Σ+
+
=
jjj
iiji
wsi pp
pp
wj
j
γ
γα
α
α
. (15)
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This ge s id o
λ
and emo es he need o any cons ain on he s'
γ
, bu al hough nea e , i
des oys he in e p e a ion o jj p
γ
Σ as an a i hme ic mean p ice index. Howe e , he de ice
will be used in sec ion IV.
Fo he demand sys em (9), co esponding o (6), all goods had o be p ice inelas ic and o
he cons an budge sha e model he elas ici y had o be –1. I is easily e i ied ha o (14)
he p ice elas ici y is
])1[(
)1(
)1(
agwsi
g
iiwsi ppw
p
w
λλ
λ
αα
+−
−
−−− ,
so ha p ice elas ic goods a e possible. The c oss-p ice elas ici y wi h espec o p ice k is
])1[(
)1(
agwsi
g
kiwsk ppw
p
w
λλ
λ
αα
+−
−
−− ,
so ha c oss-p ice elas ici ies wi h espec o k a e no cons an o e commodi ies, unlike he
si ua ion o (9) whe e hey all equalled minus he budge sha e o good k. So he weigh ed
sum o (6) and (7) does gi e a sys em wi h scope o ep esen a g ea e ange o economic
beha iou . Simila ema ks apply o he sum and p oduc combina ions ia (2) and (5), which
a e
ag
a
g
iiai
si pp
p
p
pp
w)1(
)1(
λλ
λγαλ
−+
+−
= and
a
ii
imi p
p
w
γ
λαλ
+−= )1(
espec i ely. Fo hese wo sys ems o become he same and also equal o (14) would equi e
ag pp =. Bu as is well known, a geome ic mean is always less han an a i hme ic mean
unless all commodi ies ha e he same p ice. Al hough he sys ems a e dis inc , hey ha e an
e iden similia i y – own p ice appea s explici ly and linea ly in all, while he o he p ices
(and own p ice) occu implici ly h ough he p ice indices.
The co esponding demand sys ems o combina ions o (7) and (8) a e
hg
h
i
i
gi
wsi pp
p
p
p
w
λλ
δ
λαλ
+−
+−
=)1(
)1( 2
,
jj qlog
2 A well known ela ed cha ac e is ic o hese basic demand sys ems is ha hey could ha e been
de i ed om addi i e di ec u ili y unc ions, o example,
α
Σ
in he case o cons an budge
sha e demands, o Σ in he case o (12).
jj
q
φ
/
2
8
hg
gh
i
i
hi
si pp
pp
p
p
w)1(
)1(
λλ
δ
λαλ
−+
+−
=
and
i
hi
imi p
p
w
δ
λαλ
+−= )1(
espec i ely, whe e he ha monic mean is
∑
=
j
j
h
p
p
δ
1.
Again, equali y o he sys ems equi es hg pp
=
, bu a ha monic mean is always less han a
geome ic mean unless p ices a e equal. So again he sys ems a e dis inc , wi h he simila i y
ha he ecip ocal o own p ice appea s explici ly and linea ly in all h ee, while he o he
p ices ea u e only h ough he p ice indices. As migh be expec ed, he combina ions display
mo e lexibili y in p ice elas ici ies han hei componen s did. Fo example, o he demand
sys em (10), co esponding o (8), all goods had o be p ice elas ic, bu he combina ions elax
his. The demand sys ems o combina ions o (7) and (8) can be ob ained oo and simila
commen s apply. The equa ions o a commodi y a e ound o explici ly ea u e bo h own
p ice and i s ecip ocal, which has bene i s o own-p ice lexibili y, bu o he p ices again
ea u e only h ough p ice indices.
Now conside he combina ion o u ili y unc ions (6) and (11), o demand sys ems (9) and
(12), ia (4). This leads o
qa
q
ii
ii
wsi pp
p
p
p
w
λλ
φ
λγλ
+−
+−
=)1(
)1( 2
, (16)
whe e q
p deno es he oo quad a ic p ice index
.
2
∑jj p
φ
As al eady men ioned, all he demand sys ems de i ed in his sec ion, ha e uni a y income
elas ici y and his could be seen as a se ious in lexibili y. So i is i we a e ying o model
obse ed consume demand. Howe e , i is o en conside ed a desi able p ope y in applied
gene al equilib ium s udies, some imes along wi h ex eme pa simony in pa ame e s. Fo
9
example, i (16) is u he simpli ied by aking n
iii /1
=
=
φ
γ
, whe e n is he numbe o
commodi ies, we ge (in quan i y a he han budge sha e o m) he single pa ame e sys em
])1[(
)1(
qa
q
i
wsi ppn
p
p
y
q
λλ
λλ
+−
+−
=, (17)
wi h a
p and q
p now he simples p ice mean and he squa e oo o he simple mean o
squa ed p ices. Bu (17) is he new class o demand equa ions p oposed by Da a and Dixon
(2000)3 o gene al equilib ium models, which hey belie e will also be use ul in a a ie y o
o he applica ions. F om he de elopmen he e i is e iden hei s is a sub-class o a much
wide one. They see he 'linea i y' in explici own p ice4 in (17) as a pa icula i ue, bu ha
is no unique o hei case. The sys ems esul ing om combina ions o (6) and (11) ia (2)
and (5), wi h he same imposi ion o n
iii /1
=
=
φ
γ
,
])1[(
])1[( 2
a q
i
g
a
a
q
si ppn
p
p
p
p
p
y
q
λλ
λλ
+−
+−
= and
a
i
g
a
mi pn
p
p
p
y
q
])1[( 2
λλ
+−
=
sha e he p ope y. No a e hese he only ones. The o mulae gi en ea lie o he demand
sys ems om combina ions o (7) and (8), i w i en as equa ions o quan i ies a he han
budge sha es, show ha all h ee ha e he p ope y. P esumably oo, he e will be occasions
when mo e han a single pa ame e is desi ed, so ha he mo e gene al o mulae a e
applicable. Pe haps he e may e en be be si ua ions whe e 'linea i y' in he ecip ocal o own
p ice may be desi able ins ead o , o as well as, 'linea i y' in own p ice. I is no implausible
o suspec he e a e o he use ul sys ems o use in gene al equilib ium modelling besides (17)
o be ob ained om combina ions o simple componen s sys ems.
3 They use
λ
−ins ead o
λ
, de ining i o be nega i e. O cou se, hey p o e he alidi y condi ions
di ec ly o hei sys em a he han deducing hem om he p ope ies o componen s.
4 They a gue, ci ing Dixi and S igli z (1977) ha some imes he non-linea i y due o implici in
p ice indices is unimpo an .
i
p