ii
一
l i l -t i l i-Kagawa Unzverszty Economic Revzew Vol.65 N 0..2 September 1992. 33←44
Notes
Note on t
h
e
C
o
n
t
i
n
u
i
t
y
o
f
Demand C
o
r
r
e
s
p
o
n
d
e
n
c
e
s
Takashi S
h
i
o
m
u
r
a
'
‘1 INTRODUCTION
In the Walrasian (tat叩 nemant)world
，
the balance of demands and supplies necessari1ymeans the steady state of the economy. The gap between demand and supply is ex -pected to disappear quickly. So the excess supply of labor
，
that is，
unemployment，
is to disappear. The real world，
however，
has been contrary to this theory. Fortunately or unfortun乱tely，
we can not help recognizing the disagreement between the balance ofmarkets and the steady state by the past experiences
，
especially，
through The Great Depressionin the 30s.We should thus distinguish two meanings of the word 'equilibrium': the balance of markets on the one hand
，
and the steady state on the other.. Then we m，ay construct a model which can be consistent with these two meaningsThe first approach
，
though we should carefully use the word 'the first'，
to this problem is made by Glustoff(
1
968)，
but we can not民 肝 吋a五xedpoint in his model as a Keynesian equilibrium because his model is not Keynes-inclined bu.t rather Walras・inclined in that his modellacks the core of Keynesian tools
，
that is，
the effective demand We had ωwait，
through the contributions of Barro and Grossman (1971)，
D泌氏(1975)組 dBenassy (1975) to get ω a Keynesian equilibrium.1 Indeed
，
about forty wars have p崎sedsince the appear叩 ceof Genera/ Theory. • The author wishesもothank Professor TakaoFujimoto for his helpful comments. The responsibility for any errors remains with the author 1 Although it is somewhat problematic to consider a Neo-Keynesian equilibrium as Keyn田ian one，
we will not go further into this problem.128 Kagawa University E乙onomicRevzew
34
A large number of NeかKeynesmodels have appeared in the literature
，
and thecontinuity of demand functions (or more generally
，
conespondences) has a greatsignif~Indeed
，
it is a minimum requirement icance in these models for some reasons or otherfor many problems. We will五ndthat among these models the diffemces of conditions which ensure the continuity of demand functions if we careful1y investigate.. Where can these di鉦erencesbe observed? Unfortunately
，
we can not investigate it directly from these papers because the proof of continuity is usu乱Ilyleft outj it is very tedious and lengthyIn the following
，
the systematic proofS of the continuity are provided and the relationships among the conditions which ensure the continuity are clarified. The purpωe 1 hope th乱tthis helps theof this paper was originally intended for my own exercise readers understand the related papers
TWO
TYPES OF
DEMAND
FUNCTION
2
We first explain our notation X: consumption set，
X C Rn.. U: utility function fl:om X into R p: p山evector，
p E R.
i
•
x:consumption vector，
x εx
.
ωinitial endowment vector，
ωε Rnz
:
upper bound vector on the net demand vectorx -ω，
ZjεR
.
i
主:lower bound vector on the net demand vectorxー 叫 主ER':.. θA: boundary of a set A int A: interior of a set A Here R.
i
is defined to be the non-negative ortha以 ofRn，
that is，
theset {x E RnI
x壬
O}，
and R':..is defined similarly the non-positive orthant of Rn..Throughout -i l i l i -i i i i i l l e -i i ! i g i i i i i l129 Note on the Continuity of Demand Correspondences
35-this paper
，
the 'upper (lower) hemi continuity' or 'upper(
l
ower) hemi continuous' is abbreviated asu.h c. (I..h..c. ).2Two types of demand function have been discussed in the quantity constra.Ined models.. One is called Dぬzetype
，
and the other Benassy type，
though Benassy himself calls it Crower type. The former is the solution vector of the utility maximization problem which is constra.Ined by the budget set as well as quantity constra.Ints.. Eaιh component of its vector，
that is，
the demand for a commodity，
is determined simultaneously. In other words，
it is the solutionz(p，
ω，
ζθof the following problem2
2
7
U(z) (1) subject to p(xーω)三
0ゐ~Xh一ωh~ Zh (h= 1
，
，
"
n)The Benassy type is somewhat complicated.Ifwe wish to determine all the de -mands which Benassy calls 'e貸ectivedemands'，自rstly，we have to determine the first e征伐tivedemand for the commodity labeled one..Itis the first component of the so・
lution vector of the utility maximization problem which is constrained by the budget set and all the quantity constra.Ints except one. Next
，
we have to consider similar
I
y the second effective demand for the commodity labeled two，
and continue the above proce -dure up to the commodity labeled n.. In other words，
we have to consider n constra.Ined maximization problems to get all of effective demands where the utility maxmization problem for the i-th effective demand is5
2
F
U(z)(
2
)
subド
ctto p(x-ω)孟0さh~ Xhーωh壬Zh (h
1
=
i)2 We should distinguish阻 払h.ι
∞
rrespondencewith a closed map in a manner di班erentfromA~row and Ha~n's (~971) Incid~l!tall~， in~eb!e~ (1.959)_ h，; alwaysass~mes ~ha~ t~e
range of a correspondence iscomp叫t.Therefore
，
his definition of the upper semi continuity is36ー Kagawa University EじonomiじRevieω
1
3
0
3 PROOF OF CONTINUITY
We want to show the conditions which ensure the continuity.. For this purpose，
the fact thatZh=∞
01~=一∞
f01some h is not so essential 80，
we may consider only Problem (1) Let S be the set of(p，
ω) inR+ x Rn for which the set{xε
XI
p(xーω)三O}isnot empty， and the conespondences"(，
c
fIOm S x R+ x R':..into X beγ(p
，
ω，
z
，
i)三
{xE XI
p(x-ω)~ 0，
~ ~ X h -ωh ~ Zh (h= 1，
.
.
.
，
n)}，
c(p，
ω，
2，
主
)
三 {
X
E XI
maxU(x)}，
1espectively
.
F01 convenience' sake，
let the cOIIespondenceαfrom S into X be α(p，
ω)三 {xε
X
Ip(xーω)壬O
}
，
and the conespondences
ll:om Rn x R+ x R':..intoX bes
(
ω，
z，~) 三 {xE XI
主
h~ Xhーωh壬
Zh (h = 1， .
，
吋
}
such that"((p，
ω，
ゑ
。
=α(p，
ω)nβ(ω，
Z，
~) We consider the following自veassumptions separateIy or in combination Assumption 1 X is a compact set Assumption 2 X is c/osed and bounded from be/oω Assumption 3 X is a convex set. Assumption 4 X = R+ Assumption 5 X is a c/osed set We can prove the following lemma immediately131 Note on the Continuity of Demand Correspondences
-37-Lemma 1 Under Ass叩~ption3
，
'Yis convex-valued at(p' ， ωヘ Z"， t) ε SxR+xR~Theorem 1 Un吋de引1As幻s叫mpt
i
向2)ωJ湖
ε
川 X，
then'Yis contin叫0ω，
convex and compact-ωlued at (p'，
ωヘ
グ
，
t)ε
SxR+xR~
Proof Itis well known that the couespondenceαis continuous at (p
ヘ
w')under Assumptions 1 and 3 (See，
Debreu (1959，
p..63) orFukuoka (1979，
p.58)).Now
，
let us proceed in three steps[11]
s
is百九ιandcompaιt-valued sinceX is compact.. Thelefore，
'Yis u.h. c. and compact-valued sinceα(p， ω)nβ(ω， z，~ チ。[1 2] Let x'
ε
s(w'，
Z
"
，
t
)
， 勾==
min{max(4 +ω~ ， x)
，
i
:
悲
+ωn Then，
ほ
E[4+ ω;，
弔
+ωX] (h= 1れ ，
，
n)and 5;q→♂
(q→∞)
Because of the closedness of X，
either x' E int X or x'εδx
.
In case x' E int X，
there exists a certain integerq'such that5;q E X (q主イ)泊
nceP
→
x' (q→∞)
Hence，
we can prove the 1.九Cυofs
if we define the s珂uence{
x
q} as follows: lfqくq'，
we take for xq an arbitrary point ofs
(
ωq，
，
1
Z
t
l
)
lf q2
:
q'，
we take xq=
企 qJ
N凶 W問宅c∞
O凶 釘ert恥he虫:e叫Se qイ
，
such t山ha剖twqε
intX
(ωq主qイ
的
，
う
)
since w' E intx
.
Then，
there exists uniquely an aq such that aq
三入q
w
q
+ (1 ->
.
Q
)
5;q
εθX for any 5;q
t
<
intX w here 0壬入q壬1because of the convexity ofX. 3 We should note here仙ata
Q E [4 +ωi，
認
+ωZ] (h= 1"n
)
The sequenα{
>
.
q
}
has a convergent subsequence since>
.
q
belongs to the clc崎ed ir山rval[0，
1]..Ifwe define its subse:quenceぉ{入q
'
}
such that>
.
q
'
→>"
(
q
'
→
∞
)
，
then，
See，
Scwartz (1978，
p.71) Theorem 39，
and Ni】也ido(1970，
p..195，
Corollary 1)38- Kagawa University Economic Review 132 ，¥*is uniquely determined as zero sincew~ E int
X
引 Therefore，
the original sequence {λq} also converges to zero，
that is，
aq→
x* (q→∞)
(See，
Figure 1) From the above consideration，
we can prove the /.h c. ofβif we de自nethe sequence {xq } as follows: Ifq<
q'，
we take for xq an arbitrary point of β(ωq，
i
， ♂)
:
f
•
Ifq主q'and ii;qε
intX，
we take xq = :i;q Ifq主q'and :i;q1
-
intX
，
we take xq=
aq [L3]4 Let (pq，
wq，
:
f
i
， ♂)→
(pぺ
ω'，
2
"
，
，
t
)
(q→∞)"
x.εγ(pぺ
ωぺ
"
， ，
2
t
)
.
.
Then，
there exists a sequence土q
{
}
such that:i;q Eβ(wq
，
♂，
fl)and♂→
x* (q→∞)
since β is 1 h...c.. from the result of [L2] Ifx'εωα(pぺ
ω，
つ
there existsacerfa止nintegerq'such that:i;q E intα(
P
q，
ωq) (q主 q')The巾re，
we can prove the 1 んc.of "( by taking the sequence{
X
q } as follows: Ifqくq'，
we take for Xq an arbitrary point of ^((pq，
Wq，
:
f
i
， ♂
ト
Ifq三
q'，
we takeXq=
土q Let p*(X噂 - w*) = 0.. Then，
tl悶 eexists a certain point x E X such thatまhω;;
(
h
ヂj
)
，
X
j
くω，
]
，
p事
(
ま
-w*)くoandX
Eβ(w'，
2
"
，
，
t
)
from the conditions (1) and(
2
)
Therefore，
t】悶eexists a certaiIlintegerq
'
such that pq(正一ωq)<
0(
q
とq
'
)
山 Let us define xq (q;::: q')as follows: If:i;q1
-
α(pq，
wq)，
we take for xq組 arbitraryμintin the intersection of the hyperがane pq(x -ωq) = 0 and the line x = λq:i;9+(1ー
λ勺
正
Otherwise，
xq=
土q Then，
we can define u凶quely加 がsuchthat xq Eγ(pq，
wq，
， ♂
O
1
)
.
lndeed，
for土qrtα(pq，
ωq)，
入q=----pq(正- wq. -) ， >0 pq(i:q -ω9) -pq(正一ωq) can be determined uniquely
.
Since pq(ii;q.ーω勺→〆(X*-
w*)=
0 and pq(x -wq)→
p*(x -w*)く0，
，¥q→
1，
that is，日→
x' (q→∞)
Hence，
we can pro刊 theIλc..of^(if we define the sequence{xq} as制lows:‘
The following proofisessentiallオsame描 Dreze(1975)133 N ote on the Continuity of Demand Correspondences
Ifqくq'
，
we take for xq an arbitrary point of'Y(pq，
ωq，
f
f
1
，
♂
)
Ifq
三
q'，
we take xq as stated above
-39-Summing up
，
IIom the above lemma and the results[
1
1
]
and[
1
3
，
]
the theo悶 nisproved
•
Corollary 1 Under Assur叩t附 lS2 and 3
，
げμ
リ
p'>
0 arid (2) w'ε
int X，
then'Yiscontznuo叫s
，
convex and compact-valued at (p'υ
Proo
.
f
The following proof is in two steps[21] 'Yis compact-valued because of Assumption 2 and the condition(1')
There exists an星
ε
Xsuch thatx 主主 fOIanyxε
X from Assumption 2..Let(pq
，
ωq，
f
f
1
，
♂
)
→
(p，
*
w'，
i
"
'
，
.
[
"
)
(q→∞)
and xq Eγ(pq，
ωq，
f
f
1
，
♂
)
.
Then，
there exists a certain integer q'such that pqxq壬
p'w'+
ε (q主q') for anyE>
0，
since pqxq壬
pqwq叩 dpqwq→
p'w' (q→∞)
Sincepq→
p' (q→∞)，
there exists a certain integerq"such that pqε
V(p') (q三
q
"
)
where V(p・)is叩 ycompact neighborhood of p' in intR
+
Letq・
=
=
ma.x(q'，
q")..Then，
fOI anyE>
0，
~n.p在三 pq在三 pqxq 壬p'w'+
E(
q
主ザ) l'eV(1'・)Therefore
，
xq is bounded since pq→
q・ >0(
q→∞)
and x主主Itfollows that the sequence{
x
q} has a co即 ergentsubsequence
Without loss of generality
，
we may assume thatxq→♂
(q→∞)
Then，
p事
(x'-w')
壬
0，
~壬 Xh-
ω:壬Z
，
;
j
(h= 1.
，
.
，
.
n)and x.E X from the closedness ofX Hence，
x'E 'Y(pぺ
w.，
.
，
"
'
i
[
"
)
，
that is，
'Yis.Ih.c-40ー Kagawa Unzversity Eωnomic Revzeω 134
[2.2] In case ♂ εωα(p.
，
w.)，
we c乱nprove thel.hιofγin the same way as the 五rsthalf of[
1
3
]
Let pホ(x.-w.)
=
o.Ifx・=
w'，
we take xq=
wq Otherwise，
from the condition(1')
，
there exists some j for which若く
o Then，
we can definexq in the same way as the lat ter hallf of[
1
.
3
]
Hence
，
we can prove the 1.h.c. of'YItshould be noted that we still use the resultof
[
1
司
•
Corollary 2 Under Assumption
4
，げ(1')p.>
0 and (:1)w.εX(
i
.
e.，'Y主0)，then'Y is contznuo制s，
convex and compact-叫んedat (pぺ
w*
，
i
"
，
，
t
)
ε
S x R，+xR':..Proof We proceed in three steps
[
3
.
.
1
]
We can pro刊 the叫h.c. ofY'and the compact valuedness in the same w，ay as[
2
.
.
1
]
[
3
.
.
2
]
Let x.ε s(w.， Z'，i') ， Xh三 凶n{mは(
4+ω%
， xh)，者
+ωn
Then， xqE
s(w.， Z'， l) and xq→♂ (q→∞) Hence，
βis 1 h c[
3
3
]
Consider the case ωヂO. Ifp吋
x.-ωつ
oand x事 w.，
we take xqω
q(
E
'Y(pq，ω
q， ZI，t
1
)
)
to prove thel..h.. c..of'YIf xq1
=
ω
q， tl悶 eexists some j for whichz
;
ーω
J
'
<
0 sincep市 >o. So，
there exists悶 nejfor which弓 くOandO壬
z
;
くω
J
'
because of the conditions(1')and (2')..Then，
there exists an正suchthat Xh=ωh
(hチ1)
and45
ゐ
-
w
;
:
くO.We can thus prove the 1.λιof 'Yusing the result of[
3
.
.
2
]
in the same w，ay as [L3]Let us consider the case w. 0.. Since γ(p.，w'， l) Z'， = {O}， it is sufficient to prove thel..h.cυof'Yby taking xq = wq (εγ(pq， wq， ZI，
♂
)
)
135 Note on the Continuity of Demand Correspondences -41ー
In the above theorem and corollaries
，
we allow for the case where Zh∞
or~=ー∞ forsome
h
When this case is ruled out，
we can prove thefOllowing corollaryCorollary 3 Under Assumptions3 and 5
，
ザ(1)
p.>
0 and若く
ofor some j and (2) w.ε
int X，
then'Yis conti7日LOUS，
convex and compact-valt日dαt(p. ， ωヘタ， ~)εSxR守xRn
P1'oo
.
f
We can assume thatX is a compact set becauseゐ
:
:
;
Xh一
同
:
:
;
Zh (h=1
'
，
，
n)•
Now let us consider the tιh.c. of the demand conespondence
c
.
Letr
三 SxR.
i
xR':..
，
and an element ofr
be ωTheorem 2 /f the conditions that ens叫rethe continuity and the compact va/uedness
of'Yare satisfied
，
and if the叫ti/ityfunction U is continuous，
then c(ω)三{
X
εXI
max耐 (w)U(x)
，
ωεr
}
isnon-empty，
u..h..c.and compact-va/ued atω.，
and the indirectuti/ity function U (
ご
(ω))is continuous at w./
1
U is，
in addition，
quasi-concave，
thenc
is convex-va/ued as wellFI担rther，
げU is strictly quasi-concave，
then c isa continuous也
:
f
nction Proof We can prove the first half by applying directly Berge's maximum theorem.5 The proof of the latter half is an easy exercise.•
We can similady prove the following theorem by applying the generalizedBerge's maximum theorem • As for Berge's maximum theorem and the generalized one，
see，
e.g.，
Maruyama (1980)-42- Kagawa University Eιonomic Revieω 136
Theorem 3 If the conditions that ensure the continuity and the compact valuedriess of
'Yare satisfied
，
and if the utility function V fromr
x X into R is continuous，
then ψ(ω)三 {xE XI
max"'E"((ω) V(ω，
x)，
ωεr
}
is non-er叩ty，
u h c.. and compact-叩 /ued at w'，
and the ind附 ct山lityfunctio叫V(ω，
ψ(ω))is continuoωat ω略 In additon to the continuity of V，
if V is q叫asi-concavewith respect to x，
thenψ
is conve.x-valued as well“F世rther，
if V is st7lctly quasi-concave with respect to x，
thenψ
is a contznuous:世nJct!on4 Summary
We may summarize the above resultsL One can maintain the continuity and the compaιt valuedness of'Yif one of the conditions categorized in Table 1 is satis五ed.
2. Suppose that the conditions which ensure the continuity and the compact valued -ness of'Yare satis五ed
(a) one can maintain the川 c.and the compact valuedness of the demand cor -respondence
c
(resp.ψ)，
and the continuity of the indirect utility fu恥 tion U(c(ω)) (resp.. V(ψ(ω)) if the utility fur凶ionU(x) (resp.. V(ω，
x)) is con -tinuous (b)Ifthe utility function U(x) (resp.. V(ω，
x)) is continuous and quasi-concave with respect tox，
thenc
(resp.ψ) is convex-valued as well asu.h.c and compaιt・valued (c)Ifthe utility function U(x) (resp.. V(ω，
x)) is continuous and strictly quasi -concave with respect tox，
thenc
(resp.ψ) is a continuous function，137 Note on the Continuity of Demand Correspondences -43-Figure 1 δX Table 1 condition X p
・
P-R7 w 綱" a compaιt and convex p.三
o
and pj. > 0考
，
く
O w. E int X for some j' b closed，
convex and p噂 >0 J ε int X bounded from below c R守 p・
>0 w. E Xd closed and convex p.主oand pj.> 0 all the constraints are w. E int X fOI some j' bounded and
z
:
.
く
OReferences
[lJ Anow
，
K.J.and Hahn，
F H (1971) General Comp出ti開 Analysis，
Arnsterdam:North-Holland
[2J BaI叫R.J and Grossman
，
H..J.(1971)“
A General Disequilibrium Model of
I
ncom-44 Kagawa University EwnomiιReview 138
[3] Benassy
，
1. P (1975)“
Neo必eynesianDisequilibrium Theory in Monetary Econ -omy"，
Revieωof Economic Studies，
Vo¥ 42，
pp. 502-523[4] Debeu
，
0.. (1959) Theory of Value，
New York: John Wie¥y & Sons[5] D巾e
，
J.. H (1975)“
Exister悶 ofan Exchange Equilibrium under PrIce Rigiditiesぺ
Internationα1 Economic Review
，
VoL 16，
pp.. 301-320[6] Fukuoka
，
M.. (1979) General Equilibrium Theory，
Tokyo: Sobunsha (in Japanese) [7] Gu¥sto鉦，E. (1968)“
On the Existence of a Keynesian Equilibriuぱnomic Studies
，
VoL 35，
pp.. 327-334[8] Maruyama
，
T (1976) Functional Analysis，
Tokyo: Keio-tsushin (inJapanese) [9] Nikaido，
H..(
1
970) 1πtroduction to Sets and Mappi吋 inModern Economics，
Ams-terdam: North-Holland(trby K Sato)
[10] Schwa山， L (1967)Co也1Sd M.. $ato)
