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F U N D A ç Ã O
GETUUO VARGAS
~~
FGV
EPGE
,
SEMINARIOS DE ALMOÇO
DA EPGE
Equilibria in exchange economies with
financiai constraints:Beyond the Cass
Trick
VICTOR FILIPE MARTINS-DA-RoCHA
(Université Paris - Dauphine)
Data: 05/08/2005 (Sexta-feira)
Horário: 12h 15 min
Local:
Praia de Botafogo, 190 - 110 andar
Auditório nO
1
Coordenação:
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EQUILIBRIA IN EXCHANGE ECONOMIES WITH FINANCIAL CONSTRAINTS: BEYOND THE CASS TRICK
V.F. MARTINS-DA-ROCHA AND L. TRIKI
ABSTRACT. We consider an exchange economy under incomplete financiaI markets with purely financiaI securities and finitely many agents. When portfolios are not constrained, Cass [4], Duffie [7] and Florenzano-Gourdel [12] proved that arbitrage-free security prices fully characterize equilibrium security prices. This result is based on a trick initiated by Cass [4] in which one unconstrained agent behaves as if he were in complete markets. This approach is unsatisfactory since it is asymmetric and no more valid when every agent is subject to frictions. We propose a new and symmetric approach to prove that arbitrage-free security prices still fully characterize equilibrium security prices in the more realistic situation where the financiaI market is constrained by convex restrictions, provided that financiaI markets are collectively frictionless.
KEYWORDS: Exchange economies, incomplete financiaI markets, purely financiaI secu-rities, nominaI assets, constrained portfolios, collectively frictionless financiaI markets, equilibriurn security prices, arbitrage-free security prices.
JEL CLASSIFICATION: C62, D52, GlO.
1. INTRODUCTION
We consider an exchange economy under incomplete financiaI markets with purely fi-nancial securities (nominal assets) and finitely many agents. By definition, a security price is arbitrage-free if any (unconstrained) portfolio does not yield a positive non-zero income. For frictionless financiaI markets,Cass [4], Duffie [7] and Florenzano-Gourdel [12] proved that arbitrage-free security prices fully characterize equilibrium security prices, in the sense that each equilibrium security price is arbitrage-free and each arbitrage-free se-curity price can be embedded as an equilibrium sese-curity price. This result comes from the ability of commodity prices to adjust themselves to clear both commodity and financiaI markets. The trick initiated by Cass [4] and exploited by Duffie-Shafer [8], Duffie [7], Magill-Shafer [15], Florenzano-Gourdel [12], Rahi [18], Magill-Quinzii [14] among others is based upon the fact that one agent (this agent needs to have an unconstrained portfolio set) behaves at equilibrium as if he were in complete markets.
2 INCOMPLETE MARKETS WITH FINANCIAL CONSTRAINTS
The requirement of a frictionless financial market is a highly idealized condition. In our model, the financiaI market is subject to frictions in the sense that agents are con-strained by convex restrictions on possible portfolio holdings. Luttmer [13J and EIsinger-Summer [9J illustrate that these constraints on portfolio sets are suitable for describing market frictions such as short-selling constraints and buying fioors, margin and collateral requirements. bid-ask spreads and taxes, and proportional transaction costs.
Technically the approach initiated by Cass is unsatisfactory since it is asymmetric: there must be an agent (called Arrow-Debreu agent) who is unconstrained and who is the only one to face a demand correspondence in complete markets. Moreover the Cass trick is no longer valid in markets with frictions since constrained portfolio sets preclude the presence of an Arrow-Debreu agent. This paper is dedicated to the following two linked questions raised by P. Courrege :
• Is it possible to provide a symmetric approach ?
• Under which conditions on frictions is it possible to fully characterize the set of
equili~rium security prices by arbitrage-free security prices ?
We prove in this paper that the answer to the first question is yes. We propose a new and symmetric proof: an ad hoc commodity price dependent security is artificially introduced in the financiaI market. Each agent is allowed to purchase a small amount of this security. The introduction of this ad hoc security ensures that for any commodity price and at least for one agent, his budget set has a non-empty interior. In particular the aggregate excess demand correspondence explodes if commodity prices hit the boundary of the price simplex. At equilibrium, we prove that no agent actually purchases the ad hoc security.
We also provide in this paper an answer to the second question : it is possible to fully characterize the set of equilibrium security prices by arbitrage-free security prices provided that the financiaI market is collectively frictionless in the sense that for any payoff achieved by an unconstrained portfolio there exists an agent for which this payoff is achieved by a portfolio belonging to his portfolio set. In other words, the union of alI payoff sets covers the whole space of unconstrained payoffs. It is straightforward to check that any equilibrium security price is arbitrage-free provided that the financiaI market is collectively frictionless. The main contribution of our work is to prove that the converse is true. In fact, we prove that any arbitrage-free security price can be embedded in an equilibrium, provided that the financiaI market is locally collectively frictionless in the sense that for any payoff achieved by a portfolio, there exists an agent for which a proportion of this payoff is achieved by a constrained portfolio.
There is an abundant literature on asset pricing with frictions, but only a few studies explore the existence issue: Werner [20, 21J, Siconolfi [19], Balasko-Cass-Siconolfi [1], Benveniste-Ketterer [2J, Polemarchakis-Siconolfi [17J and Cass-Siconolfi-Villanacci [5]. As far as we know, this work is the first one to investigate the validity, in markets with
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V.F. MARTINS-OA-RoCHA ANO L. TRIKI 3
frictions, of the full characterization of equilibrium security prices by arbitrage-free security prices.
The paper is organized as follows. In the next section, a model of an exchange economy with general purely financiaI markets is described and existence results are enunciated. In Section 3 the existence results are expressed in terms of the two usual examples. The last section is devoted to proving the main resulto Some proofs are referred to the appendix.
2. THE GENERAL MODEL
We propose in this section a mo deI of exchange economies with general purely financiaI markets in the sense that uncertainty is represented by a set of possible states of nature without specifying the intertemporal structure. The two-period intertemporal mo deI (see Cass [4], Werner [20, 21], Magill-Shafer [15], Magill-Quinzii [14] and Florenzano [10]) and the multi-period intertemporal model (see Duffie [7], Florenzano-Gourdel [12] and Magill-Quinzii [14]) are presented in Section 3 as special cases of our general mode!. 2.1. Exchange economies with general purely financiaI markets. We consider a
triple
(r:,
I, F) wherer:
and I are finite sets and F is a finite dimensional vector space. Each (j Er:
represents a state (of nature), each i E I represents an agent and each e E Frepresents a portfolio.
A linear operator W from F to ]RI: is called a payoff operator. For each portfolio
e
E F, we denotes the image of e, which as an element of ]RI: is denoted by (We((j), (j Er:),
each We((j) representing the payoff at state (j E
r:.
A portfolio structure is a familye·
=(e
i, i E I) where for each i E I,e
i is a subset ofF: for each i E I, the set
e
i represents the portfolio (restriction) set for agent i and the payoff set wei = {We : e Ee
i} representsthe set of payoffs available for agent i. A financial market is a pair fi = (W,
e·)
whereW is a payoff operator and
e·
is a portfolio structure.A consumption market is a triple (E, X·, u·) where E is a finite dimensional vector space, X· is a family (Xi, i E 1) with Xi subset of EI: and u· is a family (ui, i E I) with
ui real function from Xi to IR. The space E represents the commodity space and the dual
E' the price space. A vector in E represents a consumption bundle for an agent and a vector in EI: a consumption plano A vector in E' represents a spot price and a vector
in (E')I: a commodity price (system). For each agent i E I, the set Xi represents the
consumption set and the function ui the utility function. If x E Xi we denote by pi(x)
the set of strictly preferred consumption plans by agent i E I, i.e.
4 I="COMPLETE MARKETS WITH FINA="CIAL CONSTRAI="TS
Definition 2.1. An (exchange) economy (with purely financial markets) is here a pair
E
=
(EC, E!)=
(E,X·,u·,w,e·) ,
where EC
=
(E,X·,u·) is a consumption market and E!=
(W,e·) is a financiaI marketo Let E = (E, X·,u·,
W, S·) be an economyo For each commodity price p E (E')E and each consumption plan x E EE, we define the vector p O x E ]RE bypOx = ((p(O"), x(O")), O" E I:) E ]RE
where (o, o) : E' x E -> ]R is the natural dualityo The vector pOx represents the values following O" E L of the consumption plan x under the commodity price po Given a com-modity price p E (E')E, we say that a portfolio O E F finances a consumption plan x E EE
if pOx ~ W61, in the sense that
'<lO" E L, (P(O") , x(O")) ~ WO(O")o
A pair (x, O) E EE X F is called a budget feasible plan for agent í if the consumption
plan x belongs to Xi, the portfolio O belongs to
e
i and finances Xo Given a commodityprice p E (E') E, the budget set Bi (p) of agent í is the set of alI budget feasible plans for í, ioeo
Bi(p) = {(x,O) E Xi x
e
i : pOx ~ WO}oA consumption allocation x· = (xi, í E 1) is a family of consumption plans xi E EE o A portfolio aHocation O· = (Oi, í E 1) is a family of portfolios Oi E F o A budget feasible plan
(xi, Oi) for agent í is optimal if there is no other budget feasible plan (y, TI) for í such that
y is strictly preferred to xi, ioeo [pi(xi) x
e
i]n
Bi(p) =00
Definition 2.2. A triple (p, x·, O·) is an equilibrium for the economy E if x· = (xi, í E I)
is a consum pt ion alIocation, O· = (Oi, í E 1) is a portfolio allocation and p is a commodity price such that
(i) for each í E I, (xi, Oi) E Bi(p) and [pi(xi) x
e
i]
n
Bi(p) =0,
(ii) :EiEI xi = O,(iii) :EiEI
íJi
= 00If condition (iii) is replaced by the following condition (iii') :EiEI 'WOi = O,
the family (po x·, O·) is then calIed a weak equilibriumo
Obviously an equilibrium is a weak equilibriumo The folIowing Proposition 201 provides a condition for which the converse is trueo If W is a payoff operator, then we denote by
Ker W the kernel of W, ioeo Ker W
=
{O E F : WO=
O}o If Z is a closed convex subset ofF then As (Z:, denotes the asymptotic cone defined by As (Z) =
{v
E F: Z+
{v}
C Z}oProposition 2.1. Let E be an economyo
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V.F. MARTINS-DA-RoCHA AND L. TRIKI 5
(a) lf the following condition is satisfied
(2.1) KerW C UAs(ei),
iEI
then there exists an equilibrium as soon as there exists a weak equilibrium.
(b) Condition (2.1) is satisfied if eitherKerW = {O} or for every i E 1, the set
e
i2S
a closed convex subset of F containing O such that
U
iE1e
i = F.The proof ofProposition 2.1 is postponed to Appendix A.!. We refer to Won-Hahn [22] where it is shown how redundant assets contribute to risk sharing in a nontrivial manner. 2.2. Assumptions. Consider an economy (E, X·, u·, W, S·). Let X = ITiEI Xi, let
X
be the set of attainable consumption allocations, i.e.X
={X.
= (xi, i E 1) E X:L
xi =O}
iEI
and for each i E 1, let
Xi
be the projection ofX
on Xi. Assumption (C). For every agent i E 1:C.1 the consumption set Xi is closed convex and
Xi
is compact in EL.;C.2 the utility function ui is continuous and quasi-concave; C.3 the vector O belongs to the interior of Xi;
C.4 for every attainable consumption aIlocation x· E
X,
for every (7 E :E, there existsy E Xi, differing from xi only at (7, such that ui(y)
>
ui(xi).Assumption (F). For every agent i E 1, the set
we
i is a closed convex subset of ]RL.containing O.
If e is a subset of F, we let As (e) = {v E F : e
+
vc
e}. Note that if for each i E 1,ei is closed convex containing O, then Assumption F is satisfied if for each i E 1, either
ei is a finitely generated cone or Ker W
n
As (ei) = {O}.
Definition 2.3. A consumption market (E, X·, u·) is said standard if Assumption C is satisfied. A financiaI market (W, e·) is said standard if Assumption F is satisfied . An economy E = (EC, Ef) is said standard if the consumption market EC = (E, X·, u·) and the financiaI market Ef = (W, e·) are standard.
2.3. The existence resulto Before presenting the existence results, we set the definition of collectively frictionless financiaI markets. If A is a subset of F then we denote by cone A
the cone generated by A in the sense that coneA = {>.a : a E A and
>.
~ D}. If Wis a payoff operator from F to ]RL., then Im W denotes the image of W in the sense that Im W = {t E]RL. : 30 E F, t = WO}.
6 I:-;CO~!PLETE :V!ARKETS WITH FINA1'\C!AL CO!'STRAII'TS
(i) fricti.onless if
'Vi E I,
we
i = Im W;(ii) collectively frictionless if
Uwei=ImW;
iEI
(iii) locally collectively frictionless if
cone
U
we
i = Im W.iEI
In other words, a financiaI market (W, ee) is collectively frictionless if for any payoff
t
in Im W, there exists an agent i E I for which
t
beIongs to his payoff setwe
i; and it isIocally collec:ively frictionless if for any payoff t in Im W, there exist an agent k E I and
À
>
O such t,lat Àt beIongs towe
k. Note that if ee is such that UiEIei = F then anyfinanciaI market (W, ee) is collectively frictionless and if ee is such that O beIongs to the interior of Ui~Iei then any financiaI market (W, ee) is localIy collectively frictionless. Definition 2.5. A standard financiaI market EI is said viable if for every standard
con-sumption market [C, there exists a weak equiIibrium for the economy E = (EC, [I).
A vector t = (t(O"), O" E L:) in IR:E is said non-negative, denoted by t ~ O if for each
O" E L:, t(O") ~ O. The set of non-negative vectors is denoted by IR~. A vector t E IR:E is
said positive, denoted by t
>
O, if ti=
O and if it is non-negative.Definition 2.6. A standard financiaI market (W, ee) precludes arbitrage opponunities if
for each i E 1, there is no t E As
(We
i) such that t is positive, i.e.IR~
nUAs
(we
i) =
{O}.
iEI
Proposition 2.2.
lf
a standard financiai market is viabie, then it precludes arbitrage opponunitiesThe proof ·)f Proposition 2.2 is standard and postponed to Appendix A.2.
Definition 2.7. A payoff operator W is said arbitrage-free if there is no t E Im W such
that t is positive, i.e.
IR~ nIm W = {O}.
If W is an arbitrage-free payoff operator then any standard2 financiaI market (W, ee)
precludes arbürage opportunities. Furthermore, if a financiaI market (W, ee) is frictionless and precludes arbitrage opportunities, then the payoff operator W is arbitrage-free. We
prove in the following proposition that this equivaIence is still valid if financiaI markets are collectively frictionless.
2In fact it is 5Ufficient to assume that for each i E I, O E
e
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V.F. MARTINS-DA-RoCHA AND L. TRIKI 7
Proposition 2.3. lf a standard financial market (W, se) is collectively frictionless and precludes arbitrage opportunities, then the payoJJ operator W is arbitrage-free.
The proof of Proposition 2.3 is in Appendix A.3. The relationship between the con-cept of arbitrage-free and viable financiaI markets are made explicit by Theorem 2.1 and Corollary 2.1.
Theorem 2.1. Let (W, se) be a standard and locally collectively frictionless financial
market. lf W is arbitrage-free then (W, ee) is viable.
Section 4 is dedicated to the proof of Theorem 2.1. Combining Propositions 2.2-2.3 and Theorem 2.1, we get a complete description of the set of viable financial markets. Corollary 2.1. Consider a standard financial market that is collectively frictionless, then
the following assertions are equivalent :
• The financial market is viable.
• The financial market precludes arbitrage opportunities. • The payoff operator is arbitrage-free.
3. ApPLICATION TO INTERTEMPORAL MODELS
We first set some notations. If K is a finite then for every x, y in lRK, x . Y =
LkEK x(k)y(k). If L is a finite set and z belongs to lRKxL then for every k E K, we let z(k) = (z(k, f), f E L) E lRL and for every f E L, we let z'(f) = (z(k, f), k E K) E lRK.
3.1. The two-period intertemporal model. The first example of our general model of exchange economies with purely financial markets (Section 2) is the two-period intertem-poral model studied by Cass [4], Werner [20, 21], Magill-Shafer [15], Magill-Quinzii [14] and Florenzano [10]. In the two-period model the triple (I:, l, F) and the payoff operator are specified as follows:
(a) I: = {O}
u
S where S is a finite set not containing O. (b) F=
IRJ where J is a finite set.(c) W = W(q, R) where q E lRJ and R E IRSxJ is defined by
'Ve E lRJ, [We](O)
=
-q. e and 'Vs E S, [We](s)=
R(s) . e.Furthermore, the corresponding consumption market (E, Xe, ue) is the general one adapted
to the specification (a) of I: and the corresponding portfolio structure se is the general one adapted to the specification (b) of F.
This two-period intertemporal exchange economy extends over two time periods t = O and
t
= 1. Uncertainty at the second period is modelled by S. The set J is the set of financiaI nominal assets which are traded only in the first period (t = O) and pay monetary returns in units of account in the second period. Asset j E J yields R( s, j) units of account8 INCOMPLETE MARKETS WITH FINANCIAL CONSTRAINTS
the vector R represents the vector of returns. In this model, given a commodity price
p E (E/f', a portfolio () E ~J finances a consumption plan x E EL if at time t = 0,
(p(O), x(O))
+
q. () ~°
and at time t = 1, for each state s E S,
(P(s),x(s)) ~ R(s), B.
In the usual set-up of the literature the vector of returns R is given. The study concerns vectors of prices q E RJ through the relation between the arbitrage-free condition and the existence of equilibrium.
Definition 3.1. A financiaI market (W(q, R), ee) (resp. an economy (EC, W(q, R), se))
satisfying the specifications (a), (b) and (c) is caIled a two-period intertemporal financial market (resp. a two-period intertemporaI economy).
Definition 3.2. Let (W(q, R), se) be a standard two-period intertemporal financiaI mar-ket. The asset price q E ~.1 is called an equilibrium asset price if (W(q, R), ee) is viable, i.e. for every standard consumption market EC, the two-period intertemporal economy
(EC, W(q,R),ee) has a weak equilibrium.
Definition 3.3. Let (W(q, R), se) be a standard two-period intertemporal financiaI mar-ket. The asset price q E RJ is called arbitrage-free if (W(q, R), se) is arbitrage free.
As usuaIly done in the literature (see [10],[12] and [15]), we can invoke a strict separation theorem to prove that q is arbitrage-free if and only if there exists
>.
E RS such thatq =
I:
sES >'(s)R(s) and for each s E S, >.(s)>
O. If we rephrase CoroIlary 2.1, we geta complete characterization of equilibrium asset prices by means of arbitrage-free asset pnces.
Corollary 3.1. Consider a standard two-period intertemporal financial market which is collectively frictionless, then the asset price is an equilibrium if and only if it is arbitrage-free.
This coroIlary generalizes the results in Cass [4], Werner [20, 21], Magill-Shafer [15] and Florenzano [10] from frictionless financiaI markets to coIlectively frictionless financiaI markets.
3.2. The multi-period intertemporal mode!. The second example of our general mo deI of exchange economies with purely financial markets (Section 2) is the multi-period intertemporal model studied by Duffie [7], Florenzano-Gourdel [12] and Magill-Quinzii [14]. In the multi-period model the triple (2:-,[, F) and the payoff operator are specified as foIlows:
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V.F. :'vIARTINS-OA-RoCHA ANO L. TRIKI 9
(a) I: is an event tree of length T E N \ {O},3 the initial node of I: is denoted by ç and for each node u =1=
ç
at datet,
we denote by u- the uni que node which immediatelyprecedes u at date
t -
1;(b) F is the subspace ofIRExJ, where J is a finite set, defined by 0= (O(u,j), (u,j) E
I: x J) belongs to F if and only if O( u) = (O( u, j), j E J) = O for each terminal node u E I:T.
(c) W = W(S,D) where S E F and D E IRExJ, is defined by
VO E F, Vu E I:, [WO](u) = O(u-) . [S(u)
+
D(u)] - O(u) . S(u).with "O(ç-)" taken to be zero by convention. Furthermore the corresponding consumption market (E, Xe
, ue) is the general one adapted
to the specification (a) of I: and the corresponding portfolio structure
ee
is the general one adapted to the specification (b) of F.This multi-period intertemporal exchange economy extends over T
+
1 time periodst
E {O, 1, ... ,T}.
The set J represents the set of purely financial securities. A securityj is a claim to a dividend process D'(j) = (D(u,j),u E I:) E IRE where D(u,j) E IR represents the dividend paid by the security j at no de u. To each security j is associated the security price process S'(j) = (S(u,j),u E I:) E IRE where S(u,j) represents the price of the security j, ex dividend, at node u. That is, at each no de u, the security pays
its dividend D(u,j) and is then available for trade at the price S(u,j) if u ~ I:T. This
convention implies that D(ç,j) and for every terminal node u E I:T, S(u,j) play no role. For convenience we pose D(ç,j) = O and for every u E I:T, S(u,j) = O. The vectors
D = (D(u,j), (u,j) E I: x J) and S = (S(u,j), (u,j) E I: x J) are called respectively the dividend process and the security price processo
In this model, given a commodity price p E (E')E, a portfolio O E F finances a
con-sumption plan x E EE if at each node u E I:,
(p(u), x(u))
+
O(u) . S(u) :::;; O(u-) . [S(u)+
D(u)].In the usual set-up of the literature the dividend process D is given. The study concerns security price processes S E F through the relation between the arbitrage-free condition
and the existence of equilibrium.
Definition 3.4. A financial market (W(S, D), ee) (resp. an economy (EC, W(S, D), ee))
satisfying the specifications (a), (b) and (c) is called a multi-period intertemporal financial market (resp. a multi-period intertemporal economy).
Definition 3.5. Let (W(S, D), ee) be standard two-period intertemporal financiaI mar-ket. The security price process S E F is called an equilibrium security price process if
3We refer to Duffie [7], Florenzano-Gourdel [12] and Courrege-Lacroix-Matarasso [6] for precise de
10 INCOMPLETE MARKETS WITH FII'A:->CIAL CO:\"STRAIl'TS
(W(S, D), se) is viable, i.e. for every standard consumption market EC, the two-period intertempora. economy (EC, W(S, D), S-) has a weak equilibrium.
Definition 3.6. Let (W(S, D), S-) be standard multi-period intertemporal financiaI mar-ket. The security price process S E F is called an arbitrage-free security price process if
W(S, D) is arbitrage-free.
A usually done in the literature (see [12] and [14]), we can invoke a strict separation theorem to prove that S is arbitrage-free if and only if there exists >. E ]RL: such that for each rJ E 2:, ),(rJ)
>
O and"/rJ E 2: \ 2:T, >.(rJ)S(rJ) =
L
>.(rJ')[S(rJ')+
D(rJ')], u'Eu+where rJ+ der.otes the set of immediate successors of rJ. Corollary 2.1 can be rephrased in terms of equi.ibrium security price processes.
Corollary 3.2. Consider a standard multi-period interlemporal financial market which is
collectively frictionless, then the security price process is an equilibrium if and only if it is arbitrage-free.
This corollary generalizes the results in Duffie [7] and Florenzano-Gourdel [12] from frictionless financiaI markets to collectively frictionless financial markets.
4. PROOF OF THEOREM 2.1
Consider a standard economy E = (EC, E f) = (E,
x- ,
u-,
W,e-)
satisfying coneU
WSi=
Im W.iEI
We recall that for each rJ E 2:, W(rJ) is the linear form on F defined by W(rJ)O = [WO](rJ) ,
for every O E F. Suppose that W is arbitrage-free, then we can invoke a strict separation theorem to prove that there exists >. E ]RL: such that for each rJ E 2:, >.(rJ)
>
O and( 4.1)
L
>.(rJ)W(rJ) =o.
uEL:We denote by Ker>. the vector subspace of all vectors t E ]RL: such that >.. t
=
L
>.(rJ)t(rJ)=
O.uEL:
Note that from (4.1), Im W C Ker >..
The proof wil! be done in three major steps. Afier some notations, we begin by trun-cating the cO:lsumption market in order to get compact consumption sets. The second step is the core of the paper: as usual, but here symmetrically, we modify the right si de of the budget constraints leading to well behaved demand correspondences. The third and
FUl'\2' (' ',(" r:'7TUUOVARGAS BlBLiOTEU. [~iii.hlO HENRIQUE SIMONS.t:N
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V.F. :'vIARTINS-OA-RoCHA ANO L. TRIKI 11
last step consists in proving the existenee of a weak equilibrium by applying a fixed-point theorem in only (p, x-) sinee the arbitrage-free eondition allows here to find endogenously
e-.
4.1. Notations. We endow the finite dimensional spaee E with a norm /1./1. The dual norm on E' is also denoted by
11.11.
If (H,11.11)
is a normed veetor spaee (for instanee(E,II·II),
(E',11·11)
or (]R,1·1))
then the closed ball of radius r>
O on H with center O is denoted B(H, r). The spaee HL. is endowed with the counting norm still denoted 11./1 and defined byVh
= (h(u),u
E ~) E HL.,Ilhll
= LIlh(u)ll·
(TE L.
We denote (.,.) the duality on ((E')L.,EL.) defined by
V(p,x) E (E')L. x EL., (P,x) = L(p(u),x(u)).
(TE L.
A vector z = (z(u), u E ~) in ]RL. is said non-negative, denoted by z ~ O iffor eaeh u E ~, z(u) ~ O; z is said positive, denoted by z
>
O if z '" O and if it is non-negative; z is said strictly positive, denoted by z»
O if for eaeh u E ~, z(u)>
O. We denote by ]R~ (resp. ]R~+) the set of alI z E]RL. satisfying z ~ O (resp. z » O). We reeall that for every (p,x) in ]RL. x ]RL., we denote by p . x = L(TEL. p( u )x( u). If H is a veetor subspaee of ]R L. , then H.l denotes the vector subspaee of alI veetors x E ]RL. sueh that for every y in H, X· Y = O.4.2. Truncating the consumption market. The following lemma establishes that in order to prove Theorern 2.1, we ean suppose without any 1088 of generality that
eonsump-tion sets are compacto
Definition 4.1. For any r
>
O, we let t~ be the truncated consumption market definedby
t;
= (E, X; , u;)where for each i E 1, X: = Xi
n
B(EL., r) and u~ is the restrietion of ui to X:.Lemma 4.1. Let
t
c be a standard consumption market.(a) For every r
>
O, the truncated consumption market t~ is standard.(b) There exists r
>
O such that(4.2) Vi E 1,
Xi
C intB(EL.,r).(e) lf r
>
O is such that (4.2) is satisfied then for each standard financial market tI,any weak equilibrium of the economy (t~, tI) is a weak equilibrium of the economy
12 INCOMPLETE MARKETS WITH FI:-;'ANCIAL CONSTRAINTS
The proof of this lemma is in Appendix AA. Following Lemma 4.1, we can suppose
without any loss of generality that for each i E I, the set Xi is compacto Let 7í be the mapping from (E')~ to (E')~ defined by
ip
=
(p(O") , O" E ~) E (E')~, 7í(p)=
().(O")p(O") , O" E ~) E (E')~.Since ). E R~+, the mapping 7í is bijective. We restrict the commodity prices in the set TI
defined by
TI := {p E (E')~: 117í(p)
II :::;;
I}.
4.3. Modified budget sets: the symmetric approach. Let
A
be the space of contin-uous mappings from TI to IR~. If ex belongs to A, then for each p E TI, ex(p) denotes thetuple (ex(p. 0"), O" E ~) where ex(p,O") E IR. For each p E TI, we let ,(p) = (r(p, 0"), O" E ~)
be the vector in IR~ defined by ,(p, 0") = 1-117í(p)1I for each O" E ~. For each ex E A, i E I,
p E TI, let B~(p), ~(p) and d~(p) be the sets defined by
B~(p) := {x E Xi : 30 E ei, 37 E [0,1], pOx:::;; WO
+
ex(p)7+
,(p)},{3~(p) := {x E Xi : 30 E ei, 37 E [0,1], pOx«
wo
+
ex(p)7+
,(p)},~Ck(P)
=
{x E Xi : x E B~(p) and B~(p)n
pi(x)=
0}.It is to be noticed how the right side of budget constraints in B~(p) and {3~(p) includes the sum of two terms ex(p)7 and ,(p). The first one completes the basic term WO and the second one is the usual term introduced by Bergstrom [3] to deal with possibly non monotone preferences. We provide hereafter the properties of the correspondences defined above which allows the application of Kakutani's fixed-point theorem, and this for each ex E A. Then ex will be specified in order that the corresponding fixed-point be a weak equilibrium.
Lemma 4.2. For each ex E A, i E I,
(i) the correspondence B~ is upper semicontinuous on TI with compact convex values,
(ii) the correspondence B~ is lower semicontinuous on int TI,
(iii) the correspondence ~Ck is upper semicontinuous on int TI with non-empty compact convex values.
The proof of Lemma 4.2 is in Appendix A.5.
4.4. Applica.tions of Kakutani's fixed-point theorem. For each integer n ~ 1, let TIn be the compact convex subset of int TI defined by
TIn :=
{p
E (E')~: 117í(p)lI:::;; 1 - l/n}•
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V.F. MARTINS-DA-RoCHA AND L. TRIKI
and we let4 Fn be the correspondence from lIn X X to lIn X X defined by
Fn(P, X·) := <Pn(x·) X
II
cfQ(p)iEI
13
where <Pn(x·) := {p E lIn : 'Vp' E lIn, (7i(p') , LiEI xi) ~ (7i(p) , LiEI Xi)}. The cor-respondence <Pn has a closed graph. From Lemma 4.2 the correspondence Fn is upper semicontinuous with non-empty convex compact values. AppIying Kakutani's fixed-point theorem, there exists5 (Pn, x~) in lIn X X such that:
(4.3)
iEI iEI
and (4.4)
Since II x X is compact, passing to a subsequence if necessary, we can suppose that the sequence (Pn, x~)n converges6 to (p, x·) in II x X which satisfies the following properties. Claim 4.1. For each a E A,
(4.5)
iEI iEI
and for each i E f,
(4.6) xi E B~(jj) i.e. 3(Õi , fi) E ai x [0,1], pOxi ~ WÕi
+
a(p)fi+
,(p)and
(4.7)
Proof. Passing to the Iimit in (4.3) we get (4.5). Property (4.6) follows from (4.4) and
the upper semicontinuity of B~ on lI. Let us now prove (4.7). From (4.4), for each i E f,
n E N, !3~(Pn)
n
pi(x~) =0.
The correspondences ~ : II ---. Xi and pi : Xi ---. Xi haveopen graphs. It follows that !3~(p)
n
pi(xi) =0.
Now if ~(p)'1= 0
then B~(p) is the closure of !3~(p), and since pi(xi) is open, we have B~(p)n
pi(xi) =0.
OIf we Iet (Im W).l := {8 E]RE : 8· t = 0, 'Vt E Im W} then ]RE = Im W
+
(Im W).l.Definition 4.2. We Iet
• AÀ be the subset of alI mappings a E A satisfying
'Vp E lI, a(p) E Ker À. 4Note that we should write Fo.n.
14 INCO!v!PLETE MARKETS WITH F!NANCIAL CONSTRAINTS
• Ai3
be the subset of alI mappings a EA
satisfying'ip E II,
U
(3~(p)1=
0.
iEI
• A.l be the subset of alI mappings a E A satisfying
'ip E II, a(p) E (Im W).l.
Claim 4.2. The following properties are satisfied.
(i) lf a
E:
A'~ then x· EX,
i. e. LiEI xi = O.(ii) lf a E
A
Àn
Ai3
then for every O" E 1:, p(O")1=
O and for every i E 1, xi E cPQ(p).(iii) lf a E:
A
ÀnA.B
nA.l then "((p) = 0, LiEIwe
i = O and for every i E 1, a(p)fi = O.Proof of part (i). Suppose that LiEI xi
1=
O. It folIows from (4.5) that(4.8)
117r(P)!!
= 1 and (7r(p) ,L
xi)>
O.iEI
Hence "((P) = O and premultiplying by >.(0") the budget inequality (4.6) at state O" E 1:, and summing among 0", we get
(7r(p),Xi) =
>..
(pOxi ) :::;;>..
(Wei)+
[>..
a(p)Jfi.Since Im W
c:
Ker>.,
we have>..
(Wei ) = O, and since a(p) belongs to Ker>.,
(7r(p) , xi) :::;; O. Summing among i,(7r(fi),
LiEI xi) :::;; O, which yields a contradiction with (4.8). OProof of part (ii). Since a E
Ai3
there exists an agentk
E 1 for which (3~(p)1=
0.
From (4.7) the vector xk belongs to d~(fi), i.e. B~(fi)
n
pk(xk) =0.
From Assump-tion CA we get that p(O")1=
O for each O" E 1:. Therefore, in view of Assumption C.3, we ded uce tha t for every i E 1, (3~ (P)1=
O. Once again from (4.7), Xi belongs tocP
Q (p), forevery i E 1. O
Proof of part (iii). From Claim 4.2(ii), for every i E 1, xi E d~(fi). Applying
Assump-tion CA, we get that
Premultiplying by >.(0") the budget inequality at state 0", and summing among O" E 1:, we get
(7r(fi) ,
xi)=
>. .
(Wei)+
[>. .
a(p)Jfi+
li>'!!
(1 - !!fi!!).Since Im l-V' C Ker
>.,
we have>. .
(Wei) = O. Since a(p) belongs to Ker>.,
we get that(7r(p). xi)
= !>.!!
(1 - !!p!!). Summing among i, we get "((p)=
O. It folIows that for eachi E 1,
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V.F. MARTINS-DA-RoCHA AND L. TRIKI 15
Summing among i, we get
iEI iEI
Since Q E A.l, it follows that for every i E f, Q(p)fi = O.
o
It follows from CIaim 4.2 that for every Q E A-ÀnA.BnA.l,
(p,
x-, Õ-) is a weak equiIibriumof E. The proof is completed by the following Iemma. Lemma 4.3. The set A-À
n
A.Bn
A.l is non-empty.The proof of Lemma 4.3 based on the fact that the financiaI market is Iocally collectively frictionless, is postponed to Appendix A.6.
ApPENDIX A.
A.l. Proof of Proposition 2.L
Proof of pari (a) of Proposition 2.1. Let (p, x-,
e-)
be a weak equilibrium. It follows that~iEI
e
i beIongs to Ker W, and thus from (2.1) there exists and agent k E f such that - ~iEIe
i beIongs to As (ek). Consider the portfolio allocation "l- = ("li, i E f) defined byTli = ei if i
=I
k and Tlk = ()k - ~iEI ei. For each i E f, Tli beIongs to Si and it is now routine to check that (p,x-,
TI-) is an equilibrium. O Remark A.1. In Proposition 2.1, the condition (2.1) can be repIaced by the weakercon-dition
iEI iEI
Claim A.L ff for every i E f, the set
e
i is a closed convex subset of F containing O theniEI iEI
Proof. Since O belongs to e i , the cone As (ei) is a subset of e i . Hence if UiEIAs (ei) = F
then UiE1ei
=
F. Suppose now that UiElei=
F and let TI in F. For each k E N, kTlbelongs to Uiei , thus there exists i E f and an increasing sequence (kn)n of integers such that knTl belongs to e i for each n E N. Hence, for every () in ai
1 1 ·
k
n (knTl)
+
(1 - k)e Ee
t .
Now since e i is closed, passing to the limit we get
e
+
TI belongs to e i , Le. TI belongs toAs (ei
). O
16 !:-;COMPLETE MARKETS WITH FINANCIAL CONSTRAINTS
A.2. Proof of Proposition 2.2.
Proof. Let E = (EC,Ef) be a standard economy. Let (p,x·,O·) be a weak equilibrium and
suppose that Ef does not preclude arbitrage opportunities. Then there exists i E l and t E As (Wei ) such that t
>
O. It fo11ows that WOi+
t Ewe
i , i.e. there exists zi Ee
isuch that Wz i
>
WOi. But Xi satisfies the budget constraint pOxi ~ WOi , it fo11ows thatpOxi
<
Wz i . The economy E satisfies Assumption C.4, hence there exists y in pi(X) such that p O y ~ W zi. This is in contradiction with the optimality of (xi, Oi). OA.3. Proof of Proposition 2.3.
Proof. Let Ef = (w,
e·)
be a standard financiaI market that is co11ectively frictionless.Assume that W is not arbitrage-free, then there exists t E Im W such that t
>
O. For each k E N, kt belongs to Im W. Since the financial market is co11ectively frictionless,kt E Ui
we
i. Thus there exists i E l and an increasing sequence (kn)n of integers such that knt belongs towe
i for each n E N. Now let ti Ewe
i , then1 . 1 .
(1- k
n
W +
kn knt Ewe
t .
Now passing to the limit we get ti
+
t belongs towe
i, which means that t belongs toAs (Wei
). This implies that Ef does not preclude arbitrage opportunities. O
A.4. Proof of Lemma 4.1.
Proof. Let EC be a standard consumption market. Part (a) is straightforward and part (b)
fo11ows from the compactness of
Xi
for each i E l. We now prove part (c). Let E =(EC, Ef) be a standard economy and let
(p,
x·,
Õ·) be a weak equilibrium of Er = (E;, Ef).Suppose that it is not a weak equilibrium of E = (EC, Ef). Then for some i, there exists
(Xi, Oi) E X i :<
e
i such that ui(xi)>
ui(xi) and (xi,Oi) belongs to Bi(P). Recall that for each i E l, Xi belongs to int B (EE , r). Then it is easy to find°
<
.Ã ~ 1 such thatxi
+
.Ã(xi - xi) EX:
and satisfies the same budget constraints. From Assumption C.2, we also have
ui(xi
+
.Ã(xi _ xi))>
ui(xi),which yields a contradiction. A.5. Proof af Lemma 4.2.
O
Proo] o] pari (i). Let o: E A, i E l and (xn, Pn) be a sequence in Xi x II converging to
(x,p) E Xi X II and such that xn E B~(Pn). For each n E N, there exists On E
e
i andTn E
[O,
1] such that(A.I)
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V.F. MARTIN5-DA-ROCHA AND L. TRIKI 17
Passing to a subsequence if necessary, we can suppose that there exists r E [0,1] such that the sequence (rn)n converges to r. For each n E N, we let tn = WOn E Im W.
If the sequence (tn) is not bounded then passing to a subsequence if necessary, we can suppose that limn Iltnll =
+00.
Multiplying (A. 1) by I/lltnll and passing to the limit, there exists v E lR~n
Im W with Ilvll = 1. This contradicts the fact that W is arbitrage-free.It follows that the sequence (tn ) is bounded, then passing to a subsequence if necessary,
we can suppose that there exists t E lR!: such that (tn) converges to t. Since the financial market is standard,
we
i is closed and tEwe
i. In particular x belongs to B~(p). OProof of part (ii). For every p E int II, the set J3~(p) is non-empty (take x = O, r = O and
O such that WO = O) and then B~(p) is the closure of J3~(p). Since the correspondence ~ has an open graph on int II, it is lower semicontinuous and then B~ is lower semicontinuous
on intIl. O
Proof of part (iii). Note that tfÍQ(p) is the argmax of ui on B~(p). Since ui is continuous
and B~ is continuous on int II, it follows from the Berge's Maximum Theorem [3] that tfÍQ
is upper semicontinuous on int
n
with non-empty values. The convexity of d~ (p) follows from the quasi-concavity of ui. O A.6. Proof of Lemma 4.3.Proof. Since for each i E I, O belongs to the interior of Xi, there exists r
>
O such thatU := B(E!:,r) satisfies for each i E I, U C Xi. Let ~ be the set of alI vectors 8 E lR!:
such that
8
E Ker Àn
(Im W).l and11811
~ 1. Let the correspondence f from II to ~ be defined byf(p):={8E~: 3uEU, 3vEF, pOu-,(P)«Wv+8}
It is straightforward to check that the correspondence f is lower semicontinuous with convex values. In order to apply a continuous selection result (Florenzano [11, Proposi-tion 1.5.3, p.3I]), we prove that for every p in II, f(p) is non-empty. Let p E II, if ,(p)
>
O then O belongs to f(p) (take u=
O and v=
O). Suppose now that ,(p) = O, then there exists x E U such that p O x<
O. We can thus find a vector t in Ker À such that p O x«
t.Since lR!: = Im W
+
(Im W).l, it follows that there exist v E F and 8 in (Im W).l such thatt = Wv
+
8. Note that 8 belongs also to Ker À. Moreover we have pOx« t = Wv+
8.For
1/
>
O small enough, 1/X belongs to U,111/811
~ 1 and thus1/8
belongs to f(p). ApplyingProposition 1.5.3 in Florenzano [11], there exists a a continuous selection of f. Following
the definition of ~, the mapping a belongs to A>'
n
A.l.We assert that a E AI3. Indeed, let p E II and assume that ,(p)
>
O then for each i E I,Xi C J3~(p) (take r
=
O and take O Ee
i such that WO = O). Assume now that ,(p)=
O,since a(p) belongs to f(p), there exists u E U and v E F such that pOu
«
Wv+
a(p).•
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V.F. MARTIN5-DA-ROCHA AI>D L. TRlKI 19
CEREMADE, UNIVERSITÉ PARIS-DAUPHINE, PLACE OU MARÉCHAL DE LATTRE DE TASSIGNY, 75775
PARIS CEDEX 16, FRANCE
E-mail address:martinsClceremade.dauphine.fr
CERMSEM, UNIVERSITÉ PARIs-I PANTHÉON SORBONNE, 106-112 BOULEVARD DE L'HôPITAL, 75647
PARIS CEDEX 13, FRANCE
- -;---::---,.-- . ... ,.,...
..
".......
FUNDAÇÃO GETULIO VARGAS
BIBLIOTECA
ESTE VOLUME DEVE SER DEVOLVIDO A BIBLIOTECA
NA ÚLTIMA DATA MARCADA
BIBLIOTECA
M \R!Q !-TNRIQUE SIMONSEN
: : : . ' > > , '~'.' ':'.:,:. "~'_;"" V,:\,FiGAS
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56
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N.CIwD. P/EPGE SA M386e
Autor: Martins-da-Rocha, V. F.
Título: Equilibria in exchange economies with financiai
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:~69184
N" Pat.:369184
000369184
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