We construct an index theorem for smooth infinite economies with separable utilities that shows that generically the number of equilbria is odd. As a corollary, this gives a new proof of existence and gives conditions that guarantee global uniqueness of equilibria.Uniqueness; determinacy; equilibria; infinite economy; Fredholm map; equilibrium manifold; Banach manifold; index theorem; vector field; Rothe

Covarrubias, Enrique

English

Research Papers in Economics

The number of equilibria of smooth inﬁnite
economies with separable utilities
Enrique Covarrubias
School of Mathematics and
Maxwell Institute for Mathematical Sciences
The University of Edinburgh
James Clerk Maxwell Building, The King’s Buildings
Edinburgh EH9 3JZ United Kingdom
Enrique.C@ed.ac.uk
Tel. +44 (0) 131-6505079
Fax. +44 (0) 131-6506553
1Abstract
We construct an index theorem for smooth inﬁnite economies with
separable utilities that shows that generically the number of equilbria
is odd. As a corollary, this gives a new proof of existence and gives
conditions that guarantee global uniqueness of equilibria.
JEL classiﬁcation: D5, D50, D51
Keywords: Uniqueness, determinacy, equilibria, inﬁnite economy, Fredholm
map, equilibrium manifold, Banach manifold, index theorem, vector ﬁeld,
Rothe
21 Introduction
Models of competitive markets have a consumption space which may be
inﬁnite dimensional. Many authors have addressed the problem of studying
if equilibrium prices are locally unique in inﬁnite dimensions, including the
earlier work of Araujo [2] and also the papers of Kehoe, Levine, Mas-Colell
and Zame [10], Chichilnisky and Zhou [5], Shannon [12], Shannon and Zame
[13] and Covarrubias [6]. In all these cases it has become clear that there is
a trade-oﬀ between the generality of the consumption space, the generality
of utility functions and the existence and diﬀerentiability of the individual
demand functions.
However, an area that still remains largely unexplored in any such case
is that of counting the numer of equilibria. When the consumption space
is ﬁnite dimensional, Dierker [8] gave the ﬁrst solution to this problem, and
constructed an index theorem that showed that the number of equilibria is
generically odd. He does this by interpreting the excess demand function
as a vector ﬁeld on the space of prices, and noticing that equilibria are the
zeros of this vector ﬁeld. He deﬁnes the notion of index of an equilibrium
price system and shows that the sum of these indices is constant and equal
to 1. Since the number of equilibria is generically odd, in particular it can
never be zero and so Dierker’s index theorem gives a new proof of existence
of equilibria. Additionally, if the index at each equilibrium price is > 0 then
the index theorem also gives conditions for global uniqueness of equilibria.
3In inﬁnite dimensions, one of the few results on uniqueness has been
provided by Dana [7] taking into consideration a model of a pure exchange
economy where the agents’ consumption space is L
p
+(µ) and agents have
additively separable utilities which fulﬁl the (RA) assumption that agents’
relative risk aversion coeﬃcients are smaller than one. In this case, Dana
shows that one can work with the space of utility weights to avoid using the
demand approach that may not be well deﬁned. Dana ﬁnally shows that
if utilities fulﬁl the (RA) assumption then the excess utility map is gross
substitute which in turn implies existence and uniqueness of equilibrium.
In this paper, it is our aim to also consider separable utilities as in [7]
but to construct an inﬁnite dimensional analogue of Dierker’s result: that
the number of equilibria of smooth inﬁnite economies is odd and hence to
study conditions that guarantee global uniqueness of equilibria. We present
an analytical notion that has not appeared in the economic literature which
is that of a Z-Rothe vector ﬁeld. When the aggregate excess demand function
deﬁnes a Z-Rothe vector ﬁeld, it allows us to construct an index theorem on
the normalized inﬁnite dimensional price space.
In section 2 we set the market and deﬁne aggregate excess demand func-
tions in our setting; as usual, we will interpret them as vector ﬁelds on the
space of prices. In section 3 we review the basic deﬁnitions of Fredholm the-
ory, which is needed to extend diﬀerential topology to inﬁnite dimensions.
In section 4 we review the determinacy results obtained in [6] showing that
most excess demand functions have isolated zeros; that is, that equilibria are
4locally unique. This guarantees that it makes sense to actually count the
number of equilibria.
In section 5 we review the notion of Z-Rothe vector ﬁelds as developed
by Tromba [15]. When an excess demand function is Z-Rothe, we can deﬁne
a suitable index of equilibrium prices, that is, an index of zeros of a vector
ﬁeld. Then, in section 6, we construct an index theorem for smooth inﬁnite
economies. We show that the sum of indices of equilibrium prices is constant
and equal to 1. Finally in section 7, we give a corollary to the index theorem
analogous to [8], giving a new proof of existence of equilibrium and analyzing
what condition an excess demand function needs to fulﬁll to give rise to a
globally unique equilibrium.
2 The Market
We assume, following [5], that the commodity space is a subset of C(M,Rn),
where M is any compact (Riemannian) manifold. For more general commod-
ity spaces we refer to [4].
Example 1: In growth models a utility function on C(M,Rn) is a
continuous-time version of a discounted sum of time-dependent utilities. Here
M represents time.
Example 2: In ﬁnance, when the underlying parameters follow a dif-
fusion process, a utility function on C(M,Rn) is the expectation of state-
dependent utilities where M is the state space.
5This commodity space is also mathematically convenient because in order
to use diﬀerential techniques, we would like it to be a separable topological
vector space for which the interior of its positive cone (the consumption
space) is non-empty.
The consumption space is then X = C++(M,Rn), the positive cone
of C(M,Rn). Strictly speaking, prices are in the positive cone of the dual
of C(M,Rn). However, it is shown in [5] that with separable utilities only a
small subset of this space can support equilibria and we can actually consider
the price space to be S = {P ∈ C++(M,Rn) : kPk = 1} where kPk =
supt∈M kP(t)kRn with the standard metric k · kRn on Rn.
We denote by h·,·i the inner product on C(M,Rn) so that if f,g ∈
C(M,Rn) then
hf,gi =
Z
M
hf(t),g(t)iRndt
with the standard inner product h·,·iRn in Rn.
We consider a ﬁnite number I of agents. An exchange economy is
parametrized for each agent i = 1,...,I by their initial endowments ωi ∈ X
and their individual demand functions fi : S × (0,∞) → X. The maps
fi(P(t),w) are solutions to the optimization problem
max
hP(t),yi=w
Wi(y)
where Wi(x) is a separable utility function, i.e., it can be written as
6Wi(x) =
Z
M
u
i(x(t),t)dt
We assume ui(x(t),t) : Rn
++ × M → R is a strictly monotonic, concave,
C2 function where {y ∈ Rn
++ : ui(y,t) ≥ ui(x,t)} is closed. In [5] is shown
that this implies that Wi(x) is strictly monotonic, concave and twice Fr´ echet
diﬀerentiable.
In this paper we assume that the individual demand functions are ﬁxed,
so that the only parameters deﬁning an economy are the initial endowments.
Denote ω = (ω1,...,ωI) ∈ Ω = XI. For a ﬁxed economy ω ∈ Ω the ag-
gregate excess demand function is a map Zω : S → C(M,Rn) deﬁned
by
Zω(P) =
I X
i=1
(fi(P,hP,ωii) − ωi)
We also deﬁne Z : Ω × S → C(M,Rn) by the evaluation
Z(ω,P) = Zω(P)
It satisﬁes hP,Zω(P)i = 0 for all P ∈ S.
Deﬁnition 1. We say that P ∈ S is an equilibrium of the economy ω ∈ Ω
if Zω(P) = 0. We denote the equilibrium set
Γ = {(ω,P) ∈ Ω × S : Z(ω,P) = 0}
73 Fredholm Index Theory
We wish to explore the structure of aggregate excess demand functions and
since we will be using tools of diﬀerential topology in inﬁnite dimensions, we
would like our maps to be Fredholm as introduced by Smale [14].
A (linear) Fredholm operator is a continuous linear map L : E1 → E2
from one Banach space to another with the properties:
1. dim ker L < ∞
2. range L is closed
3. coker L = E2/rangeL has ﬁnite dimension
If L is a Fredholm operator, then its index is dim kerL − dim cokerL,
so that the index of L is an integer.
A Fredholm map is a C0 map f : M → V between diﬀerentiable man-
ifolds locally like Banach spaces such that for each x ∈ M the derivative
Df(x) : TxM → Tf(x)V is a Fredholm operator. The index of f is deﬁned
to be the index of Df(x) for some x. If M is connected, this deﬁnition does
not depend on x.
In our previous work [6] we have shown that the excess demand function
Zω : S → C(M,Rn) of economy ω ∈ Ω is a Fredholm map of index zero.
84 Determinacy of equilibria
Since we would like to count the number of price equilibria of an economy,
the ﬁrst result that we need to establish is that generically equilibria will be
isolated. Below we remind the reader the notion of a regular economy and
of a regular price system.
Deﬁnition 2. We say that an economy is regular (resp. critical) if and
only if ω is a regular (resp. critical) value of the projection pr : Γ → Ω .
Deﬁnition 3. Let Zω be the excess demand of economy ω. A price system
P ∈ S is a regular equilibrium price system if and only if Zω(P) = 0
and DZω(P) is surjective.
In our previous work [6] we showed the relation between regular economies
and regular equilibrium prices.
Proposition 1. [6] The economy ω ∈ Ω is regular if and only if all equilib-
rium prices of Zω are regular.
Proposition 2 showed that for most economies, its aggregate excess de-
mand function will have isolated zeros. Hence, it makes sense to try to count
them.
Proposition 2. [6] Almost all economies are regular. That is, the set of
economies ω ∈ Ω that give rise to an excess demand function Zω with only
regular equilibrium prices, are residual in Ω.
9Since we haven shown that for most excess demand functions Zω will have
isolated zeros, we will drop the explicit dependence on ω and will simply write
Z.
5 Z-Rothe vector ﬁelds
Knowing that the excess demand function is a vector ﬁeld on the price space,
and that is a Fredholm map for which we know its index, we would like to
give it the structure of a Z-Rothe vector ﬁeld as developed by Tromba [15].
In section 6 it will become clear that we need a vector ﬁeld that is outward
pointing, so we insist −Zω to be Z-Rothe.
Let E be any Banach space and L(E) be the set of linear continuous
maps from E to itself. Denote by GL(E) the general linear group of E; that
is, the set of invertible linear maps in L(E). Let C(E) be the linear space of
compact linear maps from E to itself.
We write S(E) ⊂ GL(E) to denote the maximal starred neighborhood of
the identity in GL(E). Formally,
S(E) = {T ∈ GL(E) : (αT + (1 − α)I) ∈ GL(E),∀α ∈ [0,1]}
The Rothe set of E is deﬁned as
R(E) = {A : A = T + C,T ∈ S(E),C ∈ C(E)}
10and its invertible members by GR(E) = R(E) ∩ GL(E).
A C1 vector ﬁeld X on a Banach manifold M is Z-Rothe if whenever
X(p) = 0, DX(P) ∈ R(TPM)
Proposition 3. The negative of the excess demand function, −Z : S → TS
is a Z-Rothe vector ﬁeld.
Proof. If we want −Z to be a Z-Rothe vector ﬁeld, we need to check that
whenever Z(P) = 0, DZ(P) ∈ R(TPS). That is, we need to write DZ(P)
as the sum of an element of S(TPS) and an element of C(TPS).
Chichilnisky and Zhou [5] have shown that for each agent i, his individual
demand function Dfi can be written as the sum of the ﬁnite rank operator
−
λhP(t),(ui
xx)−1DP(t)i + hDP(t),fii
hP(t),(ui
xx)−1P(t)i
(u
i
xx)
−1P(t) (1)
and the invertible operator
(ui
xx)−1P(t)
hP(t),(ui
xx)−1P(t)i
Dw + λ(u
i
xx)
−1DP(t) (2)
Adding over agents in equation (1), deﬁne ZC be the sum of the ﬁnite
rank operators, that is,
ZC =
I X
i=1
−
λhP(t),(ui
xx)−1DP(t)i + hDP(t),fii
hP(t),(ui
xx)−1P(t)i
(u
i
xx)
−1P(t) (3)
Then ZC has ﬁnite rank, and hence ZC ∈ C(TPS). Now add over agents
in equation (2) and deﬁne ZR to be
11ZR =
I X
i=1
(ui
xx)−1P(t)
hP(t),(ui
xx)−1P(t)i
Dw + λ(u
i
xx)
−1DP(t) (4)
The matrix (ui
xx) is negative deﬁnite, and every negative deﬁnite matrix is
invertible and its inverse is also negative deﬁnite. So ZR has to be invertible
and hence ZR ∈ GL(TPS)
All we need to show then is that ZR ∈ S(TPS), that is, that
α
"
−
I X
i=1

(ui
xx)−1P(t)
hP(t),(ui
xx)−1P(t)i
Dw − λ(u
i
xx)
−1DP(t)
#
+ (1 − α)I
is invertible for all α ∈ [0,1]. But this sum is just a homotopy of positive-
deﬁnite operators.
6 The Index Theorem of Smooth Inﬁnite Econ-
omies
Knowing that most economies are regular we need to ﬁnd a right way of
counting the number of equilibria. With an excess demand function that
is a Fredholm map, we may use tools from inﬁnite-dimensional diﬀerential
topology that resembles the ﬁnite dimensional case.
Below we review the notion of index of a zero of a Z-Rothe vector ﬁeld.
12We also review the Euler characteristic, which is the topological invariant
that we would like our index theorem to be equal to.
6.1 Euler Characteristic
A zero P of a vector ﬁeld X is nondegenerate if DX(P) : TPM → TPM is
an isomorphism.
Suppose that a Z-Rothe vector ﬁeld X has only nondegenerate zeros, and
let P be one of them. Then, DX(P) ∈ GR(TPM). Tromba [15] shows
that GR(TPM) has two components; GR+(E) denotes the component of
the identity. Deﬁne
sgnDX(P) =

  
  
+1, if DX(P) ∈ GR+(TPM)
−1, if DX(P) ∈ GR−(TPM)
The Euler characteristic is then given by the formula
χ(X) =
X
P∈Zeros(X)
sgnDX(P)
Tromba also shows that this Euler characteristic is invariant under ho-
motopy of vector ﬁelds. All we have to do is to construct a vector ﬁeld on S
that has only one singularity and that is homotopic to the aggregate excess
demand Z.
136.2 The Index Theorem of Smooth Inﬁnite Economies
Suppose that the excess demand satisﬁes the ‘boundary assumption’ of Dierker
[8], namely that if Pn ∈ S and Pn → P ∈ ∂S, then kZ(Pn)k → ∞. Suppose
also that Z is bounded below. Then −Z is an outward-pointing vector ﬁeld.
Finally, assume that there are only ﬁnitely many zeros.
We are now ready to introduce our main result.
Proposition 4. Suppose that an aggregate excess demand function Z is
bounded from below and that it satisﬁes the boundary assumption. Suppose
also that Z has only ﬁnitely many singularities and that they are all nonde-
generate. Then,
X
P∈ZerosZ
sgn[−DZ(P)] = 1
Proof. For any ﬁxed Q ∈ C++(M,Rn) deﬁne the vector ﬁeld ZQ : ¯ S → TS
given by
Z
Q(P) =

Q(t)
hP(t),Q(t)i

− P(t)
By construction, ZQ(P) has only one zero and is inward-pointing. Its
derivative DZ
Q
(P) : T ¯ S → T(TS) is given by
DZ
Q
(P)(h) = −
Qhh,Qi
hP,Qi2 − h
where h → −
Qhh,Qi
hP,Qi2 is compact and h → −h is invertible; then DZQ ∈
14R(TPS). Now let
−
Qhh,Qi
hP,Qi2 − h = h
0 (5)
We need to solve for h. Then,
Qhh,Qi + hhP,Qi
2 = −h
0hP,Qi
2
Acting Q on both sides we get,
hQ,Qihh,Qi + hh,QihP,Qi
2 = −hh
0,QihP,Qi
2
Solving for hh,Qi we get
hh,Qi =
−hh0,QihP,Qi2
hQ,Qi + hP,Qi2
where the denominator never vanishes since Q ∈ C++(M,Rn). Substi-
tuting hh,Qi in 5 we then get
h = h
0 +
Q
hP,Qi2

hh0,QihP,Qi2
hQ,Qi + hP,Qi2

This shows that DZQ is invertible and therefore DZQ ∈ GR(TPS). Fur-
thermore, since it is not in the same component of the identity it has to be
in GR−(TPS) and its only zero has index -1. The vector ﬁeld ZQ is inward
pointing so reversing orientation will make outward poiting with index of +1.
157 Concluding Remarks
We conclude from Proposition 4 that the number of equilibria of smooth
inﬁnite economies generically is odd. In particular, it can never be zero so
this gives a new proof of existence.
Also, as a corollary of Proposition 4, we can also provide an inﬁnite
dimensional analogue of [8]; Dierker shows
Proposition 5. [8] If the Jacobian of the excess supply function is positive
at all Walras equilibria, then there is exactly one equilibrium.
We show that:
Proposition 6. If the sign of the derivative of the excess supply function is
positive at all Walras equilibria, then there is exactly one equilibrium.
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A Third Remark on the Number of Equilibria of an

An Inﬁnite Dimensional Version of Sard’s Theorem.

Determinacy of equilibria of smooth inﬁnite economies. Working Paper

Determinacy of Equilibrium in Large-Scale Economies.

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Quadratic Concavity and Determinacy of Equilibrium.

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Transversal Maps and Flows.

Two Remarks on the Number of Equilibria of an Economy.

The number of equilibria of smooth infinite economies with separable utilities

The number of equilibria of smooth infinite economies with separable utilities

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