Documentos de trabajo
Notes about uniqueness of equilibrium for infinite
dimensional economies
Elvio Accinelli
Documento No. 10/93
Diciembre, 1993
Rotes about uniqueness of
equilibrium for infinite
dimensional economies
Elvio Accinelli
Documento No. 10/93
Octubre, 1993
..
'
El autor es investigador del Instituto de Matemática de la Facultad de Ingeniería
Resumen
En el presente artículo presentamos condiciones suficientes para la unicidad del equilibrio
walrasiano, para economías con espacios de consumo de dimensión infinita.
Mediante la función exceso de utilidad, transformamos el problema de hallar condiciones de
unicidad en un espacio de dimensión infinita, en un problema equivalente en un espacio de
dimensión finita.
Las propiedades de dicha función permiten usar técnicas conocidas de la topología diferencial,
para obtener condiciones de unicidad.
La función exceso de utilidad aparece como una herramienta poderosa para caracterizar el
conjunto de los equilibrios, especialmente en aquellos espacios donde la función exceso de
demanda no surge como un resultado natural del proceso de maximización.
1
Introduction
Conditions for uniqueness of equilibrium in economies with finite dimensional consumption space
are well know, wltile uniqueness result in economies with infinite dimensional consumption space
are scarcer.
Dana (1991) was the first one to obtain an uniqueness result in infinite dimensional consumption spaces. She considers tite case of a pure excltange economy wltere tite agent's consurnption
space is f,~(ll) and agents have additively separable utilities.
We consider a pure exchange econorny with cousurnptions spaces that are a finite cartesian
product of measurable functions. Utility functions are additively separable (see Section 1).
Tite properties of tite excess utility function (see Section 2) allow us to apply degree theory
to tite considered ecouomics. We prove that t he cardinality of equilihrium for these economies is
odd and we obtain suíicient rondltions for uniqueness of equilibrium to hold.
By mean s of the excess ut ility function, we transforming an infinite dimensional optirnization
problern in a finite dimensional one,
In section 3 we illustrate the developed theory with sorne examples.
In tite last section the proof of theorems are given.
2
The Model
Let us cousidcr apure excltange economy with n agents and I al each state of the world.
The stale sct is a mcasure space (nA, v) .
•• wish to thank Aloisio Araujo, Paulo K. Monteiro and Ricardo Marchesini for usefu] comments and suggestions
and also Rose Aune Dana for scnding lile her preprint about Uniqucllcss under Gross Substrtute hypathesis.
1
=
We assume that each agent k has the same consurnption space, A1
the space of all nonnegative measurable functions defined
011
n~=IMj where A1jis
(n, A, v)
Following Mas-Colell (1990), we consider the space A of the C2(R~+) utility functions U.
Utilities are strictly monotones, differentiably strictly concaves and propers.
This space is endowed with t he topology of (,'2 uuiform couvcrgence in compact ,
More precisely we say that {Un}
IIUn
-
UIIK
l/ if for each cornpact ¡.."
-+
e
R~+
=
+ ¡OUn(s, z ) -
ess sup max.(IUn(s, z) - Ut s, z)1
sEO z E K
Ol/(s, z)¡ + I¡PUn(s, z) - 02U(s, z)1
go to zero with n.
Agent k is characterized by his utility function
From
1I0W
and by his endowment
Uk
UJk.
on we will work with economies with the following characteristics:
a) Tite utility functions
'Uk : Xk
-+
R are separable and they are represented by
'Uk(X)
= ¡Ch(s,
x(s»dv(s)
U
(] )
with x E M,
b) the Uk(S,.) belongs to a fixed cornpact subset of A,
e) the agents' endowments,
E M are bounded aboye and bounded away from zero in any
Wk
component, Le. there exists
l
and 1J with
l
<
UJkj
< H
The following definitions are standard.
Definition 1 A TI allocation 01 commodities is a list x
; j
= {l, ... , n}
= (Xl, ••• , x n )
where
z E Rl we denote
(p, z)
=
¡
n
p(s)z(s)dv(s)
O
Definition 3 The pair (p, x) is an equilibrium if:
X
is
(11l
ii) (p, Xi)
n -+
R/
is measurable, atul Li~1 x;(s) ~ L~1 UJ¡(s)
Definition 2 A commodity pricc system is a mesurable [unction p:
i)
Xj :
allocation,
~ (p,
'W¡) <
'ti i
00
iii) if ip, z) ~ (p, w¡) with
z:
n -+
R~+, then
[ U¡(s,x¡(s»dv(s);::: [ U¡(s,z(s»dv(li) 'ti i.
Jo
Jn
2
-+
R~+, and for any
3
The Excess Utility Function
In order to obtain our results we introduce the excess utility Iuuction.
For all A in the (n - 1) dimensional open simplex, ~"-I, and for all s in
n, .there is a solution
in R':I\, x(s, A), of the problem
maxL¡ A¡U¡(s,X¡(s»
subject to L¡ x¡( s) ::; L¡ W¡( s) and X¡( s) 2: O.
(2)
See Mas-Colell (1990) for more details.
For this solution, the following identitics hold
A¡8U¡(s,x(s,A),w)
Wherc A¡~
="j
with i
= ,,(s,A,w)
Vi
= I, ... ,n
and Vs En.
(3)
= {l,oo.,n} andj = {1,oo.,l}
Let us now define the excess utili ly function.
Definition 4 We say that e: R+
e¡(A)
= A¡\1
r--r
R" e(A)
l,(s,
II
= (el(A), ...,e,,(A»,
A)[x;(s, A) - W¡(S)]dll(S), i
untl:
= 1, ... , n.
(4 )
is the ezcess utility [unction.
Each cquilibrium (p,x) can be charactcrizcd by an unique A E 6,,-1 such t hat e(A)
This is shown in the next proposi tion
= O.
Proposition 1 The pair (p, x) is an equilibrium iff pes) = l'(s, A); x(s) = x(s, A) with e(A)
Proof: For each A E 6"-1 there corresponds an uuique ,,(s, A)
x(s)
= O.
R~+ andan alocation
= x(s, A) such that (3) follows.
Ir we call A an equilibriurn whenever (p(S),X(8» is an equilibriurn, we have that A is an
equilibrium iff e(A) = O.
Its convenient to introduce the following definition.
Definition 5 Let us define by E the set:
E
=P
E ~n-l
3
:
c(A)
= O}.
that will be called, equilibrium seto
As we have that
esssup
ID(J¡(s,w¡(s»1 <
00
sEO
then E is a non empty set, see Araujo-Monteiro (1988)
Let us !I0W consider the product space A x M of the characteristics (U, w), with tite C'J. uniform
convergence in compacto
That is if(Un'w n) ...... (U,w)iffor each rompe!' K E Jl~+
lI(Un,w,,) - (U,w)IIK -, O with n.
Where II(U,w)IIK
= II(U)IIK + IIwll =
This is a metrizable space and the induced metric can be taken as:
Where
J(N
II(lJ,w)IIK
= IIUIIK
N
= {z
+
E Rl
:
~
:s Z¡ :s N}
IIwll = ess sup
lllax.(IU/+ jiJU/ +
sEO z E h
liJ'J.lll +
Ilwlll
Let r be the set of econornies with characteristics in A x M such that zero is a regular value
of its excess utility function.
Mas-Colell (1990) preves that r is open and dense in the set of economies.
Frorn now on we will work with econornies in I'.
As for economies in
r,
zero is a regular value of the e(A) we have for all A E E that
J(e(A» maps T)..S++l ...... 1\S++I.
Hence the rank of .I(e(A» is n-lo
The determinant of this ruap ís equal to:
See Mas Collel (1985) ll.5.2.
Definition 6 Then we put signJ (e( A»
= (+ 1) 4
1 accordinq to whether det[n TJ( eeA»]( > O)
<O
We are now in condition of statinq our main result:
Theorem 1 Consider an econonuj wilh injinitdy dimensional cousumption set, separable utilities
satisfying the conditions in section 1), then:
(1) The cardinaliuj of E is finite atul odd,
(2) /f signJ(e(A)) is constant in E, there ezists a1l unique equilibrium, iahere J(e(A)) denote the
Jacobian of the ezcess utilily [unction.
4
Examples of Economies With Uniqueness
In this section we consider sorne examples with uniqueness of equilibria.
Let [J(e(A)));j be the term in the row i and column j of the Jacobian of the excess utility
function.
'fhen
[J(e(A»));j
where8U;
=
fu
8{oUi(s , x;(s, A~1:;(S' A) - 'Uli(S}j} dv(s)
8 U;
8U;)
= ( UXl
~, ... ,~
UXI
and X;(S,A)
= (X;¡(S,A), ... ,X;I(S,A».
8+f;;'} = [8 a{Xq 8 2U; with [8 a:{Aq = (~, ... ,~)
We have t.hat
82U;
=
[
/)2 U;/8x 1 2
¡)2 Ui] OX]OX2
{J2U;j~X20Xl
o2U;jox2 2
(5)
and
'" 8~lI'/fJXl¡)xI]
. .. ()2l!,/ iJX20X I
8 2U;/¡)x/¡)Xl
.
¡)2U./OXI 2
Then, we obtain
[J(e(A»);j
4.1
=
fu ;~; [ü
2
U;(x ;(s , A) [x;(s, A) - w;(s)¡tr
+ (8U;(s, Xi(S.A»tT] dv(s). .
(6)
Economies With Gross Substitute Pr-oper-ty,
Following Dana (1991) we say that the excess utility function displays the Gross Substitute property if satisfies the next definition.
5
Definition 7 The excess utiJity function díspJays the so called "Gross Substituta" property, if:
De j (>,. )( > O) < O 1,'fe'1
~
)
= J')'.,J.'
1 r J.
Proposition 2 lf the ezcess utiJity [unction has the Gross Substitute property, then signJ( e( >.))
is consiant,
Proof: Let A be (n - 1)
X
(n - 1) northwest submatrix of {J(e(>')) + [.J(t(>.))Jtr}.
From Gross Substitute property we can prove that A has dominant diagonal positive. See
Lugon (1991).
,
~ '. ~
¡
Let V 1I = (O, ... , 1), as rank J( c(>')) is '( TI '-1): then for al! vetor z such that
that zJ(e(>.))z > O.
Now take
11
:¡: O with >'v =
O and let v'" = v
+ a>'; if a
= -~, then
= O we have t hat 1IaJ(c(>'))ver = 1IJ(c(>')) > O.
That is if e( >.) = O then J( e( >.)) as a map from T>. to TA
VaV
n
ZV
n
= O we have
= O
lf e(>')
its determinant has sign (--1 )n-l.
Now the Theorem 1 guarantees the uniqueness.
4.1.1
Economies With One' Good in Each State.
Economies with one good in each state of the world and utility functions with the next property,
¡PU
Dx 2 (x - w)
+
mt
ax
? O,
(*)
have Gross Substitute property. See Dana (1991)
For the following two examples the above condition is satisfied.
Example 1 Suppose an economy with individual's utilities :
Uj(x)
=
in
U¡(x(s))g¡(s)dJ.t(s), withg¡;
n ......
R+ and i
= {l,. ,.,n}.
lf lI¡(x) satisfy (*) titen we liane uuiquencss.
For instauce: u¡(x)
= Jo x(s)7e-rsdjl(S) with O <
Q
< 1 and r > ()
Example 2 Let us nOlll consider ecoTlOmies untl: the [ollountuj utilitics:
U¡(x)
=.1
in[ai(s) + b¡(s)x(s)J<>'djl(S).
Where ajes) > -w(s) bj(s) > () arul 0<
n¡
FOT' these economies (*) is satiesficd.
6
<
Hemark: The property a[x8U] ~
o is equivalent to risk
aversion srnaller than one.
Example 3 11 tite eeonomy has one good in eaeh state 01 lite uiorld í.e. x :
n .....
R, and 11 eaeh
aqent Itas risk aversion is smaller than one, titen the economu display Gross Substitute propertu.
Remark: When x( s, o) : 6. -. R, the Iact that
Xi(S,
A.,
is increasing for Ai and decreasing if Ak,
000'
Ak-" o, Ak+1>
k"#
000'
A,,)
i has econornical sense because x( A) is a solution of
the social choice problem,
, '.'
Ir the social weight of i - lit agent is increased, then the consumption bundle of the agent must
also increase,
4.2
Separable Goods Economies
1
Consider economies with good-separable utility functions i.e,
a2Uh
ax'B x k
=0 VhE {l,oo.,n}andi, k E {l, ..o,l}, i
"#
k,
Proposition 3 Lid U be a uli/ity [unction that is botli additi\lcly separable atul qood separable. 11
(7)
VIt E {1,o.o,n}a71ds En,
then we have uniqueness.
Proof:
From the first order conditions, (2), we have that
that is
ou;
{)[JI
Alaxk =: o.o=: A"ax k
Vk E {l,o.o,l}
where
XI.
= (xJ., ... xt)
Taking deri vati ves with respoct to Ai (j E {I,
that
000'
1 ::s: It::s: no
11},) aud recallillg t hat
{)2{lh/ i)xiiJx k
= Oit follows
(8)
7
where
¡PUh
ahk
= Dx k2
oo:
and bhk = Dx k
'
Let wk(s) be the total endowruent of good k.
Frorn xf(s, A) + ...
+ x~(s, A) = wk(s)
we obtain that
axf
DAj
iJx~
+... + OAj = O.
(9)
From (8) we obtain the foilowi ng equation
Alalk ax~
üx~,
--- = Ahahk DAj
DAj
(10)
Vh f:. j.
Replacing (10) in (9) and without loss of generality supposing that j
f:.
1 and j
f:.
h
::f:
.
,
1 give us
or equivalently
(11 )
From (8)
ax k
- A¡aIk
DxJ;
a/ + Ajajk DA}
= -b j k.
J
( 12)
}
Finally, replacing (12) in (11) we obtain
Dxl;
' < O Vk and i f:. j .
OAj'
(13 )
From the fact that X(A) is homogeneous of degree zero we obtain that
axJ;
iJ>"~ > () Vk and j.
Then (1:1) and (14) are sufficicnt conditions to obtain (8) of Proposition 2.
The following example ilustrates this proposition ,
8
( 14)
Example 4 Economies untl: ulilily [unciions such that
(15 )
hace a unique equilibrium price.
In order to prove the proposition rocall that:
1)
Wi( s)
is positi ve for all i and s E
n,
2) the Hessian is a diagonal mat rix with negativo entries.
For inst ance, economies with the following utility functions
x
x,?
with
= XI,. ",X n Uj(Xj) =
and O
Then we have Gross Substitute property.
4.3
<
P
< 1 ,0 <
(tj
< 1
Economies With Two Goods and Two Agents.
Let E be an economy with two goods an two agents, (Ui, w¡), i
{1,2}
Frorn the first order condition we have that:
where
Xk
= (x¡'xt).
Taking the derivative with rospect to
>'1
in the aboye identity we obtaln:
(16)
where
ij
ak
[)2Uk
= ox'{jxJ
-r-r-v-- .
i
and b]
= al!]
~j
vx'
k
i,j,'
= 1,2.
Let wi(s) be the endowment of good i. Then
(17)
Hence
(l8)
9
Replacing (18) in (16)¡ the below equations follows
Then
+ A2a~2
+ A2a~2
I
:~: = ~I ~::~: t ~~:~: =:~
l.
A¡aF
A¡af:.!
and
( 19)
(20)
where \l is the determinant of a 2 x 2 hessian matrix of a convex combination of differntiably
strictly con cave functions, then \l is non negativo. We suppose that \l > O.
From the fact that X(A) is homogeneous of degree zero [see Property 2,( Sec 1)] we obtain that
ox i
sgn-1.
OA)
ox i.
= -sgn--)
OAk
i
= 1,2
and j f:.k.
(21 )
Frorn (17), we obtain that:
Oxi
oxi
-)
-o,.
OA
J
A
i= 1,2 andj f:. k.
(22)
J
The next proposition follows.
ox'
Proposition 4 Jf ~ V i
Á)
= 1,2
and j
= 1,2 has
lhe sanie sign atul if
(23)
then uniqueness [ollouis,
Proof: 111 t his conditions Gross Substituto proper ty follows ..
A suflcient condition to obtain (23) is that:
Example 5 RCOTlOlIIics ioitb. a~ > Oi f:. j, ami
equilibriuni.
10
k
=
1,2 Mltisfyiug (2;'1), hane uniqueuess of
Example 6 Suppose an ecotiomg with
=
UI(X)
lo
(a(s)x(s)
U~(X) =
Where X
= (x, y)
+ b(s)y(s)tdJI(,~)
r (x(s)fJ + y(~P)(LJl(~).
Jo
O < {a,,8, I} < 1 with a and b inteqrable [unciions :
The endounncnts are
We obtain that
W¡
=
+
(}2U2-CX)
{}2U2w2
=
n
{}U2
0(0' - 1)(ax
= a 2(ax + by)<>-I{a,b}
+
by)<>-2{aw¡
> O
+ bW2,bw¡ + aW2} <
l'hen
[iPU 2 ](x - 111) + i)(f2 > O
From (19) and (20) it foLLows that
{}x I
(}A¡
a
.
= - \7o(ax¡ + bU.)"-
¡.
(1-2
A2¡3(¡j - l)x 1
> O
Those conditious are stisfied [or U2 because it is a separable utility [unctioti,
Uniqueness follows.
Example 7 The sanie result is obtained with
UI(X)
utul
5
u~
->
{WiJ.., Wiy}
=
fn
Log[a(s)x(s)
+ b(s)y(s)dJI(S).
a separable utiLity [uuction.
Proof of Theorems
In order to ¡HOVe those theorerns, we nocd tite following lemmas:
Lemma 1 The excess utility [unction is (,' l.
11
O
R++.
Proof:
"}'( s, A, w) and x.( s, A, w) are
el
with with respect to A
1'0 see that this is truth, observe that if x(s, A) is a solution for (2) then we have the following
identity:
n
n
.=1
i=1
L x.(.~, A, 111) = L lIIi(8).
Now let
liS
(2·1 )
consider the sistern of equations:
= "}'(
AiiJU.( s; x( s, A), UJ)
s, A, 111)
L:i=1 x¡(s,A,w) = L:i';"l W¡(s).
(;J)
(4)
Frorn the irnplicit function theorem, taking derivatives in the above sistern, with respect to x
and "}', we obtain a matrix with the following form:
M _ [ .4
Bl
B ]
()
Wherc 11 is a(nl) x (ni) matrix,
.4=
oi;
{fA
()
O
(11/
(I/}
O
O
O
O
O
O
O
()
O
O
U:'I
lJíi
v«
Ulí
alld
1 O
O 1
B=
()
O
1 O
O 1
U.
O
1
O
O
O O
That is IJ is a I x (ni) matrix
Claim There is not a vector z = (v, 111) =f. () such that M z
Proof Let v such that M z = O, thcn
= O.
(25)
12
•
and
Av
+ Bw = ()
(26)
Then for (4) and (5), we have that
=O
v t r Av
If v E ker tr: then
+
'02 +
'01+1
VI
vI
vl+2
+
V21
+
+
+
+
(27)
=O
v(n-I)I+2 = O
v(n_I)I+1
+ ... +
'Vil I
=
O
Observe that
n
a¿>.ilJi
=
;=1
IOUl
.m»
I()UI
VXI
VX2
VXI
"oUn
nolJ"
{>. ~,>. ~, ... ,>' ~"",>' -;:)-,>' -;:¡-, ... ,
= {71, /'2, .. ·,71, .. "
VXI
(fX2
>.n o u n}
!:l
=
VXI
/'1, /'2, ... , /'1, .. ',71> /'2, ... , 7d
Then
n
o { ¿ >.iUTv
=
/'I(VI
+ VI+I + ... + 'v"-I)I+d + ... + 71CVI + V21 + ... + 'Vnd
=O
i=1
Because 2::'=1 »u! is diferentiably strictly convex, (6) and (7) we have that v
B is a injective matrix then, from (6) w = O.
We have that z
= O.
Proving our claim.
From the claim and the fact that
(lJi( 5, .),
is in an campad set of A, the lemma follows.
Lemma 2 The cxcess utility [uuction has tite Jollowiny properties:
1) e( >.) is homoqeneous oJ deqree
2) >'l:(>') = O,
ZC1"O;
V>' E R++;
3) therc ezists k E R such that c( >.)
4) lI e(>')II-....
00
as
= O.
« k1.
>'i -.... O [or any i E {l, ... ,n},.
5) ae(>') : Tp(s)n-I -....l'¡,(s)n-l.
13
(28)
Proof: '1'0 prove 1) note that x¡(s, w,·) is horuogeneous of degree zero Vsi E
Properties 2, and 4 are imrnediate. Property
f)
n.
follows from the fact that zero is a regular
value of e.
To prove Property 3, note that from equation (2) we can writc
e¡(>') =
r {)U¡(s, X¡(>'»)[X¡(s, >.) -
ln
'W,(s)]dv(s).
From the concavity of U¡ it follows that:
U¡(s, x(s, >.)) - U¡(s, w(s» 2: au¡(s, x(s, A»(X¡(s, >.) - w(s)).
Therefore,
C¡(>') S
r ll¡(s,x¡(s,>'»- U¡(lII¡(s»dv(s) SlU¡('tWj(s»dV(s),
VA.
j=1
ln
¡¡
If we let
k¡
=
r U¡(L,"=n W¡(s»
lrn
dv(s) and
1
k
=
sup k¡
l~¡~n
Property 3) follows.
We can
IlOW ¡HOVe
the following lerruna:
Lemma 3 : The ezcess utility [unction is an outuiard pointing vector field on the tangenl space
01 S++1 = {>. E R++ : 11>'11 = 1}.
Proof: With a straightforward application of Property 2 of lernma (2) we obtain that for all
>. E
8++
1'0
1
,
e(>.) E T>.S.
prove that e(>') is an outward pointing vector field, let us
1l0W
define Z¡
By Property 3 we know that there exists k E R such that e¡(>') S k and by Property 4,
lIe(>')1I . . . .
oo. Then we condude that Z¡ SO.
Furt hermore, Z¡ couJd be differeut from zero only if >.¡ were zero, This Iollows Irorn the fact
that if >.¡ is differcnt Irom zero, then we can wri te
k*
Letting k'
= -k*/>'m¡, we have that k' S e,(>'m) S k.
14
Hence Z¡
= O.
Strictly speaking, we have proved that we have a continuous outward pointing vector field for
almost any point in the boundary of S~+l. Theexcess utility function has similar properties to
those of the excess demand funct.ion. Mas-Colell (198.'») proves that for excess demand Iunctions
there is an homotopic inward vector field for all points of the boundary
8++
1
•
In our case, with
au analogous proof, wc can obtain an homotopic outward vector Iield for excess utility functions.
Proof of Theorem 1
From the fact that e(>.) is homogeneous of degree zero we can define the equilibrium set as
From property 4) in lemrna (2) and from the coutinuity of e we have that if {An}
E
E, and An -r--r A, then A E E.
Then E is a com pact set in Rn-I.
Moreover, from the fact that zero is a regular value of e, we have that E is a finite seto
Since
8++
1
is homeomrphic to the (n - 1)-dimensional disk, its Euler characteristic is one.
On the other hand, e(>.) ls an outward vector field on the taugcnt space of
8++
follows, from the Polncaré-Hopf theorern.
Now item 2) is a straightforward aplications of t he Poincaré-Hopf thcorern.
Because:
1
=
L
signdetJ(e(A»).
{A:c(\=O)}
15
1
,
then item 1)
REFERENCES
Araujo, A. and Monteiro, P.K., (1989), "Equilibrium Without Uniform Conditions", Journal
o/ Econotnic Theory, 48,
..
416-427.
Daua, R.A., (1991), "Existeuce and If niqueuess of Equilibrium When Preferences are Additively Separable and Gross Substitute", tó appear in Ecollolllctrica (199:l).
Kehoe, T., J., Levine, D., K., Mas-Colell, A.,Zame, W.,R.(1989) "Determinacy of Equilibrium in Large Scale Econornies". Jourtuil o/ .Halhematical EcoTlOmics" 18 231-262.
Lugon, A.,(1991) Sobre a Unicidade Do Equilibrio Walrasiano" se. thesis, I.M.P.A. Rio de
Janeiro Brazil.
Mas-Colell, A., (1985), "The Theory of General Equllíbríum: A Differentiable Approach",
Cambridge Univcrsity Press,
Mas-Colell, A., (1991), " Indeterminacy in Incomplete Market Economies" Economic Theory
1 45-62.
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