『経済研究』（明治学院大学）第 140・141 合併号

2008 年

Regular Economies with Infinitely Many
Commodities
Takashi SUZUKI

Ⅰ．Introduction
In a seminar paper, Debreu [1970] started the study of the local uniqueness (finiteness) ofcompetitive equilibria of exchange economies. He considered a topological space of all (smooth)
economies and concluded that the economies with locally finite and structurally stable equilibria are open
and dense subset of the space. Such economies are now called the regular economies. This result has
been extended to several directions, for example, to production economies, the exchange economies witha continuum of traders and so on by Smale [1974], Dieker [1975], Mas-Colell [1975, 1977], Kehoe [1983],
Rajan [1977] and many others. The reader can consult the book of Mas-Colell [1985] for related works andliteratures.
On the other hand, Kehoe and Levine [1985], Balasko [1997] and Chichilnisky and Zhou [1998] andothers have extended Debreuʼs result to economies with infinite dimensional commodity spaces. This
note is a modest contribution to this direction of research.
When the commodity space is infinite dimensional, the study of the local uniqueness gives rise to amathematically subtle problem. Debreuʼs original approach is to invoke excess demand functions andapply the differential calculus technique (the Sardʼs theorem) to it. In the case of infinite dimensionalcommodity spaces, however, the story does not proceed so smoothly as in the finite dimensional cases.
First of all, Araujo [1985] observed that in general Banach spaces of commodities, the continuous demand
functions exist only when the space is reflexive. Hence for 0 p cases, the case p/y is excluded. Moreover,in such spaces the smooth demand functions exist only when the dual spaces coincide with the spaces
themselves. Therefore in 0 p, only in the case p/2, differential calculus via excess demand functions is
possible. In order to rescue this difficulty, people resort to the Negishi approach to equilibrium analysis,
1

『経済研究』（明治学院大学）第 140・141 合併号

characterizing competitive equilibrium using a social welfare function rather than the excess domainfunctions. For instance, Balasko [1997] studied regular economies on the space 0  , the set of boundedsequences with supremum norm is a natural candidate for the commodity space (See Bewley [1972]). He
parameterized "the equilibrium manifold" by weights of a social welfare function (the weighted sum of
individual utility functions) on Pareto optimal allocations, rather than (as usual) by the price vectors
through the excess demand equations. This clever trick made him to avoid the fundamental difficultyarising in the infinite dimensional settings mentioned above. Chichilnisky and Zhou [1998] dealt with
C([0,T]) as the commodity space. It is the set of continuous functions on [0,T] (continuous time interval).
Their strategy to handle the problem is that they discarded price vectors which do not support equilibria
and worked on a smaller subspace rather than the positive cone itself.
But this is not the end of the story. As Shannon [1999] pointed out, in the infinite dimensionalcommodity spaces, the boundary conditions on the indifference surface ensuring that equilibrium
allocations sit within the interior of consumption sets are not so trivial compared to the finite dimensional
cases. The condition that indifference surface does not intersect the boundary of the consumption set
does not work. We have to handle carefully the tail of allocation vectors. This problem is relatively not so
serious in both of the above cases where the topology is defined by the sup norm of the vectors. In such
spaces, for instance, the vectors bounded away from 0 can be included as allocation vectors and we canrestrict our attention to such well behaved consumption bundles.
In this note we will work on the Hilbert space 0 2, the set of square summable sequences. The 0 2
space has been found to be of great importance of applications such as economic analysis of financial
markets. We follow the strategy of Balasko to use the Negishi approach and avoid the problem of thesmoothness of excess demand functions. In order to handle the problem of boundary condition of theindifference surface, we have to find appropriate boundary conditions fitted to the topology of this space
and they have not been known so far. Furthermore, the proof of showing smoothness of a technical key
concept (the saving function or Negishi function) is not trivial at all. The next section presents the modeland results. In appendix, we collect the mathematical concepts and related results used in our analysis.

Ⅱ．The model and results
Consider an exchange economy with m consumers. The commodity space of our economy is theHilbert space

H=0 2= x=(x t) 6 *x t* 2?y .





t=0



It is well known that H is a separable Banach space, therefore it is a Lindelöf space.
The consumer iʼs consumption set X i is defined by
2(1)

Regular Economies with Infinitely Many Commodities

X i={x=(x) t*x t>a i, t=0,1,...}, a i?0,(2)

namely that he/she can supply the commodity up to ,a i unit at every period. The economic
interpretation of this consumption set is natural enough, and seems to be even more appropriate than thecase a i=0 where the consumption set is positive orthant in the case of production economies, although
we focus on the exchange economies in the paper. The mathematical advantage of this consumption set
for our analysis is that it is a convex and open subset of 0 2, which is much suitable for differential calculus
on it. Indeed, let x=(x ti)X i . Since x0 2 , there exists r>0 such that *x t*?1/2 for t>r . Let
x q+1=min{x t+a i*0CtCr}, and set X=min{x q+a i,1/2}. Then it is easy to verify that yX i for all ysuch that x,y?X. Hence X i is open subset of 0 2 in the norm topology. The convexity is obvious.
The consumer iʼs initial endowment vector is denoted by w i=(w 0i ,w 1i ,...)X i. The total resourcer0 2++={x0 2*x t>0, t,0,1...} is assumed to be fixed. Let `={(w 1...w m)7 m
i=1X i*6 iw i=r} be theset of initial endowment vectors satisfying the resource constraint. As will be seen, we identify each
vector (w 1...w m)` as an economy.
The consumer iʼs preference is represented by a utility function
u i : X iR,
R xu i(x), i=1...m.

(3)

We assume that the utility function satisfies the following conditions.
(U‑1) u i is twice continuously (Fréchet) differentiable, namely that of class C 2 on X i.
Let Du i(x)=( 0u i(x), 1u i(x),...) be the derivative (tangent map) of u i at xX i, and 0 0u i(x)  0 1u i(x) 

D u i(x)=  1 0u i(x)  1 1u i(x) 
2



(4)

 

be the second derivative, where  tu i(x)=lim h0(1/h)(u i(x 0,...,x t+h,...),u i(x 0,...,x t,...)), and so on.
For every xX i, D 2u i(x) is considered to be a linear map from 0 2 to itself. We assume for every i,(U‑2) u i is strictly differentiably monotone, i.e., Du i(x)]0 for every xX i.
(U‑3) D 2u i(x) is a nondegenerate, negative definite bilinear form on 0 2,
i.e., (y, D 2u i(x)y)C0 for every y0 2 and the equality holds only when y=0.
(U‑4) for every sequence x n=(x tn)X i such that x tna i, for some t, it follows that u i(x),y.
Note that the assumption (U‑3) implies that the linear map D 2u i(x) is a linear isomorphism of X i
to itself. The assumption (U‑4) will be sometimes called Inada condition.

3

『経済研究』（明治学院大学）第 140・141 合併号

The list {u i,w i} m
i=1 is called an economy and denoted by e. From now on, we fix the utility
functions and parameterize economy by the endowment vectors. Therefore the set `={(w 1...w m)
7 i` i*6 iw i=r} is the space of all economies.
t 2
2
A price vector in the economies is a sequence p=(p t)]0 with 6 
t=0*p * ?+y, an element of 0 ++.

For a given commodity vector x=(x t) , the value of x evaluated by the price p is denoted by
t t
px=6 
t=0p x . An m‑tuple of consumption vectors (x 1,...,x m)7 iX i is called an allocation. The

allocation is said to be feasible if 6 ix i=6 iw i(=r).
Definition 1: An allocation (x i) is said to be Pareto optimal if it is a solution of the following social welfare
maximization problem;mGiven l=(l 2...l m)R
R m-1++ , maximize u 1(x 1)+6 i=2l iu i(x i) subject to 6 ix=r.

We denote the solution associated with l by x i(l). Obviously, x i(l)/(x ti(l))X i. Indeed, sincex(l)0 2, we have x ti(l)0 (ty). Hence x ti(l)>a i for all t large enough. By the Inada condition (U‑4),
it is impossible that x ti(l)=a i for any t.
The Negishi equation (Balasko [1997a]) then express the property that the equilibrium allocation ofa given economy are the Pareto optimal that satisfy the budget constraints of consumers, namely that(5)n i(l,w)=Du i(x i(l))(x i(l),w i)=0, i=2...m.
Of
6

coursethefirstconsumerʼsbudgetequation

followsfromtheWalruslaw;

Du i(x i)(x i,w i)=0 and the first order conditions of the maximization problem, namely thatmi=1

Du 1(x i)=l iDu i(x i), i=2...m. The next definition is due to Balasko [1997a].
Definition 2: A pair (l,w)R
R m-1++ -` is an equilibrium if it is a solution of the Negishi equation systems(5).
Example: There exists one consumer. The consumption set is given by
X/{x=(x t)0 2*x t>,1, t=0,1,...},(6)and the utility function is of time separable form,
u(x)=6 d tlog(x t+1), 0?d t?1.
t=0

(7)

Then we have,Du(x)=(d1,,...),
x 0+1 x 1+1

and
4

(8)

Regular Economies with Infinitely Many Commodities1(x 0+1) 2

0

0 

0

d
(x 1+1) 20 .

D 2u(x)=,     (9)LetGRR m‑1++-` denote the set of equilibria. The natural projection p : G` is the restriction to
G of the projection (l,w)w.
Theorem 1: The Negishi function n=(n 2...n m) is smooth.
Proof: First we shall show that the map lx i(l), i=2...m is smooth. Note that the function isdetermined implicitly by the first order condition of the social welfare maximization problem (5)with a resource constraint;(10)

6x i,r=0, Du 1(x 1),l iDu i(x i)=0, i=2...m.
i

It is sufficient to check that the Jacobian matrix of the above equation systemII,D 2u 1 l 2D 2u 2

,D 2u 1

0

I



I

0



0

l 3D 2u 3 0      ,D 2u 1

0





(11)

 l mD 2u m

0

is invertible (Theorem A1). This matrix is equivalent to
I

0

0



0

,D 2u 1D 2u 1+l 2D 2u 2

D 2u 1

D 2u 1,D 2u 1D 2u 1D 2u 1+l 3D 2u 3 

D 2u 1       

,D 2u 1D 2u 1D 2u 1

Therefore it suffices to show that the matrix

5

 D 2u 1+l mD 2u m



.

(12)

『経済研究』（明治学院大学）第 140・141 合併号

D 2u 1+l 2D 2u 2M=

D 2u 1

D 2u 1D 2u 1D 2u 1+l 3D 2u 3 

D 2u 1D u1

 

2  
D 2u 1D 2u 1



(13)

 D 2u 1+l mD 2u m

is invertible. Let u=(u 2u m)(0 2) m-1 and consider the equation Mv=0. Then we have(D 2u 1+l jD 2u j)v j+D 2u 1 6 v k =0, j=2...m.

r



kj

(14)

Making the inner product with the vector v j then yields

rv ,(D u +l D u )v +rv ,D u r 6 v =0, j=2...m,
2

j

j

1

2

j

j

2

j

k

1

kj

(15)

which simplifies intom(l jv j,D 2u jv j)+ v j,D 2u 1 6 v k =0, j=2...m.

r

r

k2



(16)

We add up these equations for i from 2 to m,mm

m

6 l j(v j,D 2u jv j)+ 6 v j , D 2u 1 6 v j =0.

rr 

j2

r 

j=2

j=2

(17)

Since D 2u j are negative definite, each term in the sum of the left hand side of (14) is C0, hence
each term must be 0,mm

rr 6 v , D u r 6 v =0, l (v ,D u v )=0, j=2...m.
j

j=2

2

j

1

j

j

2

j

j

j=2

(18)

Therefore we have v 2==v m=0. This shows that the orthonormal basis {(1,0,0,...),(0,1,0,...),...} maps to an independent set, which can be transformed to an orthonormal basis by
the Gram-Schmidt procedure. It is well known that the two Hilbert spaces with the bases of thesame cardinality are isomorphic, hence the matrix M is invertible. The smoothness of n i follows
from the smoothness of Du i and the inner product.
Theorem 2: The set of equilibria G is a smooth submanifold of R m-1-` of codimension m,1.
Proof: We shall prove this proposition by applying the infinite dimensional version of the regular value
theorem (Theorem A2, appendix), which amounts to prove that the map n, which is smooth by
Proposition 1, is transversal to the closed submanifold {0}R
R m-1. Let Dn(l,w) denote the tangent
map (derivative) of n at (l,n)G. In order to show the transversality, we have to verify (i) themap Dn(l.w): R m-1-T w`R
R m-1 is onto, and (ii) the kernel N=Dn(l,w) -1(0) has a closed
complement in R m-1-T w`, i.e., there exists a closed subspace H such that N.H/R
R m-1-T w`.
First we consider the topological complement of the tangent map, which is defined by the m,1
6

Regular Economies with Infinitely Many Commodities

coordinate mappings of the derivatives of n i with respect to (l,w 1,...,w m). Note that the partial
derivative of n i with respect to w i(j4i) is 0. The derivative of n i is then written as

Dv i(l j,w i)(m,x)=n i(l,w i)
n i(l,w i) t
x
m+6
l
t=0
w ti

(19)

for mR
R m-1 and x=(x t)0 2. The linear functional Dn i(l,w i) is continuous for the norm topology,
so its kernel is a closed hyperplane in the Banach space R m-1-T w`. The kernel of the tangent
map Dn=(Dn 1,...,Dn m) is therefore the closed subspace of finite codimension defined by theintersection of m,1 closed hyperplanes. Hence it has a topological complement.
We now consider the surjectivity of the tangent map. It suffices that we prove thesurjectivity of the restriction of this map to some subspace of R m-1-T w` . Pick an arbitrary
period t. The Jacobian matrix of Dn with respect to (w t2...w tm) is tu 2(x 2(l))



0

0



0

 tu 3(x 3(l))

0  



,

(20)

   0  tu m(x m(l))

0which is obviously of rank m,1 by the assumption (U‑2).
By Proposition 2, it follows that the projection map p : G`,(l,w)w is smooth, since it is acomposition of smooth maps, the canonical embedding GR m-1++ -` and the canonical projection map
R m-1++ -``.
Definition 3: An economy w` is said to be regular if it is a regular value of the projection
p : G`,(l,w)w.
Our main result reads as follows.
Theorem 3: The set of regular economies is a dense G d subset of `.
Proof: It follows from theorem A3 in Appendix that the natural projection p : G` is Fredholm map ofindex 0, since the codimension of G is equal to m,1, hence by theorem A4, the set of regulareconomies, which is the set of regular values of p is the residual subset of `.

Ⅲ．Appendix
In this appendix, we collect mathematical concepts and results which are used in the text forreaderʼs convenience.
7

『経済研究』（明治学院大学）第 140・141 合併号

Let E and F be Banach spaces and UE be open. A map f : UF is (Fréchet) differentiable atxU if there is a continuous linear map Df(x) : EF such that
lim

h

f(x+h),f(x),Df(x)
=0
h

(21)

Let L(E,F) denote the Banach space of continuous linear map from E to F under the operator
norm, namely that for TL(E,F),
T=sup Tx xC1 .







(22)

For every integer kB0, the map f is said to be of class C k at xU if the map
D kf(x)6D(D k-1f )(x):UL(E,L k-1(E,F))6L k(E,F)(23)

is continuous. Let E=E 1.E 2 (direct sum), where E 1 and E 2 are Banach spaces, and UE is open. For amap of class C KF : UF, the partial derivative with respect to E 1 is defined by
D 1f(x)e 16Df(x)(e 1,0), e 1E 1.

(24)

The partial derivative with respect to E 2 is defined similarly. The next theorem is of central
importance in the differential calculus on manifolds (Abraham et al [1981, p.121]).
Theorem A1 (Implicit function theorem): Let U 1E 1, U 2E 2 be open and f : U 1-U 2F be of class
C k(kB1). For some x 0U 1 and y 0U 2, assume that D 2f(x 0,y 0):E 2F is an isomorphism. Then there
are neighborhoods U 0 of x 0 and VF of f(x 0,y 0) , and a unique map ":U 0-VU 2 such that for all(x,z)U 0-V,
f(x,g(x,z))=z.
A manifold M modeled by a Banach space E is defined similarly as in the case of E ＝R
R n; A topologicalspace M is a Banach manifold if there exist an open cover {U a} and local isomorphisms f a on U a to Ekksuch that f b~f -1
a : f a(U aU b) f b(U aU b) is of class C for each a and b. Let pM. A C curvethrough p is a C k map from (,X,X) to M such that c(0)=p. Two curves c and d are equivalent if and only
C(0). The equivalence class is denoted as [c] and called a tangent vector at p. Let
if Bc(0)(6dc(0)/dt)=d
T pM be the set of all tangent vectors at p and call the tangent space at p. It is easy to see that T p is avector space. A map f from a Banach manifold M to a Banach manifold N is said to be of class C k if theis of class C k . The tangent map (derivative) at pM of a C k map f is a linear map
map f b~f~f -1
aDf(p) : T qT qN defined by Df(p)([c] p)=[f~c] f(p).
Let F be a closed subspace of a Banach space E. F splits in E if there exists a closed subspaceGE such that E=F.G. Let f : MN be a smooth map between two Banach manifolds. A point qN
8Regular Economies with Infinitely Many Commodities

is a regular value of f (or f is transversal to {q}) if for every pf -1(q), Df(p), is surjective and the kernel
D -1f(p)({0}) splits in T pM.
Theorem A2 (Abraham et al [1981, p. 202]): Let f : MN be a smooth map and qN a regular value of f.
Then f -1(q) is a submanifold of N. If N is finite dimensional, then dimf -1(q) (=dimension of f -1(q)) ＝
dim N.
A linear continuous map l : EF is called a Fredholm operator if Ker(l)6l -1(0) is a finite
dimensional subspace of E,and Im(l)6l(E) has a finite codimension in F. The index of the Fredholm
operator, ind(l) is defined by(25)

ind(l)=dimKer(l),codimIm(l)A C k map f is called a Fredholm map if Df(p) is the Fredholm operator at every p. We have
(Abraham and Robbing [1967, pp. 45‑46]),
Theorem A3: Let E and F be a Banach spaces with dimE=n, and assume that p : E-FF is the
projection operator. If G is a closed subspace of E-F of codimension q, then the restriction p : GF is a
Fredholm operator of index n,q.
The next result which is an infinite dimensional analogue of Sardʼs theorem is due to Smale(Abraham and Robbing [1967, pp. 42‑43]).
Theorem A4: Let f : EF be a C k Fredholm map between C k Banach manifolds where E is a Lindelöf
space. If at every pE, k>max{0,ind(Df(p)}, then the set of regular values of f is a residual subset of F.
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（2007 年９月 26 日

10

産業経済研究所受理）