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Refinements to the generic existence of equilibrium in
incomplete markets
Lionel Boisdeffre
To cite this version:
Lionel Boisdeffre. Refinements to the generic existence of equilibrium in incomplete markets. 2019. �halshs-02181033�
Documents de Travail du
Centre d’Economie de la Sorbonne
Refinements to the generic existence of equilibrium in incomplete markets
Lionel de BOISDEFFRE
Refinements to the generic existence of equilibrium in incomplete markets
Lionel de Boisde¤re,1
(June 2019)
Abstract
The paper demonstrates the generic existence of general equilibria in incomplete markets with well behaved price properties. The economy has two periods and an ex ante uncertainty over the state of nature to be revealed at the second period. Se-curities pay o¤ in cash or commodities at the second period, conditionally on the state of nature to be revealed. They permit …nancial transfers across periods and states, which are insu¢ cient to span all state contingent claims to value, whatever the spot price to prevail. Under smooth preference and the standard Radner (1972) perfect foresight assumptions, equilibrium is shown to exist, except for a closed set of measure zero of endowments and securities. The proof provides additional argu-ments and insights to Du¢ e-Shafer’s (1985) on the same subject and re…nes it in two ways. First, equilibrium is shown to exist generically for any norm values of commodity prices on any spot market, and for any collection of state prices. Second, assets need no longer pay o¤ in commodities, but may in any mix of cash and goods.
.
Key words: sequential equilibrium, temporary equilibrium, perfect foresight,
exis-tence, rational expectations, …nancial markets, asymmetric information, arbitrage. JEL Classi…cation: D52
1University of Paris 1-Panthéon-Sorbonne, 106-112 Bd. de l’Hôpital, 75013 Paris.
1 Introduction
This paper demonstrates the generic existence of equilibrium in incomplete …nan-cial markets with di¤erential information. It presents a two-period pure exchange economy, with an ex ante uncertainty over the state of nature to be revealed at the second period.
When assets pay o¤ in goods, equilibrium needs not exist, as shown by Hart (1975). His example is based on the collapse of the span of assets’ payo¤s, that occurs at clearing prices. Attempts to restore the existence of equilibrium noticed that the above "bad " prices could only occur exceptionally, as a consequence of Sard’s theorem. These attempts include Mc Manus (1984), Repullo (1984), Magill & Shafer (1984, 1985), for potentially complete markets (i.e., complete for at least one price), and Du¢ e-Shafer (1985, 1986), for incomplete markets. These papers build on di¤erential topology arguments, and demonstrate the generic existence of equilibrium, namely, existence except for a closed set of measure zero of economies, parametrized by the assets’payo¤s and the agents’endowments.
The current model extends Du¢ e-Shafer’s (1985) in two ways. First, its …nancial structure may cover any mix of nominal and real assets. Second, it normalizes (to arbitrary values) equilibrium prices on every spot market. In Du¢ e-Shafer (1985), the value of one particular consumer’s endowment is normalized across all states of nature. The aim is to prove existence of equilibrium under the perfect foresight assumption. The relevance and the means of inferring correct prices are no issues.
In the current paper, however, normalizing price anticipations in every state of nature to relevant values is an important issue, because it is a step towards dropping the perfect foresight assumption, also called the rational expectation assumption.
This standard assumption, privileged by Radner (1972), states that agents know the map between the state of nature and the spot price to obtain. It is seen as unreal-istic by most theorists, including Radner (1982) himself, for whom the assumption "seems to require of the traders a capacity for imagination and computation far beyond what is realistic".
Yet, to our best knowledge, no de…nition of a sequential equilibrium (as opposed to the temporary) was given so far, which dropped the assumption. The temporary equilibrium drops the assumption, but allows agents to form erroneous forecasts. When agents have no price model, they typically face endogenous uncertainty over future spot prices and need focus their anticipations on sets of relevant values, so that one of them be self-ful…lling ex post. The current paper is a step in this direction, which we develop in a companion paper.
The current proof uses standard di¤erential topology arguments, introduced by Debreu (1970, 1972) for the study of general equilibrium. Following Du¢ e-Shafer (1985), we de…ne so-called "pseudo-equilibria" and prove their full existence from modulo 2 degree theory. The generic existence of equilibria is then derived from Sard’s theorem and Grassmannians’properties. Relative to Du¢ e-Shafer’s, the proof provides additional arguments and insights. It re…nes the latter by exhibiting well behaved properties of the equilibrium prices.
The paper is organized as follows: Section 2 introduces the model and its con-cepts of equilibrium, pseudo-equilibrium, Grassmannians and their main properties. Section 3 presents the pseudo-equilibrium manifold and states and proves the exis-tence theorems. An Appendix proves a technical Lemma.
2 The model
Throughout the paper, we consider a pure-exchange economy with two periods,
t 2 f0; 1g, and an uncertainty, at t = 0, upon which state of nature will randomly
prevail, at t = 1. Consumers exchange goods, on spots markets, and assets of all
kinds, on typically incomplete …nancial markets. The sets,I,S,LandJ, respectively,
of consumers, states of nature, consumption goods and assets are all …nite. The
state of the …rst period (t = 0) is denoted bys = 0 and we letS0:= f0g [ S. Similarly,
l = 0 denotes the unit of account and we let L0:= f0g [ L.
2.1 The commodity and …nancial markets
Agents consume or exchange the consumption goods,l 2 L, on both periods’spot
markets. Commodity prices on spot markets are restricted to the positive quadrant,
P := fp := (ps) 2 RL S
0
++ : kpsk = 1; 8s 2 S0g. Normalization to one is assumed for
convenience but non restrictive. In any state, s 2 S0, that bound could be replaced
by any positive value without changing the model’s results.
Consumers may operate transfers across states by exchanging, at t = 0, …nitely
many assets, j 2 J (with #J 6 #S), which pay o¤, at t = 1, conditionally on
the realization of the anticipated states, s 2 S. These conditional payo¤s may be
nominal or real or a mix of both. The generic payo¤s of an asset, j 2 J, in a state,
s 2 S, are thus a bundle, vj(s) := (vlj(s)) 2 RL
0
, of the quantities, v0
j(s), of cash, and
vl
j(s), of each good l 2 L, which are delivered if states prevails.
These payo¤s de…ne a (S L0) J real matrix, V, identi…ed to a map, V : S RL
+!
RJ, relating the forecasts of a state and its spot price,! := (s; p) 2 S RL
+, to the row
Thus, at asset price q 2 RJ, agents may buy or sell portfolios of assets,z = (z
j) 2 RJ,
for q z units of account att = 0, against the promise of delivery of a ‡ow, V (!) z, of
conditional payo¤s across forecasts, ! 2 S RL
+. For notational purposes, we let:
R J be the set of all J matrices for = S L0 or = S;
V0(s) 2 RJ, for every matrix,V02 RS J, and state,s 2 S, be the matrix’sthrow;
V0(!) 2 RJ, for every V0 2 R(S L0) J
, and forecast, ! := (s; p) 2 S RL
+, be the
matrix’row of cash payo¤s if! obtains;
V0
p 2 RS J, for every V0 2 R(S L
0) J
and every p := (ps) 2 P, be de…ned by
V0
p(s) := V0(s; ps), for every s 2 S;
G := f V0 2 RS J; rank V0 = #J g and G := f < V0 > : V0 2 G g, in which < V0 >
denotes the span inRS of the matrix’columns.
p x 2 RS, for every pair (p; x) 2 P RL S0
, be the vector, whose components
are the scalar products, ps xs, for every s 2 S.
S be identi…ed to #S, L to#L, J to #J,I to #I, whenever needed;
v := J:S:(L + 1) = dim R(S L0) J;
v := (S J ):J;
e := I:L:(S + 1);
l := (S + 1):(L 1) = dim P.
2.2 The consumer’s behaviour and concept of equilibrum
Each agent, i 2 I, receives an endowment, ei := (eis), granting the commodity
prices, p := (ps) 2 P, for commodities, and q 2 RJ, for securities, and given the
endowment and payo¤ collections, (e := (ei); V ) 2 RL S
0 I
R(S L0) J, the generic ith
agent’s consumption set isX := RL S++ 0, and her budget set is:
Bi(p; q; ei; V ) := f (x; z) 2 X RJ: p0(x0 ei0)6 q z and p (x ei)6 Vp z g;
Each consumer,i 2 I, has preferences represented by a utility function,ui: X ! R
and optimizes her consumption in the budget set. The above economy is denoted
by E(e;V ) = f(I; S; L; J); V; e := (ei)i2I; (ui)i2Ig. Given (e0; V0) 2 XI R(S L
0) J
, we de…ne
the economy E(e0;V0) = f(I; S; L; J); V0; e0; (ui)i2Ig in the same way as above, and its
equilibrium and pseudo-equilibrium concepts as follows:
De…nition 1 Given the endowments,e0:= (e0
i) 2 XI, and payo¤ matrix,V02 R(S L
0) J
,
a collection of prices, (p; q) 2 P RJ, and strategies, (x
i; zi) 2 Bi(p; q; e0i; V0), for each i 2 I, is an equilibrium of the economy, E(e0;V0), if the following Conditions hold:
(a) 8i 2 I; xi2 arg max ui(x), for (x; z) 2 Bi(p; q; e0i; V0); (b) Pi2I (xi e0i) = 0;
(c) Pi2I zi= 0.
The state prices, = ( s) 2 RS++, are said to support an equilibrium of the economy,
E(e0;V0), if the equilibrium prices, (p; q) 2 P RJ, meet the condition: q =Ps2S sVp0(s).
The economy is called standard if it meets the following conditions:
Assumption A1 : 8i 2 I; ui is C1;
Assumption A2 (Inada Conditions): 8(i; s; l; x := (xl
s)) 2 I S0 L X, @ui(x)=@xls2 R++, limxl
s!0@ui(x)=@x
l
s= 1 (where xls! 0 stands for "xls tends to zero
while other components of xare …xed"), limxl
s!1@ui(x)=@x
l
s= 0 (where xls! 1
stands for "xl
s tends to in…nity at other components of x …xed"), limxl
s!0ui(x) = 0;
Assumption A3 (strict concavity): 8i 2 I, hTD2u
De…nition 2 Let := ( s) 2 RS++ be given. The collection of a scalar,y 2 R++, prices, p := (ps) 2 P, a payo¤ matrix, V0 2 R(S L
0) J
, a vector space, G 2 G , consumptions,
xi := (xis) 2 X, and endowments, e0i := (e0is) 2 X, for each i 2 I, is said to be a
-pseudo-equilibrium of the economy, E(e0;V0), if the following conditions hold:
(a) x12 arg max u1(x), forx 2 f x := (xs) 2 X : p0 (x0 e010) + P
s2S sps (xs e01s) = 0 g;
(b) for every i 2 Inf1g, xi2 arg max ui(x),
for x 2 f x := (xs) 2 X : p0 (x0 e0i0) + P s2S s ps (xs e0is) and p (x e0i) 2 G g; (c) < V0 p> G; (d) Pi2I (xi e0i) = 0; (e) p0 e010+ P s2S s ps e01s= y.
Given (e0; V0) 2 XI R(S L0) J, we say that (y; p; G) 2 R
++ P G is a
-pseudo-equilibrium, if there exists x 2 XI, such that(x; y; p; G; e0; V0)is a -pseudo-equilibrium
along Conditions (a)to (e), above. We let E be the pseudo-equilibrium manifold, or
the set of collections,(y; p; G; e0; V0), such that(y; p; G)is a -pseudo-equilibrium, given
(e0; V0). We de…ne a projection map, : (y; p; G; e0; V0) 2 E 7! (e0; V0) 2 XI R(S L0) J
. Remark 1 We chose to de…ne pseudo-equilibria and equilibria with reference to …nancial structures mixing both nominal and real assets. This is no restriction. All arguments and results of this paper hold if assets pay o¤ in goods or cash only.
Claim 1 Let := ( s) 2 RS++ be given and (x := (xi); y; p; G; e0 := (e0i); V0) be a
-pseudo-equilibrium of a standard economy, E(e0;V0), such that < Vp0 > = G. Then, the
economy, E(e0;V0), has an equilibrium, (p; q; [(xi; zi)]), supported by the state prices, .
Proof Let := ( s) 2 RS++ be given and let (x; y; p; G; e0; V0) be a
-pseudo-equilibrium such that < V0
p > = G. From Condition (b) of De…nition 2, there exists
zi2 RJ, for eachi 2 Inf1g, such that p (xi e0i) = Vp0 zi. Letz1:= Pi2Inf1gzi. Then,
P
from above: p (x1 e01) = ( P
i2Inf1gps (xis e0is))s2S= Pi2Inf1g Vp0 zi= Vp0 z1.
Letz := (zi) 2 RJ I,q :=Ps2S sVp0(s)andC := (p; q; x; z)be given from above. From
De…nition 2 and above the relations (xi; zi) 2 Bi(p; q; e0i; V0) hold, for every i 2 I, and
the collection, C := (p; q; x; z), meets Conditions(b)-(c) of De…nition 1 of equilibrium.
Leti 2 Inf1g be given. From Assumption A2, the budget setBi(p; q; e0i; V0) can be
replaced by B0
i(p; q; e0i; V0) := f(x; z) 2 X RJ : p0(x0-e0i0) = q z and p (x-e0i) = Vp0 zg in
De…nition 1 at no cost. From the de…nition of q, the pseudo-equilibrium budget set,
B0
i := fx 2 X : p0(x0 e0i0) + P
s2S sps(xs e0is) = 0 and p (x-e0i) 2 Gg, coincides
with Bi := f x 2 X : 9z 2 RJ; (x; z) 2 B0
i(p; q; e0i; V0) g. Since xi is optimal in Bi0 = Bi,
the strategy (xi; zi) is optimal in Bi(p; q; e0i; V0) from above. We show similarly that
(x1; z1)is optimal in B1(p; q; e01; V0). Then, C also meets Condition (a) of De…nition 1,
and, from above, de…nes an equilibrium of the economy E(e0;V0).
2.3 A characterization of the set G
The set,G , of#J-dimensional subspaces ofRS, is referred to as a Grassmannian.
To present the topological properties of the Grassmannian, we need introduce a distance between its elements. A traditional approach to this problem builds on the
concept of principal angles between the elements of G , along De…nition 3:
De…nition 3 The principal angles, j2 [0; =2], for j 2 f1; :::; #Jg, between two vector
spaces, (G; G0) 2 G 2, are de…ned by #J pairs of vectors, (u
j; vj), for j 2 f1; :::; #Jg,
solving the problems cos j := uj vj = max(u;v)2G G0u v, subject to kuk = kvk = 1, and
u uj0 = v vj0 = 0 for eachj02 f1; :::; j 1g. They de…ne a distance, d : (G; G0) 2 G 27!
d(G; G0) =qP j2J sin
2
j, and a related topology, , on G , referred to throughout.
or Calderbank and alii (1999). A distance had to be de…ned on G , which is no Euclidean space, before characterizing this set, along the following Claim 2.
We de…ne Z := fW 2 R(S J ) S : the rows of matrix W form an orthonormal set g and
Z := fW 2 RS J : 9W0 2 Z, such that W0:W = 0, and the columns of W are orthonormal g.
We can characterize the Grassmannian manifold owing to the latter vector spaces:
Claim 2 Let G be a sub-vector space of RS. The following Assertions hold:
(i) (G 2 G ) , (9W 2 Z : G = f z 2 RS : W z = 0 g); (ii) G = f < W > : W 2 Z g;
(iii) G is compact.
Proof Assertion(i)LetW 2 Z andG = f z 2 RS : W z = 0 gbe given. Since all rows of
W are orthonormal and their number isS J,Gis a J-dimensional sub-vector space
ofRS. That is,G := fz 2 RS : W z = 0g 2 G , wheneverW 2 Z. Conversely, letG 2 G be
given. SinceGis aJ-dimensional vector space, we may construct a(S J ) S matrix
W, whose rows form an orthonormal set and such thatG = f z 2 RS: W z = 0 g.
Assertion(ii) stems tautologically from Assertion (i). First, let W 2 Z be given.
There exists W0 2 Z, such that W0:W = 0. Then, each column of W belongs to
G := f z 2 RS : W0z = 0 g, which implies, from Assertion(i): < W > G 2 G . Hence,
the inclusion f< W > : W 2 Z g G holds.
Conversely, let G 2 G be given. From Assertion (i), there exists W0 2 Z, such
that G = f z 2 RS : W0z = 0 g. SinceG is J-dimensional, there exists a set, fWjg, of J
orthonormal vectors of G. Then, the matrix, W, whose columns are the vectors Wj
(forj 2 J) belongs toZ and is such that< W > = G. Hence,G f < W > : W 2 Z g.
Assetion(iii) LetfGkg
(ii), there exists a representing sequence, fWkg
k2N, of elements of Z , i.e., such that
< Wk > = Gk, for all k 2 N. Since Z is compact in an Euclidean space, we may
assume that fWkg
k2N converges, say to W 2 Z , for the Euclidean norm. From
Assertion (ii), the relation G := < W > 2 G holds, whereas, from Calderbank and
al. (1999, p. 130), d(Gk; G) = kWk:TWk W:TW k =p2 holds for allk 2 N. Hence, from
above,limk!1d(Gk; G) = 0, that is, G is compact for the topology .
2.4 The other main properties of the set G
Let be the set of permutations between states, s 2 S. For every 2 , we let
P 2 RS S be the corresponding permutation matrix. That is, for every V0 2 RS J,
P :V0 2 RS J is obtained by permuting the matrix’rows along . From the de…nition
of G, for every V0 2 G, there exists 2 , which needs not be unique, such that the
last J rows of P :V0 are linearly independent. Thus, for each 2 , we let:
G := fV02 RS J: P :V0 = 0 B B @ W V 1 C C
A 2 RS J, with W 2 R(S J ) J and V 2 RJ J invertibleg
and G := f < V0> : V02 G g.
Given 2 , the generic vector spaceG 2 G , admits, from above, a unique matrix
representation of the form P 1:
0 B B @ (G) I 1 C C
A, where (G) 2 R(S J ) J takes arbitrary
values when G varies. We de…ne the map, : G 2 G 7! P 1:
0 B B @ (G) I 1 C C A, and let:
[ I j (G) ]be the(S J ) S matrix, whose …rst(S J )columns are those of the
identiy matrix,I 2 R(S J ) (S J ), followed by the columns of (G) 2 R(S J ) J;
K : P G R(S L0) J! R(S J ) J be the map de…ned byK (p; G; V0) := [ I j (G) ]:P :V0 p.
Claim 3 Let 2 and G 2 G be given. The following Assertions hold:
(i) fG g 2 is an open cover of G; (ii) G = fz 2 RS : [ I j (G) ]:P z = 0g; (iii) is a homeomorphism;
(iv) fG g 2 is an open cover of G ; (v) G is a manifold without boundary;
(vi) the map (p; V0) 2 P V ! K (p; G; V0) 2 R(S J ) J is C1;
(vii) the sets Im K , G , G and G are manifolds of dimension v := (S J ):J; the
derivative DV0 K (p; G; V0) has full rank, v .
Proof We set 2 as given and, to simplify, we will assume w.l.o.g. that = Id,
unless stated otherwise. Assertion (i)results from the de…nitions.
Assertion(ii)LetG 2 GId be given. The relation[ I j Id(G) ]
0 B B @ Id(G) I 1 C C A = 0holds
from the de…nition. Let z 2 G be given. From above, there exists z0 2 RJ, such that
z = 0 B B @ Id(G) I 1 C C
A z0. Hence, the relation[ I j Id(G) ] z = [ I j Id(G) ]
0 B B @ Id(G) I 1 C C A z0 = 0
holds and the relation G fz 2 RS : [ I j Id(G) ] z = 0g follows.
Conversely, letz := 0 B B @ z1 z2 1 C C
A 2 RS be such that [ I j Id(G) ] z = 0, where z12 RS J
and z2 2 RJ. From the above de…nitions, the relation [ I j Id(G) ] z = 0 is written
z1 = Id(G) z2, that is, z = 0 B B @ Id(G) I 1 C C A z2 2 < 0 B B @ Id(G) I 1 C C A > = G. The converse relation, fz 2 RS
: [Ij Id(G)] z = 0g G, follows, and Assertion(ii) holds fom above.
Assertion (iii)Assume w.l.o.g. that = Id. From the above de…nitions, the map
Id : G 2 GId 7! Id(G) = 0 B B @ Id(G) I 1 C C
We show it is bicontinuous for the two topologies. The continuity of Id is obvious
from De…nition 3. To show that 1
Id is continous we let W = Id(G) and G 2 GId be
given. As an immediate corollary of Claim 2, there exists a neighbourhood, U, of
W in Id(GId), a scalar, K > 0, and a map, W 2 U 7! VW 2 RJ J, such that, VW is
invertible, (W ) := W :VW 2 Z and kVWk < K, for every W 2 U. It follows that the
map, : U ! Z , is (K-Lipschitzian) continuous. Then, the continuity of Id1 at W
stems from the following relations:
d( Id1(W ); Id1(W )) = k (W ):T (W ) (W ):T (W )k =p2,
which hold, for everyW 2 U, from Calderbank and alii (1999, p. 130).
Indeed, letWk 2 U, for everyk 2 N, be such thatlim
k!1kW Wkk = 0. The
Calder-bank relations and the continuity of imply: limk!1d( Id1(Wk); Id1(W )) = 0. That is
1
Id is continuous atW 2 GId, hence, continuous.
Assertions(iv)and(v)result from Assertions(i)-(iii)and the de…nition ofG .
Assertion(vi)results from the de…nition ofK .
Assertion (vii)Let 2 be given. From Assertion (iii), G is homeomorphic to:
fV0 = P 1: 0 B B @ W I 1 C C A, W 2 R(S J ) Jg, whose dimension is v := (S J ):J.
Hence, from the de…nitions, Assertions (i) (iv)and above, G , G , G and Im K
are all manifolds of the same dimension,v .
LetJ be the set of last J states. To check that DV0KId(p; G; V0) has full rank
(hence, alsoDV0K (p; G; V0)for 2 ), for every (p; G; V0) 2 P G V, we write:
where V0
p(SnJ ) is the (S J ) J matrix, whose rows are the top (S J ) rows of V0
p and Vp0(J )is the J J matrix, whose rows are the last J rows of Vp0.
The derivatives of KId(p; G; V0) with respect to payo¤s, for s 2 SnJ, are of the form
of a (S J ) (S J ) block diagonal matrix,P, of diagonal elements:
the J J (L+1)matrices P (s) = 0 B B B B B B B B B B @ (1; ps) 0 0 :: 0 0 (1; ps) 0 :: 0 : : : 0 0 0 :: (1; ps) 1 C C C C C C C C C C A , for everys 2 SnJ.
The matrix P, therefore, has rank (S J ):J. It follows from above that the
deriva-tive DV0KId(p; G; V0)has maximal rank, v = (S J )J.
3 The pseudo-equilibrium manifold and existence theorems
This Section de…nes demand correspondences, characterizes the pseudo-equilibriummanifold,E , given 2 RS
++, and states and proves the existence Theorems.
3.1 The demand and excess demand correspondences
We recall the notations of sub-Section 2.1 and let = ( s) 2 RS++ be given,
throughout. For agent i = 1, we de…ne her demand, D1 : R++ P ! X, by D1(y; p) :=
arg max u1(x), for x 2 f x 2 X : p0 e010+ P
s2S s ps e01s = y g. In the latter problem, y > 0 is taken as given. As classical results, in a standard economy,D1 is aC1 map,
such that, given y,limp!pkD1(y; p)k = +1whenever p 2 @P nf0g.
The demand correspondence, Di : P G X ! X, de…ned by Di(p; G; e0
i) := arg max ui(x), forx 2 f x 2 Xi: p0(x0 e0i0)+
P
s2S sps(xs e0is) = 0 and p (x e0i) 2 G g,
Using Walras’law, we pick up one good, say l = 1. We recall that dim P = l := (S + 1)(L 1). For every i 2 I, and every consumption xi 2 X, we denote by xi := (xis) 2 Rl
++, the extracted vector, which drops consumptions in all goods l = 1. We
denote similarly (with stars) the extracted demands in Rl . Given (y; p; G; e0:=(e0
i)) 2
Rl +1++ G Re++, the excess demand (in Rl ) is then:
Z (y; p; G; e0) := D 1 (y; p) + P i2Inf1g Di (p; G; e0i) P i2I e0i .
It de…nes a demand correspondence, Z : Rl +1++ G Re++ ! Rl . It follows from
above that Z is a C1 map, whose (partial) derivative satis…es D
e1Z (y; p; W; (e
0 i)) =
I, whereI is thel l identity matrix. We notice from the limit property ofD1 that
lim(y;p;G;e0)!(y;p;G;e0)kZ (y; p; G; e0)k = +1whenever(y; p; G; e0) 2 R++ @(Rl++)nf0g G Re++.
3.2 The pseudo-equilibrium manifold’s characterization and properties
We consider the following mappings, for each 2 :
h : (y; p; e0
1) 2 Rl +1++ X17! h (y; p; e01) := (p0 e010+ P
s2S sps e01s y) 2 R;
K : (p; G; V0) 2 P G R(S L0) J7! [ I j (G) ]:P :V0
p 2 Rv (as de…ned above);
H : (y; p; G; e0; V0)2Rl +1
++ G Re++ R(S L
0) J
7! (h (y; p; e0
1); Z (y; p; G; e0); K (p; G; V0))2Rl +1+v .
From the de…nition and Claim 3, the pseudo-equilibrium manifold,E , is[ 2 H (0) 1.
The manifold’s properties stem from those ofH (for 2 ), which, following Du¢
e-Shafer (1985), are summarized hereafter.
Claim 4 Given 2 , the image 0 is a regular value of the map H , which is C1
Proof Let 2 be given. The fact that H is C1 is standard (see Du¢ e-Shafer,
1985, pp. 292-293). To show that 0 is regular, consider the derivative of H with
respect to y,e0 1 and V0: D(y;e0 1;V0) H (y; p; G; e 0; V0) : = 0 B B B B B B @ Dyh (y; p; e01) = 1 DyZ (y; p; G; e0) 0 0 De1Z (y; p; G; e 0) = I 0 0 0 DV0K (p; G; V0) 1 C C C C C C A :
This matrix has full rank, 1 + l + v , from the above Claim 3. Claim 4 follows.
Claim 5 E is a submanifold of Rl +1++ G Re++ Rv without boundary of
dimen-sion e + v . Hence, is a map between manifolds of the same dimension.
Proof From Claims 3 and 4 and the pre-image theorem, the -pseudo-equilibrium
set, E = [ 2 H (0) 1, is a submanifold (of Rl +1++ G Re++ R(S L
0) J
) without
boundary of dimension (l + 1 + v + e + v ) (1 + l + v ) = e + v .
Claim 6 The map : E ! Re
++ R(S L
0) J
is proper, that is, the inverse image by
of a compact set is compact. Moreover, the set, R , of regular values of is
open and of zero Lebesgue measure complement.
Proof LetY be a compact subset of Re
++ R(S L
0) J
and let a sequence of elements of (Y ) 1, fCk:= (yk; pk; Gk; (ek
i); Vk)gk2N, be given. Since Y is compact, we may
assume that the sequence f(eki); Vk)g converges and denote((e0i); V0) 2 Y its limit.
From the limit relation on Z and the relation E = [ 2 H (0) 1, we may assume
that f(yk; pk)g converges, say to(y; p) 2 R
++ P. From Claim 2, the sequence fGkg
may be assumed to converge, say to G 2 G .
Let C := (y; p; G; (e0
i); V0) := (limk!1 Ck) be given. From Claims 3 and 4 and
Def-inition 2, there exists 2 , such that the relations H (Ck) = 0 hold, for k 2 N big
converges to C 2 E , which makes (Y ) 1 compact. Thus, is proper. Since is
proper, its set of singular values, Rc, is closed, that is, R is open. From Claim 5
and Sard’s theorem (see Milnor, 1997, p. 10), Rc is of zero Lebesgue measure.
Lemma 1 There exists a regular value, (e ; V ), of , such that # (e ; V ) 1= 1.
Proof See the Appendix.
3.3 The existence Theorems
We now state and prove the main existence results.
Theorem 1For every collection of endowments and payo¤s,(e0; V0) 2 Re
++ R(S L
0) J
,
and every 2 RS++, a standard economy, E(e0;V0), admits a -pseudo-equilibrium.
Proof From Lemma 1, there exists at least one -pseudo-equilibrium, so the map
is well de…ned. As standard from modulo 2 degree theory, iff : X ! Y is a smooth
proper map between two boundaryless manifolds of same dimension, with Y being
connected, the number, #f 1(y), of elements x 2 X, such thaty = f (x), is the same,
modulo 2, for every regular value y 2 Y. In particular, if one regular value, y, of f,
is such that #f 1(y) = 1, then, f 1(y) is non-empty for every y 2 Y. Indeed, y 2 Y is
regular by de…nition if f 1(y) = ?. From Claims 2 to 6 and Lemma 1, the map, ,
meets all above condititions for X := E and Y := Re
++ R(S L
0) J
. Hence, for every
(e0; V0) 2 Re
++ R(S L
0) J
, a standard economy, E(e0;V0), has a -pseudo-equilibrium.
Theorem 2 For every = ( s) 2 RS++, there exists an open set of null
comple-ment, R (along Claim 6), such that, for every (e0; V0) 2 , the state prices
Proof Let = ( s) 2 RS++ and (e ; V ) 2 R be given, a non-empty set from Lemma
1. From Claims 3 and 4, Theorem 1, the implicit function theorem and the
de…n-ition of a regular value, there exists a pseudo-equilibrium, C := (y ; p ; G ; e ; V ) 2
(e ; V ) 1, and two open sets, W E and U R , containing C and (e ; V ),
respectively, which are mapped homeomorphically by =W.
We assume w.l.o.g. that G0 2 GId whenever C0 := (y0; p0; G0; e0; V0) 2 W. The price
map,(e0; V0) 2 U 7! f
1(e0; V0) 2 P, and the map,(e0; V0) 2 U 7! f2(e0; V0) 2 GId, de…ned by (f1(e0; V0) e01; f1(e0; V0); f2(e0; V0); e0; V0) 2 W, for (e0; V0) 2 U, are continuous from above.
The map : (y0; p0; G0; e0; V0) 2 W 7! (p0;
Id(G0); e0; V0) is a homeomorphism from
Claim 3 and we letW := (W ) be its image. Following Du¢ e-Shafer (1985), we let:
H : C0 := (p0; E0; e0; V0) 2 W 7! (Z (p0 0e010+
P
s2S sp0se01s; p0; Id1(E0); e0); KId(p0; Id1(E0); V0)).
Di¤erentiating the relationH (C0) = 0, which holds from the de…nition for every
C02 W , yields: [D
(p0;E0)H (C0)] [D (f1; f2)(e0; V0)]+D(e0;V0)H (C0) = 0, for everyC02 W .
The latter equation states that the rows ofD(e0;V0)H (C0)are linear combinations
of those of D (f1; f2)(e0; V0). From the proof of Claim 4, D(e0;V0)H (C0)has full rank,
l + v , and so does, from above, D (f1; f2)(e0; V0). In particular, rank Df1(e0; V0) = l
holds, for every (e0; V0) 2 U. We now de…ne the following maps and set:
: (e0; V0) 2 U 7! (f 1(e0; V0); V0) 2 P R(S L 0) J ; : (p0; V0) 2 P R(S L0) J 7! V0 p0 2 RS J; Q := 1(G).
The setG is (relatively) open, and of null complement from Sard’s theorem. The
so, the maps and are submersions. Since is a submersion and G is open and
of null complement, so is Q := 1(G) inP R(S L0) J. Let (U ) be the image set of
U by . Then, Q0 := Q \ (U)is open and of null complement in (U ), which is open.
By the same token, U := 1(Q0)is open and of null complement inU. From the
above de…nitions, Theorem 1 and Claim 1, for all (e0; V0) 2 U, a standard economy,
E(e0;V0), admits a -pseudo-equilibrium, which yields an equilibrium supported by .
Applying a classical local to global argument, there exists an open subset, ,
of R , with null complement in R , hence, in Re
++ R(S L
0) J
, such that, for every
(e0; V0) 2 , a standard economy,E
(e0;V0), admits an equilibrium supported by .
Moreover, from Lemma 1, for all(e0; V0) 2 , the number of equilibria in (e0; V0) 1
is odd, and each of these equilibria is, from above, a continuous function of(e0; V0).
Appendix
Lemma 1 There exists a regular value, (e ; V ), of , such that # (e ; V ) 1= 1.
Proof We may choose a price,p := (ps) 2 P, and a matrix,V 2 R(S L0) J, such
that the last J rows of Vp form the identity matrix,I 2 RJ J. We letG := < V
p >.
From Assumption A2, we may choose endowments such that rui(ei)=p holds for
each i 2 I. Then, in a standard economy,(p ; G ; (ei); V ) de…nes a tradeless pure
spot market equilibrium by construction, whose allocation, (ei), is Pareto optimal.
For any 2 RS
++, this equilibrium also de…nes a -pseudo-equilibrium.
Let (xi) be a -pseudo-equilibrium allocation. From the de…nition, the relation
from Assumption A3 and above, for n 2 N large enough,([xi
n +
(n 1)ei
n ]) is attainable
and Pareto dominates (ei), in contradiction with above. Hence, # ((ei); V ) 1= 1.
The last part of the proof of Lemma 1 is to show that((ei); V )is a regular value of
. This is equivalent to showing that the derivative,D(y;p;E)HId(y ; p ; Id1(E ); (ei); V ),
has full rank, l + 1 + v , where - following Du¢ e-Shafer (1985, p. 296) - we let
E := Id(G ) and E := Id(G), for everyG 2 GId, and consider the maps:
h (y; p) := p0 e10+ P
s2S s ps e1s y; Z (p; E) = Z (y ; p; Id1(E); e );
K (p; E) = KId(y ; p; Id1(E); V );
H (y; p; E) = ( h (y; p); Z (p; E); K (p; E) );
D H (y ; p ; E ) := 0 B B B B B B @ Dy h (y ; p ) = 1 Dp h (y ; p ) 0 0 Dp Z (p ; E ) DE Z (p ; E ) 0 Dp K (p ; E ) DE K (p ; E ) 1 C C C C C C A .
We showD H (y ; p ; E ), hence,D(y;p;E) HId(y ; p ;
1
Id(E ); (ei); V ) have full rank.
The arguments are similar to Du¢ e-Shafer’s (1985) and recalled for completeness.
We show, …rst, that rank DE K (p ; E ) = v . Denoting by J the set of last J
states, we recall that Vp (J ) = I 2 RJ J is the identity matrix.The derivative of K
with respect toEat theS J rows ofE is the(S J ) (S J )block diagonal matrix,
P, of common block diagonal elementI 2 RJ J. The rank ofP is, therefore,(S J )J.
Then, from above, the matrix DE K (p ; E ) has maximal rank,v = (S J )J.
Second, we show that DE Z (p ; E ) = 0. Let E 2 R(S J ) J be given. By
construc-tion, the gradient’s condiconstruc-tion, rui(ei) = p 2 RL S++ , holds for each i 2 I, making (ei)
Pareto optimal. Since a¤ordable to any agent and optimal, (ei)is the demand
allo-cation. Hence,Z (y ; p ; Id1(E); (ei)) = 0, for allE 2 R(S J ) J, andD
The proof thatDp Z (p ; E )is non-singular, as a last step, relies on Lemmata 1:
Lemmata 1 Let (p; G; i) 2 P G Inf1g be given. The following Assertions hold:
(i) pTD p(D1(y ; p)) = D1(y ; p)T; (ii) hTDp(D1(y ; p)) h < 0, 8h 2 RL S 0 nf0g, such that h D1(y ; p) = 0; (iii) p TDp(Di(p ; G; ei)) = 0;
(iv) Dp(Di(p ; G; ei)) is negative semi-de…nite.
Proof of Lemmata 1Assertion (i)results from di¤erentiating p D1(y ; p) = y .
Assertion (ii) From …rst order conditions, the …rst agent’s gradient and pricep 2 P
are colinear at D1(y ; p). Then, Assertion(ii)is standard from Assumption A3.
Assertion (iii) The satiated budget constraint at theith agent’s demand is written:
p Di(p; G; ei) = p ei = p Di(p ; G; ei). Di¤erentiating this atp=p yields Assertion(iii).
Assertion (iv)Given (p; G) 2 P G , the relationDi(p ; G; ei) = ei holds from above.
The relations p Di(p; G; ei)> p ei and p Di(p; G; ei) = p ei hold from the de…nitions.
Then, (p p ) (Di(p; G; ei) Di(p ; G; ei))6 0 holds for every (p; G) 2 P G . So, the
map Di(p; G; ei) is non-increasing inp 2 P and C1 and Assertion (iv) holds.
We can now complete the proof of Lemma 1. Assume, by contraposition, that
Dp Z (p ; G ) h = 0 holds for some h 6= 0. The vector h belongs to Rl := R(L 1)S
0 ,
from the de…nition of the model’s excess demand, Z , hence, of Z . We let h be
the vector of RLS0
, which coincides withh on Rl and whose other components (in
good l = 1) are all zeros. Then, it follows from the de…nition of Z and above that:
0 = DpZ (p ; G ) h = Dp(D1(y ; p )) h + P
i2Inf1g Dp(Di(p ; G ; ei)) h(forhset above).
0 = p TD
p(D1(y ; p )) h +Pi2Inf1g p TDp(Di(p ; G ; ei)) h = D1(y ; p )T h .
The above relation, D1(y ; p )T h = 0 (for h 6= 0), implies, from Lemmata 1-(ii):
h TDp(D1(y ; p)) h < 0. Then, the relation hTDp Z (p ; G ) h < 0holds from Lemmata
1-(iv) and above and contradicts the fact that Dp Z (p ; G ) h = 0, assumed above.
This contradiction proves thatDpZ (p ; G )and, from above,DH (y ; p ; G )have full
rank, i.e., ((ei); V ) is a regular value of . This completes the proof of Lemma 1.
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