Cass existence approach of financial equilibria when portfolios are constrained1 Bernard Cornet2 and, Ramu Gopalan3
Abstract It is well known that equilibrium asset prices will not offer arbitrage opportunities to individuals. Using an approach that dates back to Cass (1984) [4], we seek to isolate arbitrage free asset prices that are also equilibrium asset prices. However we do this when each agent’s portfolio choice is restricted to a closed, convex set containing zero (as in Siconolfi [23]). In the presence of such portfolio restrictions we need to confine our attention to strong-arbitrage-free asset prices. We also describe a considerably weak condition on the space of income transfers that ensure these asset prices to be part of a financial equilibrium. Staying in the Cass [4] framework, we consider an exchange economy with nominal assets, but allow for multiple (finite) periods and very general preference relations. Moreover we show the existence theorem using the approach of Martins Da-Rocha and Triki [6], and hence do not resort to the Cass trick.
Keywords: Exchange Economies, incomplete markets, financial equilibrium, constrained portfolios, multiperiod models, arbitrage free asset prices. JEL Classification: C62, D52, D53, G11, G12.
1
Current version: November 2008 Paris School of Economics, Paris France and University of Kansas, Lawrence, KS 3 University of Texas at Dallas, Dallas, TX (
[email protected]) 2
1
Contents 1
Introduction
3
2
The T-period financial exchange economy
5
2.1
Time and uncertainty in a multiperiod model . . . . . . . . . . . . . . . . .
5
2.2
The stochastic exchange economy . . . . . . . . . . . . . . . . . . . . . . . .
6
2.3
The financial structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4
Financial equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.5
Arbitrage and equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3
Existence of equilibrium
10
3.1
The main existence result
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.2
Quasi-equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
4
Existence of a weak equilibrium
13
5
Proof of existence under additional assumptions.
14
5.1
Additional assumptions (K): . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
5.2
Preliminary lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
5.2.1
The fixed point argument . . . . . . . . . . . . . . . . . . . . . . . .
15
5.2.2
Properties of (¯ x, w, ¯ p¯, q¯). . . . . . . . . . . . . . . . . . . . . . . . . .
18
5.2.3
Limit argument when ε = equilibrium.
5.3
6
1 n
converges to zero and quasi-weak-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Proof in the general case (without additional assumptions) . . . . . . . . .
22
5.3.1
Enlarging the preferences as in Gale and Mas-Colell . . . . . . . . .
22
5.3.2
Truncating the economy . . . . . . . . . . . . . . . . . . . . . . . . .
24
Appendix: Proof of lemma 5.2
26
2
1
Introduction
Investors facing restrictions on the portfolios that they can trade, is more of a norm than an exception. We consider a model in which investors’ portfolio sets are constrained. As in Balasko, Cass and Siconolfi [2] these constraints are exogenously given (probably arising due to some institutional reasons). Moreover, we consider very general restrictions on portfolio sets as in Siconolfi [23], where each agent’s portfolio set is assumed to be closed, convex and contains zero. This paper primarily examines the existence of a financial equilibrium in a multiperiod model when investors face such general portfolio restrictions. In two date (one period) models without restrictions on portfolio sets, the existence issue has been extensively studied. Cass ([4]) and Werner ([25], [26]) showed existence with nominal assets. Duffie and Shafer ([9]) showed a generic existence result with real assets. This second approach has been extensively used. Magill and Shafer [15] provide a good survey of financial markets equilibria and contingent markets equilibria. Another approach to prove existence in a differentiable economy is to show existence in a numeraire asset economy and infer the existence in the nominal asset economy (See Villanacci et al. Villanacci et al and Magill and Quinzii [16]). Multiperiod models are better equipped to capture the evolution of time and uncertainty and are a necessary step before studying infinite horizon models. Following Debreu’s [7] pioneering model we consider an event-tree to represent the evolution of time and uncertainty. Magill and Quinzii ([16]) and Angeloni and Cornet ([1]) are great references for the treatment of multiperiod financial models. Each node in the event tree represents a date event. Given information on asset prices and spot prices at all date events, consumers will choose a consumption and a portfolio of assets (assumed to be constrained here), such that the node specific value of the consumption does not exceed the node specific value of their endowments and the net returns from the portfolio. In the absence of such portfolio restrictions the notion of absence of arbitrage is clear if there is no portfolio that yields nonnegative net returns in all nodes and strictly positive returns in some node. However in the case where all agents face restrictions in their asset market participation, the notion of arbitrage and its absence at the individual level may differ from that at the aggregate level. Angeloni and Cornet (2006)[1] make this distinction. Given asset prices, an agent does not have arbitrage opportunities if she cannot find a portfolio within her constrained portfolio set that yields nonnegative net returns in all nodes and strictly positive returns in some node. On the other hand, there are no arbitrage opportunities in the aggregate, if there is no portfolio in the set of pooled 3
portfolio sets of all agents that yields nonnegative net returns in all nodes and strictly positive returns in some node. At an equilibrium there must be no arbitrage at the individual level. A natural question then is will any asset price at which there is no arbitrage be an equilibrium asset price. The objective of this paper is to explore this characterization under general portfolio constraints. In the absence of portfolio constraints, Cass ([4]), Duffie ([8]) and Florenzano and Gourdel ([10]) show this characterization of equilibrium and arbitrage free asset prices. In the presence of such constraints, the approach initiated by Cass ([4]), where one agent has an unconstrained portfolio set, facilitates the existence proof. This approach has been extensively used to show existence ever since, Magill and Shafer ([15]), Florenzano and Gourdel ([10]), Magill and Quinzii ([16]), Angeloni and Cornet ([1]) among others. This approach of Cass ([4]), breaks the symmetry of the problem and hence it is not possible to give a symmetric existence (symmetric with respect to the agents’ problem). More recently in a working paper, with such general portfolio restrictions, Da-Rocha and Triki ([6]) have been able to show the characterization between equilibrium and arbitrage free asset prices without the use of the Cass approach. In this paper we explore this characterization issue by showing that any strong-arbitragefree asset price (where the span of the aggregate payoff space does not intersect the interior of the positive quadrant) can be supported as an equilibrium asset price. The approach here is similar to that in Da-Rocha and Triki ([6]), however the notion of absence of arbitrage and the assumptions on the set of income transfers are weaker that those in Da-Rocha and Triki ([6]). With Cass [4] as a motivation, the condition we require in this paper is that the cone generated by the aggregate income transfer possibilities be contained in the cone generated by the union of individual income transfer possibilities. In the Cass approach, the unconstrained agent behaves like in an Arrow-Debreu economy and is able to accommodate the equilibrium excess demand for assets. The attainable consumption and asset allocation are then bounded. We observe that if the set of attainable income transfers is bounded then we can guarantee the existence of a weak equilibrium, which differs from an equilibrium only in the requirement that instead of asset market clearing, there is accounts clearing in the asset markets. The notion of weak equilibrium is useful when redundant assets exist. Section 2 describes the T-period model and the notion of a financial equilibrium. Section 3, states the main result and discusses the notion of a weak equilibrium and its existence. Section 5 gives a detailed proof of the central result in this paper. 4
2
The T-period financial exchange economy
2.1
Time and uncertainty in a multiperiod model
We4 consider a multiperiod exchange economy with (T + 1) dates, t ∈ T := {0, . . . , T}, and a finite set of agents I = {1, ..., I}. The stochastic structure of the model is described by a finite event-tree D of length T and we shall essentially use the same model as Angeloni and Cornet [1], (we refer to [16] for an equivalent presentation with information partitions). The set Dt denotes the nodes (also called date-events) that could occur at date t and the family (Dt )t∈T defines a partition of the set D; for each ξ ∈ D we denote by t(ξ) the unique t ∈ T such that ξ ∈ Dt . Also we denote the cardinality of the set D by D. At each date t 6= T, there is an a priori uncertainty about which node will prevail in the next date. There is a unique non-stochastic event occurring at date t = 0, which is denoted ξ0 , (or simply 0) so D0 = {ξ0 } . Finally, every ξ 6= ξ0 in the event-tree D has a unique predecessor denoted pr(ξ) in D. The predecessor mapping pr : D \ {ξ0 } −→ D satisfies pr(Dt ) = Dt−1 , for every t 6= 0. The element pr(ξ) is called the immediate predecessor of ξ and is also denoted ξ − . For each ξ ∈ D, we let ξ + = {ξ¯ ∈ D : ξ = ξ¯− } be the set of immediate successors of ξ; we notice that the set ξ + is nonempty if and only if ξ ∈ D \ DT . S −1 Dt we define, by induction, prτ (ξ) = Moreover, for τ ∈ T \ {0} and ξ ∈ D \ τt=0 pr(prτ −1 (ξ)) and we let the set of (not necessarily immediate) successors and the set of predecessors of ξ be respectively defined by D+ (ξ) = {ξ 0 ∈ D : ∃τ ∈ T \ {0} | ξ = prτ (ξ 0 )}, D− (ξ) = {ξ 0 ∈ D : ∃τ ∈ T \ {0} | ξ 0 = prτ (ξ)}. If ξ 0 ∈ D+ (ξ) [resp. ξ 0 ∈ D+ (ξ) ∪ {ξ}], we shall also use the notation ξ 0 > ξ [resp. ξ 0 ≥ ξ]. We notice that D+ (ξ) is nonempty if and only if ξ 6∈ DT and D− (ξ) is nonempty if and only if ξ 6= ξ0 . Moreover, one has ξ 0 ∈ D+ (ξ) if and only if ξ ∈ D− (ξ 0 ) and similarly ξ 0 ∈ ξ + if and only if ξ = (ξ 0 )− . 4
In this paper, we shall use the following notations. A (D × J)−matrix A is an element of RD×J , with
entries (a(ξ, j))ξ∈D,j∈J ; we denote by A(ξ) ∈ RJ the ξ−th row of A and by A(j) ∈ RD the j−th column of A.We recall that the transpose of A is the unique (J × D)−matrix t A satisfying (Ax) •D y = x •J (t Ay), for every x ∈ RJ , y ∈ RD , where •D [resp. •J ] denotes the usual scalar product in RD [resp. RJ ]. We shall e ⊂ D and e e ×e denote by rankA the rank of the matrix A. For every subsets D J ⊂ J, the (D J)−sub-matrix e ×e e ×e e with entries e of A is the (D J)−matrix A a(ξ, j) = a(ξ, j) for every (ξ, j) ∈ D J. Let x, y be in Rn ; we shall use the notation x ≥ y (resp. x y) if xh ≥ yh (resp. xh yh ) for every h = 1, . . . , n and we let n n n Rn + = {x ∈ R : x ≥ 0}, R++ = {x ∈ R : x 0}. We shall also use the notation x > y if x ≥ y and
x 6= y. We shall denote by k · k the Euclidean norm in the different Euclidean spaces used in this paper and the closed ball centered at x ∈ RL of radius r > 0 is denoted BL (x, r) := {y ∈ RL : ky − xk ≤ r}.
5
2.2
The stochastic exchange economy
At each node ξ ∈ D, there is a spot market where a finite set H = {1, ..., H} of divisible physical goods is available. We assume that each good does not last for more than one period. In this model, a commodity is a couple (h, ξ) of a physical good h ∈ H and a node ξ ∈ D at which it will be available, so the commodity space is RL , where L = H × D. An element x in RL is called a consumption, that is x = (x(ξ))ξ∈D ∈ RL , where x(ξ) = (x(h, ξ))h∈H ∈ RH , for every ξ ∈ D. We denote by p = (p(ξ))ξ∈D ∈ RL the vector of spot prices and p(ξ) = (p(h, ξ))h∈H ∈ RH is called the spot price at node ξ. The spot price p(h, ξ) is the price paid, at date t(ξ), for the delivery of one unit of the physical good h at node ξ. Thus the value of the consumption x(ξ) at node ξ ∈ D (evaluated in unit of account of node ξ) is X p(ξ) •H x(ξ) = p(h, ξ)x(h, ξ). h∈H
Each agent i ∈ I is endowed with a consumption set Xi ⊂ RL which is the set of her Q possible consumptions. An allocation is an element x ∈ i∈I Xi , and we denote by xi the consumption of agent i, that is the projection of x onto Xi . The tastes of each consumer i ∈ I are represented by a strict preference corresponQ dence Pi : j∈I X j −→ Xi , where Pi (x) defines the set of consumptions that are strictly preferred by i to xi , that is, given the consumptions xj for the other consumers j 6= i. Thus Pi represents the tastes of consumer i but also her behavior under time and uncertainty, in particular her impatience and her attitude towards risk. If consumers’preferences are represented by utility functions ui : Xi −→ R, for every i ∈ I, the strict preference correspondence is defined by Pi (x) = {¯ xi ∈ Xi | ui (¯ xi ) > ui (xi )}. Finally, at each node ξ ∈ D, every consumer i ∈ I has a node-endowment ei (ξ) ∈ RH (contingent to the fact that ξ prevails) and we denote by ei = (ei (ξ))ξ∈D ∈ RL her endowment vector across the different nodes. The exchange economy E can thus be summarized by E = [D; H; I; (Xi , Pi , ei )i∈I ].
2.3
The financial structure
We consider finitely many financial assets and we denote by J = {1, ..., J} the set of assets. An asset j ∈ J is a contract, which is issued at a given and unique node in D, denoted by ξ(j) and called the emission node of j. Each asset j is bought (or sold) at its emission node ξ(j) and only yields payoffs at the successor nodes ξ 0 of ξ(j), that is, for ξ 0 > ξ(j). We denote by v(ξ, j) the payoff of asset j at node ξ. Since we consider only 6
nominal assets this payoff does not depend on the spot prices. For the sake of convenient notations, we shall in fact consider the payoff of asset j at every node ξ ∈ D and assume that it is zero if ξ is not a successor of the emission node ξ(j). Formally, we assume that v(ξ, j) = 0 if ξ 6∈ D+ (ξ(j)). With the above convention, we notice that every asset has a zero payoff at the initial node, that is v(ξ0 , j) = 0 for every j ∈ J . Furthermore, every asset j which is emitted at the terminal date T has a zero payoff, that is, if ξ(j) ∈ DT , v(ξ, j) = 0 for every ξ ∈ D. For every consumer i ∈ I, if zij > 0 [resp. zij < 0], then |zij | will denote the quantity of asset j ∈ J bought [resp. sold] by agent i at the emission node ξ(j). The vector zi = (zij )j∈J ∈ RJ is called the portfolio of agent i. We assume that each consumer i ∈ I is endowed with a portfolio set Zi ⊂ RJ , which represents the set of portfolios that are admissible for agent i. The price of asset j is denoted by qj and we recall that it is paid at its emission node ξ(j). We let q = (qj )j∈J ∈ RJ be the asset price (vector). To summarize, the financial asset structure F = (J , (ξ(j), V j )j∈J , (Zi )i∈I ) consists of - a set of assets J , - each asset j ∈ J is defined by a node of issue ξ(j) ∈ D and the vector of payoffs across all nodes V j = (v(ξ, j))ξ∈D ∈ RD , - the portfolio set Zi ⊂ RJ for every agent i ∈ I. The payoff matrix of F is the D × J matrix V = (v(ξ, j))ξ∈D,j∈J , and it satisfies the condition v(ξ, j) = 0 if ξ 6∈ D+ (ξ(j)) as explained above. The full payoff matrix WF (q) is the (D × J)−matrix with entries wF (q)(ξ, j) := v(ξ, j) − δξ,ξ(j) qj , where δξ,ξ0 = 1 if ξ = ξ 0 and δξ,ξ0 = 0 otherwise. So, for a given portfolio z ∈ RJ (and asset price q) the full flow of payoffs is WF (q)z and the (full) financial payoff at node ξ is [WF (q)z](ξ) := WF (q, ξ) •J z =
X
v(ξ, j)zj −
j∈J
=
X
v(ξ, j)zj −
{j∈J | ξ(j)ξ(j)
7
In the following, when the financial structure F remains fixed, while only prices vary, we shall simply denote by W (q) the full matrix of payoffs. In the case of unconstrained portfolios, namely Zi = RJ , for every i ∈ I, the financial asset structure will be simply denoted by F = (J, (ξ(j), V j )j∈J ).
2.4
Financial equilibrium
We now consider a financial exchange economy, which is defined as the couple of an exchange economy E and a financial structure F. It can thus be summarized by (E, F) := [D, H, I, (Xi , p, ei )i∈I ; J , (ξ(j), V j )j∈J , (Zi )i∈I ]. Given the price (p, q) ∈ RL × RJ , the budget set (resp. quasi-budget set) of consumer i ∈ I is5 Bi (p, q) = {(xi , zi ) ∈ Xi × Zi : ∀ξ ∈ D, p(ξ) •H [xi (ξ) − ei (ξ)] ≤ [WF (q)zi ](ξ)} = {(xi , zi ) ∈ Xi × Zi : p
(xi − ei ) ≤ WF (q)zi }.
˘i (p, q) = {(xi , zi ) ∈ Xi × Zi : p resp. B
(xi − ei ) WF (q)zi } .
When F is fixed we can drop the subscript F from the budget set. We now introduce the equilibrium notion. Definition 2.1 An equilibrium of the financial exchange economy (E, F) is a list of strategies and prices x ¯, z¯, p¯, q¯ ∈ (RL )I × (RJ )I × RL \ {0} × RJ such that (a) for every i ∈ I, (¯ xi , z¯i ) maximizes the preferences Pi in the budget set Bi (¯ p, q¯), in the sense that
X
(b)
i∈I
x ¯i =
X
(¯ xi , z¯i ) ∈ Bi (¯ p, q¯) and [Pi (¯ x) × Zi ] ∩ Bi (¯ p, q¯) = ∅; X ei and z¯i = 0.
i∈I
i∈I
Definition 2.2 A list of strategies and prices x ¯, z¯, p¯, q¯ ∈ (RL )I × (RJ )I × RL \ {0} × RJ is a quasi-equilibrium of the financial exchange economy (E, F) if p¯ 6= 0 and 6 (a i) for every i ∈ I, ˘i (¯ (¯ xi , z¯i ) ∈ Bi (¯ p, q¯) and [Pi (¯ x) × Zi ] ∩ B p, q¯) = ∅; 5
For x = (x(ξ))ξ∈D , p = (p(ξ))ξ∈D in RL = RH×D (with x(ξ), p(ξ) in RH ) we let p
D
R . 6
For x = (x(ξ))ξ∈D ∈ RD , we denote x(−ξ) = (x(ξ 0 ))ξ0 ∈D\ξ .
8
x = (p(ξ)•H x(ξ))ξ∈D ∈
(a ii) for every i ∈ I, p¯ (¯ xi − ei ) = W (¯ q )zi ; (a iii) for every ξ ∈ D, (xi (−s), xi (s)) ∈ Pi (¯ x) implies p¯(s) •H (xi (s)) ≥ p¯(s) •H (¯ xi (s)); X X X (b) x ¯i = ei and z¯i = 0. i∈I
2.5
i∈I
i∈I
Arbitrage and equilibrium
In the case where portfolio sets are constrained the absence of arbitrage opportunities at the individual level will differ from that at the aggregate level. As outlined in Angeloni and Cornet [1] we have the following definition. Definition 2.3 Given the financial structure F = (J, (ξ(j), V j )j∈J , (Zi )i∈I ), (i) A portfolio z¯i ∈ Zi is said to have no arbitrage opportunities or to be arbitrage-free for agent i ∈ I at the price q ∈ RJ if there is no portfolio zi ∈ Zi such that WF (q)zi > WF (q)¯ zi , that is, [WF (q)zi ](ξ) ≥ [WF (q)¯ zi ](ξ), for every ξ ∈ D, with at least one strict inequality, or, equivalently, if WF (q) (Zi − z¯i ) ∩ RD + = {0}. (ii) We say q is an arbitrage free asset price or the financial structure F is said to be arbitrageP free at (q) if there exists no portfolios zi ∈ Zi (i ∈ I) such that WF (q)( i∈I zi ) > 0, or, equivalently, if: X WF (q) Zi ∩ RD + = {0}. i∈I
When asset market participation is restricted, as we will show in the course of this paper, there are fewer asset prices that can be supported as equilibrium asset prices than those that do not offer arbitrage. We will thus consider a subset of the no arbitrage asset prices that are given by the following definition. Definition 2.4 We say that the asset price q is aggregate arbitrage-free if one of the following equivalent conditions hold 7 D P E (i) WF (q) ∩ RD + = {0}. i∈I Zi D P E (ii) There exists λ ∈ RD Z . i ++ such that λ •D w = 0 for all w ∈ WF (q) i∈I If q is an aggregate arbitrage-free asset price then the financial structure is arbitrage free at q. Let the financial structure F be arbitrage-free at q, and let z¯i ∈ Zi (i ∈ I) such P that i∈I WF (q)¯ zi = 0, then it is easy to see that, for every i ∈ I, z¯i is arbitrage-free at q. P S The converse is true, when i∈I WF (q)Zi ⊂ cone [ i∈I W (q)(Zi − z¯i )]. The later is true 7
Given a subset A ⊂ Rn we denote hAi := Span A.
9
in particular when some agent’s portfolio set is unconstrained, that is, Zi = RJ for some i ∈ I. Consider the following non-satiation assumption: Q P P Assumption NS (i) For every x ¯ ∈ i∈I Xi such that i∈I x ¯i = i∈I ei , Q (Non-Satiation at Every Node) for every ξi ∈ D, there exists x ∈ i∈I Xi such that, for each ξ 6= ξi , xi (ξ) = x ¯i (ξ) and xi ∈ Pi (¯ x); (ii) if xi ∈ Pi (¯ x), then [xi , x ¯i [⊂ Pi (¯ x). It is well known that if preferences are non-satiated then there is no arbitrage at the individual level. In particular, under (NS), if (¯ x, z¯, p¯, q¯) is an equilibrium of the economy (E, F), then z¯i is arbitrage-free at q¯ for every i ∈ I (see Angeloni and Cornet [1]).
3
Existence of equilibrium
3.1
The main existence result
We will prove that when agents’ portfolio sets are constrained, any aggregate strong-noarbitrage asset price can be characterized as an equilibrium asset price. Our approach however does not cover the general case of real assets which needs a different treatment. Let us consider, the financial economy (E, F) = [D, H, I, (Xi , Pi , ei )i∈I ; J , (ξ(j), V j )j∈J , (Zi )i∈I ]. Define the set of attainable consumptions by Y X X b = x∈ X Xi | xi = ei i∈I
i∈I
i∈I
bi be the projection of X b on Xi . and for each i ∈ I, let X We introduce the following assumptions. Assumption (C) (Consumption Side) For all i ∈ I and all x ¯∈
Q
i∈I
Xi ,
bi is compact8 in RL ; (i) Xi is a closed and convex subset of RL and X (ii) [Continuity] the preference correspondence Pi :
Q
i∈I
Xi → Xi , is lower semicontinuous9
and Pi (¯ x) is open in Xi (for its relative topology); 8 9
bi is compact if Xi is bounded below. Note: X A correspondence ϕ : X −→ Y is said to be lower semicontinuous at x0 ∈ X if, for every open set
V ⊂ Y such that V ∩ ϕ(x0 ) is not empty, there exists a neighborhood U of x0 in X such that, for all x ∈ U , V ∩ϕ(x) is nonempty. The correspondence ϕ is said to be lower semicontinuous if it is lower semicontinuous at each point of X.
10
(iii) [Convexity] Pi (¯ x) is convex; (iv) (Irreflexivity) x ¯i 6∈ Pi (¯ x); P P (v) (Non-Satiation of Preferences at Every Node) if i∈I x ¯i = i∈I ei , for every ξ ∈ D there Q exists x ∈ i∈I Xi such that, for each ξ 0 6= ξ, xi (ξ 0 ) = x ¯i (ξ 0 ) and xi ∈ Pi (¯ x); (vi) (Strong Survival Assumption) ei ∈ intXi . Note that these assumptions on Pi are satisfied in particular when agents preferences are given by a utility function that is continuous, monotonic increasing, and quasi-concave. Assumption (F) (Financial Side) (i) For every i ∈ I, Zi is closed, convex and contains zero; (iii) For every asset price q ∈ RJ , for every i ∈ I, W (q)Zi is a closed subset of RD ; We can now state the main theorem characterizing equilibrium prices with arbitrage free prices under the appropriate compatibility condition. Theorem 3.1 (a) Suppose the financial exchange economy (E, F) satisfies C and F. Let q¯ ∈ RJ be an aggregate arbitrage-free asset price such that the following conditions hold: (i) Closed Cone (ii) (WEQ)
S
is convex; and X X − Zi ∩ Ker W (¯ q) ⊂ (A(Zi ) ∩ Ker W (¯ q )) . q )(Zi ) i∈I W (¯
i∈I
i∈I
Then there exists (¯ x, z¯, p¯) with p¯ 6= 0, such that (¯ x, z¯, p¯, q¯) is a quasi-equilibrium. (b) If we additionally assume that (i) Closed Cone
S
q )(Zi ) i∈I W (¯
is linear,
then there exists (¯ x, z¯, p¯) with p¯(ξ) 6= 0, for all ξ ∈ D such that (¯ x, z¯, p¯, q¯) is an equilibrium. Assumption WEQ is taken from Martins - Triki [6] and allows us to show that existence of a weak equilibrium implies the existence of an equilibrium. It is worth noticing that the two assumptions in the theorem are satisfied in the following examples. Example 3.1 For every i ∈ I, Zi is closed, convex and contains zero and
S
i∈I Zi
is a vector
space. Example 3.2 (Cass Condition) For every i ∈ I, Zi is closed, convex and contains zero and S for some i0 ∈ I, Zi ⊂ Zi0 and Zi0 is a vector space. Then i∈I Zi = Zi0 is a vector space and we are in the structure of Example 3.3. 11
Example 3.3 Let I = I1 ∪ I2 and ∀i ∈ I1 ,
Zi = {z | z · e1 ≥ 0, e1 = (1, 0)}
∀i ∈ I2 ,
Zi = {z | z · e2 ≥ 0, e2 = (0, 1)}.
That is every agent in I1 can buy the first asset and every agent in I2 can sell the first asset. It is also worth noticing that in Condition (i) in Theorem 3.1, taking
S
q )Zi i∈I W (¯
to be
‘closed cone’ instead of a linear space allows us to consider the following example. Let I = 1, 2 and Z1 = {z ∈ R2 | z2 ≥ (z1 )2 } Z1 = {z ∈ R2 | z2 ≤ −(z1 )2 } Example 3.4 Give counterexample here.
3.2
Quasi-equilibrium
Gottardi and Hens consider a two date incomplete markets model without consumption in the first date. Their definition of a quasi-equilibrium was suitably modified by Seghir et al. to include consumption in first date. They define a quasi-equilibrium as in the following definition. The lemma that follows shows that in a two date model, a quasiweak-equilibrium in our model implies a quasi-weak-equilibrium in the sense of Seghir et al. Definition 3.1 (Seghir et al.) A list of strategies and prices x ¯, z¯, p¯, q¯ ∈ (RL )I ×(RJ )I ×RL ×RJ is a quasi-equilibrium of the financial exchange economy (E, F), if p¯ 6= 0, (a i) for every i ∈ I, xi ∈ Pi (¯ x) and p¯
1 (xi − ei )
≤ V zi ⇒ p¯(0) •H (xi (0) − ei (0)) + q¯•J zi ≥ 0;
(a ii) for every i ∈ I, p¯ (¯ xi − ei ) = W (¯ q )¯ zi ; (a iii) for every ξ ∈ D, (xi (−s), xi (s)) ∈ Pi (¯ x) implies p¯(s) •H (xi (s)) ≥ p¯(s) •H (¯ xi (s)); X X X x ¯i = ei and z¯i = 0. (b) i∈I
i∈I
i∈I
Let ei (p, q) = {(xi , zi ) ∈ Xi × Zi | p(0) •H (xi (0) − ei (0)) < −q •J zi ; p B
1 (xi
− ei ) ≤ V zi }
ei (¯ Notice that (a i) of this definition is true if an only if (Pi (¯ x) × Zi ) ∩ B p, q¯) = ∅. Comparing Definitions 2.2 and 3.1 we see that the quasi-equilibrium conditions a( ii), (a iii) and (b) are the same. The following lemma shows how (a i) in Definition 2.2 is related to (a i) of Definition 3.1. 12
Lemma 3.1 Under the assumption10 (F0): ∃ˆ zi ∈ A (Zi ) such that V zˆi >> 0, ˘i (¯ ei (¯ (Pi (¯ x) × Zi ) ∩ B p, q¯) = ∅ ⇒ (Pi (¯ x) × Zi ) ∩ B p, q¯) = ∅. Proof of Lemma 3.1. Suppose on the contrary there is some (xi , zi ) ∈ (Pi (¯ x) × Zi ) ∩ ei (¯ B p, q¯) 6= ∅. Then p¯(0) •H (xi (0) − ei (0)) < −¯ q •J z i
p
1 (xi
− ei ) ≤ V zi .
Take zˆi ∈ AZi with V zˆi >> 0. Then for all t > 0 small enough p¯(0) •H (xi (0) − ei (0)) < −¯ q •J zi − q¯ •J (tˆ zi ) = −¯ q (z + tˆ zi )
p
1 (xi
− ei ) ∩RD + = {0}. Then there exists λ ∈ RD ++ such that: hWi ⊂ λ⊥ := {t ∈ RD | λ •D t = 0} 14
11 Given λ ∈ RD ++ as in Lemma 5.1, let
BL = {p ∈ RL | ||λ
p|| ≤ 1}.
Given p ∈ BL , let ρ(p) ∈ RD with ρ(p, ξ) = (1 − ||λ
p||) for all ξ ∈ D.
Given p ∈ BL and γ : BL → RD , for all i ∈ I define n Biγρ (p) = (xi , wi ) ∈ Xi × W Zi : ∃τi ∈ [0, 1], p
o (xi − ei ) ≤ wi + τi γ(p) + ρ(p) ,
n B˘iγρ (p) = (xi , wi ) ∈ Xi × W Zi : ∃τi ∈ [0, 1], p
o (xi − ei ) wi + τi γ(p) + ρ(p) .
We state a lemma, which extends a previous result by Da-Rocha and Triki [6], the proof of which is in the appendix. Lemma 5.2
(i) For every ε > 0, there exists a continuous mapping γ : BL → RD such that,
∀ p ∈ BL , λ •D γ(p) = 0 and ∀ w ∈ W, w •D γ(p) ≤ 0 and ||γ(p)|| ≤ ε. (ii) Assume that the closed cone spanned by W (¯ q )(∪I∈I Zi ) is convex, then γρ ˘ ∀ p ∈ BL , ∃ i ∈ I, such that B (p) 6= ∅ . i
5.2.1
The fixed point argument
Let γ : BL → RD be the continuous mapping associated by Lemma 5.2 to some ε. For Q Q (x, w, p) ∈ i∈I Xi × i∈I W Zi × BL , we define the correspondences Φi for i ∈ I0 := {0} ∪ I as follows: n Φ0 (x, w, p) = p0 ∈ BL | λ
o X (p0 − p) •L (xi − ei ) > 0 , i∈I
and for every i ∈ I, {(ei , 0)} if (xi , wi ) ∈ / Biγ (p) and B˘iγρ (p) = ∅, Φi (x, w, p) = Biγρ (p) if (xi , wi ) ∈ / Biγρ (p) and B˘iγρ (p) 6= ∅, ˘γρ Bi (p) ∩ (Pi (x) × W Zi ) if (xi , wi ) ∈ Biγρ (p). The existence proof relies on the following fixed-point-type theorem due to Gale and MasCollel ([12]). 11
For x ∈ Rn , ||x|| denotes the euclidean norm.
15
Theorem 5.1 Let I0 be a finite set, let Ci (i ∈ I0 ) be a nonempty, compact, convex subset of some Q Euclidean space, let C = i∈I0 Ci and let Φi (i ∈ I0 ) be a correspondence from C to Ci , which is lower semicontinuous and convex-valued. Then, there exists c¯ ∈ C such that, for every i ∈ I0 either c¯i ∈ Φi (¯ c) or Φi (¯ c) = ∅. We now show the sets C0 = BL , Ci = Xi × W Zi (i ∈ I) and the above defined correspondences Φi (i ∈ I0 ) satisfy the assumptions of Theorem 5.1. Claim 5.1 For every c¯ := (¯ x, w, ¯ p¯, ) ∈
Q
i∈I
Xi ×
Q
i∈I
W Zi × BL ,
(i) Φi (¯ c) is convex (possibly empty); (ii) p¯ 6∈ Φ0 (¯ c), and for all i ∈ I, (¯ xi , w ¯i ) 6∈ Φi (¯ c); (iii) for every i ∈ I0 , the correspondence Φi is lower semicontinuous at c¯. Proof of Claim 5.1: Let c¯ := (¯ x, w, ¯ p¯) ∈
Q
i∈I
Xi ×
Q
i∈I
W Zi × BL be given.
Proof of (i): Clearly Φ0 (¯ c) is convex. For every i ∈ I, recalling that Pi (¯ x) and W Zi are convex sets, by Assumption C and F, we have Φi (¯ c) is a convex set. Proof of (ii): Clearly, p¯ 6∈ Φ0 (¯ c) and (¯ xi , w ¯i ) 6∈ Φi (¯ c) follows from the d efinitions of these sets and the fact that x ¯i 6∈ Pi (¯ x) (from Assumption C). Proof of (iii): We need to show that Φi is lower semicontinuous for all i ∈ I0 . Since Φ0 has an open graph, clearly it is lower semicontinuous. To show lower semicontinuity of Φi for i ∈ I, let U be an open subset of Xi × W Zi such that Φ(¯ c) ∩ U 6= ∅ and we will distinguish three cases: Case(1): (¯ xi , w ¯i ) ∈ / Biγρ (¯ p) and B˘iγρ (¯ p) = ∅. Then Φi (¯ c) = {(ei , 0)} ⊂ U . The set Ωi = {(xi , wi , p) | (xi , wi ) ∈ / Biγρ (p)} is an open subset of Xi × W Zi × BL (by Assumptions C and F). To see this, let {(xi , wi , p)} be such that (xi , wi ) ∈ Biγρ (p) and (xi , wi , p) → (x, w, p). Since for all n, (xi , wi ) ∈ Biγρ (p), there exists τi ∈ [0, 1] such that p
(xi − ei ) ≤ wi + τi γ(p) + ρ(p)
In the limit we have p
(x − ei ) ≤ w + τ γ(p) + ρ(p)
Where τ = limn→∞ τi ∈ [0, 1]. Thus (x, w) ∈ Biγρ (p). Thus Ωi contains an open neighborhood O of c¯. Now, let c = (x, w, p) ∈ O. If B˘iγρ (p) = ∅ then Φi (c) = {(ei , 0)} ⊂ U and so Φi (c)∩U is nonempty. If B˘γρ (p) 6= ∅ then Φi (c) = B γρ (p). i
But Assumptions C and F imply that (ei , 0) ∈ Xi ×W Zi , hence (ei , 0) ∈
i γρ Bi (p) (with τi
and noticing that ρ(p) ≥ 0). So {(ei , 0)} ⊂ Φi (c) ∩ U which is also nonempty. 16
=0
Case (2): c¯ = (¯ xi , w ¯i , p¯) ∈ Ωi := {c = (xi , wi , p) : (xi , wi ) ∈ / Biγρ (p) and B˘iγρ (p) 6= ∅}. Then the set Ωi is clearly open (since its complement is closed). On the set Ωi one has Φi (c) = Biγρ (p). We recall that ∅ = 6 Φi (¯ c) ∩ U = Biγρ (¯ p) ∩ U . We notice γρ γρ γρ γρ that B (¯ p) = cl B˘ (¯ p) since B˘ (¯ p) 6= ∅. Consequently, B˘ (¯ p) ∩ U 6= ∅ and we chose a i
i
i
i
point (xi , wi ) ∈ B˘iγρ (¯ p) ∩ U , that is, (xi , wi ) ∈ [Xi × W Zi ] ∩ U and for some τi ∈ [0, 1], p) + ρ(¯ p). p¯ (xi − ei ) wi + τi γ(¯ Clearly the above inequality is also satisfied for the same point (xi , wi ) and the same τi when p belongs to a neighborhood O of p¯ small enough (using the continuity of ρ(·) and γ(·)). This shows that on O one has ∅ = 6 B˘γρ (p) ∩ U ⊂ B γρ (p) ∩ U = Φ(c) ∩ U . i
Case (3): (¯ xi , w ¯i ) ∈
Biγρ (¯ p).
i
By assumption we have
∅= 6 Φi (¯ c) ∩ U = B˘iγρ (¯ p) ∩ [Pi (¯ x) × W Zi ] ∩ U. By an argument similar to what is done above, one shows that there exists an open neighborhood N of p¯ and an open set M such that, for every p ∈ N , one has ∅ = 6 M ⊂ B˘γρ (p)∩U . i
Since Pi is lower semicontinuous at c¯ (by Assumption C, there exists an open neighborhood Ω of c¯ such that, for every c ∈ Ω, ∅ = 6 [Pi (x) × W Zi ] ∩ M , hence ∅= 6 [Pi (x) × W Zi ] ∩ B˘iγρ (p) ∩ U ⊂ Biγρ (p) ∩ U, for every c ∈ Ω. Consequently, from the definition of Φi , we get ∅ = 6 Φi (c) ∩ U , for every c ∈ Ω. γρ ˘ The correspondence Ψi := B ∩ (Pi × W Zi ) is lower semicontinuous on the whole set, i
being the intersection of an open graph correspondence and a lower semicontinuous correspondence. Then there exists an open neighborhood O of c¯ := (¯ x, w, ¯ p¯) such that, for every (x, w, p) ∈ O, then U ∩ Ψi (x, w, p) 6= ∅ hence ∅ = 6 U ∩ Φi (x, w, p) (since we always have Ψi (x, w, p) ⊂ Φi (x, w, p)). Given ε > 0, in view of Claim 5.1, we can apply the fixed-point Theorem 5.1. Thus there Q Q exists c¯ = (¯ x, w, ¯ p¯) ∈ i∈I Xi × i∈I W Zi × BL such that, for every i ∈ I0 , Φi (¯ x, w, ¯ p¯) = ∅. Written coordinatewise, this is equivalent to saying that:
(5.1)
∀ p ∈ BL , (λ
p) •D
X
(¯ xi − ei ) ≤ (λ
i∈I
(5.2)
p¯) •D
X
(¯ xi − ei )
i∈I
∀i ∈ I, (¯ xi , w ¯i ) ∈ Biγρ (¯ p) and B˘iγρ (¯ p) ∩ (Pi (¯ x) × W Zi ) = ∅. 17
5.2.2
Properties of (¯ x, w, ¯ p¯, q¯).
We first prove that x ¯ = (¯ xi )i∈I satifies the market clearing condition. Claim 5.2
P
¯i i∈I x
=
P
i∈I ei .
deduce that (λ
p¯) =
P
xi − ei ) i∈I (¯ (¯ x −e ) i i Pi∈I || i∈I (¯ xi −ei )|| and ||λ
Proof of Claim 5.2: Suppose P
(5.3)
(λ
6= 0. From the fixed-point assertion (5.1) we p¯|| = 1. So
X
p¯) •L
(¯ xi − ei ) > 0.
i∈I
Recalling that (¯ xi , w ¯i ) ∈ Biγρ (¯ p), for all i ∈ I, by the fixed-point assertion (5.2), hence there exists τ¯i ∈ [0, 1] such that p¯ (¯ xi − e i ) ≤ w ¯i + τ¯i γ(¯ p) + ρ(¯ p). Summing up over i we get: X
p¯
(¯ xi − e i ) ≤
i∈I
X
w ¯i + (
i∈I
X τ¯i )γ(¯ p) + (#I)ρ(¯ p). i∈I
Taking the scalar product with λ we get, (λ
p¯) •L
X
(¯ xi − e i ) ≤ λ •D
i∈I
X i∈I
On the right hand side, we have λ •D Lemma 5.2), and ρ(¯ p) = 0 (since ||λ
w ¯i + (
X τ¯i )λ •D γ(¯ p) + (#I)λ •D ρ(¯ p). i∈I
P
= 0 (by Lemma 5.1), λ •D γ(¯ p) = 0 (by P p¯|| = 1). Thus (λ p¯) •L i∈I (¯ xi − ei ) ≤ 0, which ¯i i∈I w
contradicts (5.3). Claim 5.3 The following conditions hold: (i) If for some i ∈ I, B˘γρ (¯ p) 6= ∅ then (¯ xi , w ¯i ) ∈ B γρ (¯ p) and B γρ (¯ p) ∩ (Pi (¯ x) × W Zi ) = ∅; i
i
i
(ii) For all ξ ∈ RD , p¯(ξ) 6= 0; (iii) For all i ∈ I, (¯ xi , w ¯i ) ∈ Biγρ (¯ p) and Biγρ (¯ p) ∩ (Pi (¯ xi ) × W Zi ) = ∅. Proof of Claim 5.3: Proof of (i): The first assertion that (¯ xi , w ¯i ) ∈ Biγρ (¯ p) is a consequence of the Fixed-Point Assertion (5.2). We now show that Biγρ (¯ p) ∩ (Pi (¯ xi ) × W Zi ) = ∅. Indeed suppose that B γρ (¯ p) ∩ (Pi (¯ x) × W Zi ) contains an element (xi , wi ). Since B˘γρ (¯ p) 6= ∅, we i
i
let (¯ xi , w ¯i ) ∈ B˘iγρ (¯ p). Suppose first that x ¯i = xi , then, from above (xi , w ¯i ) ∈ [Pi (¯ x) × W Zi ] ∩ B˘iγρ (¯ p), which contradicts the fact that this set is empty by Assertion (5.2). Suppose now that x ¯i 6= xi , from Assumption C (iii), (recalling that xi ∈ Pi (¯ x)) the set [¯ xi , xi [∩Pi (¯ x) is nonempty, 18
hence contains a point xi (λ) := (1 − λ)¯ xi + λxi for some λ ∈ [0, 1[. We let wi (λ) := (1 − λ)w ¯i + λwi and we check that (xi (λ), wi (λ)) ∈ B˘iγρ (¯ p) (since (xi , wi ) ∈ Biγρ (¯ p) and γρ γρ ˘ ˘ (¯ xi , w ¯i ) ∈ B (¯ p)). Consequently, B (¯ p) ∩ (Pi (¯ x) × W Zi ) 6= ∅, which contradicts again i
i
Assertion (5.2). Thus (¯ xi , w ¯i ) ∈ Biγρ (¯ p) and Biγρ (¯ p) ∩ (Pi (¯ x) × W Zi ) = ∅
(5.4)
(¯ p) 6= ∅. Suppose Proof of (ii): From Lemma 5.2 (ii), there exists i0 ∈ I such that B˘iγρ P P0 there exists ξ ∈ D such that p(ξ) = 0. From Claim 5.2, i∈I x ¯i = i∈I ei , and from the x) such that Non-Satiation Assumption at node ξ (for Consumer i0 ) there exists xi0 ∈ Pi0 (¯ 0
0
0
(¯ p) and, recalling ¯i0 ) ∈ Biγρ x i0 , w ¯i0 (ξ ) for every ξ 6= ξ; from Assertion (5.2), (¯ xi0 (ξ ) = x 0 (¯ p). Consequently, ¯i0 ) ∈ Biγρ that p¯(ξ) = 0, one deduces that (xi0 , w 0 Biγρ (¯ p) ∩ [Pi0 (¯ x) ∩ W Zi0 ] 6= ∅, 0 which contradicts Condition 5.4. Proof of (iii): From Part (ii) of this Claim we have, for all ξ ∈ RD , p¯(ξ) 6= 0, hence ¯i = 0, τ¯i = 0 and x ¯i = ei − t¯ p, for t > 0 small p¯(ξ) · p¯(ξ) > 0 and (¯ p p¯ >> 0. Taking w enough, we get x ¯i ∈ BL (ei , r) ⊂ Xi (from the Survival Assumption). Then p¯ (¯ xi − ei ) = γρ −t(¯ p p¯) 0 ≤ 0 + ρ(¯ p). This shows that (xi , 0) ∈ B˘ (¯ p) 6= ∅. i
Claim 5.4 The following conditions hold: (i) ρ(¯ p) = 0 (ii) If closed cone
S
q )Z)i i∈I W (¯
is linear then for all i ∈ I, τ¯i = 0 and
(iii) If closed cone
S
q )Z)i i∈I W (¯
P is linear then || i∈I w ¯ i || ≤ (#I)||γ(¯ p)||
P
¯i i∈I w
= 0.
Proof of Claim 5.4: Proof of (i): We first prove that the modified budget constraints are binding, that is (5.5)
p¯ (¯ xi − ei ) = w ¯i + τ¯i γ(¯ p) + ρ(¯ p), ∀ i ∈ I
Indeed, suppose that there exists i ∈ I such that p¯ (¯ xi − e i ) < w ¯i + τ¯i γ(¯ p) + ρ(¯ p) That is there exist ξ ∈ D such that p¯(ξ) •H (¯ xi (ξ) − ei (ξ)) < w ¯i (ξ) + τ¯i γ(¯ p)(ξ) + ρ(¯ p) 19
But by Claim 5.2,
P
¯i i∈I x
=
P
and by the nonsatiation assumption C (v) for
i∈I ei
consumer i, there exists xi ∈ Pi (¯ x) such that xi (ξ 0 ) = x ¯i (ξ 0 ) for every ξ 0 6= ξ. Consequently, we can choose x ∈ [xi , x ¯i ) close enough to x ¯i so that (x, w ¯i ) ∈ Biγρ (¯ p). But, from the local non-satiation (Assumption K (ii)), [xi , x ¯i ) ⊂ Pi (¯ x). Consequently, Biγρ (¯ p) ∩ (Pi (¯ x) × W Zi ) 6= ∅ which contradicts Claim 5.3. This ends the proof of Equation (5.5). Summing up over i ∈ I in the equalities (5.5) and using the market clearing condition (Claim 5.2) we get:
(5.6)
0=
X
w ¯i + (
X τ¯i )γ(¯ p) + (#I)ρ(¯ p)
i∈I
i∈I
Taking above the scalar product with λ yields: X X 0 = λ •D ( w ¯i ) + ( τ¯i )λ •D γ(¯ p) + (#I)λ •L ρ(¯ p) i∈I
i∈I
P P ¯i ∈ W ⊂ < W >⊂ λ⊥ by Lemma 5.1. Moreover ¯i ) = 0 since i∈I w But λ •D ( i∈I w 0 = λ •D γ(¯ p) by Lemma 5.2. Consequently ρ(p) = 0. Proof of (ii): From Assertion (5.6) and the fact that ρ(p) = 0 we get
X X w ¯i = −( τ¯i )γ(¯ p).
(5.7)
i∈I
i∈I
P
¯i on both sides and using the fact the closed Taking the scalar product with i∈I w P S ¯i = 0. Again from Assertion q )Z)i is linear in Lemma 5.2 we get i∈I w cone i∈I W (¯ 5.7 we get (
X τ¯i )γ(¯ p) = 0. i∈I
Since each τ¯i ≥ 0 we can take the scalar product in the above equation with γ(¯ p) to conclude that for all i ∈ I, τ¯i = 0. Proof of (iii): From Assertion 5.7, taking scalar product on both sides with P both sides and recalling that τi ∈ [0, 1] we get i∈I τ¯i ≤ #I hence
P
¯i i∈I w
on
X ¯i || ≤ (#I)||γ(¯ p)|| || w i∈I
If closed cone
S
q )Z)i i∈I W (¯
is a linear space then (¯ x, z¯, p¯, q¯) is a weak-equilibrium as in
Theorem 3.1 as consequence of Claim 5.2, Claim 5.3 and Claim 5.4 (i and ii). S q )Z)i is a convex set then we can go to the limit as ε If however, closed cone i∈I W (¯ goes to zero and verify that the limit of the sequence will be quasi-weak-equilibrium. 20
5.2.3
Limit argument when ε =
1 n
converges to zero and quasi-weak-equilibrium.
In the previous section we have associated to every ε > 0, the list (¯ x, w, ¯ p¯) and we will now take ε = 1/n and denote the associated list by (¯ xn , w ¯ n , p¯n ), to make the dependence on n explicit. Since BL and each of Xi and W Zi are compact, without any loss of general¯, w, ¯ p¯). Let ity we can assume that the sequence (¯ xn , w ¯ n , p¯n ) converges to some element (x z¯i be such that w¯i = W (¯ q )z¯i . We first recall the following sets: Bi (p, q) = {(xi , zi ) ∈ Xi × Zi : p ˘i (p, q) = {(xi , zi ) ∈ Xi × Zi : p B
(xi − ei ) ≤ W (q)zi }. (xi − ei ) 0 such that, for every agent i ∈ I, X q ) ⊂ intBL (0, r) ˆ i (¯ and W q ) ⊂ intBD (0, r). The truncated economy (Eˆr , F r ) is the collection (Eˆr , F r ) = [D, H, I, (Xir , pˆr , ei )i∈I ; J , (ξ(j), V j )j∈J , (Zir )i∈I ], 24
where,
Xir = Xi ∩ BL (0, r), Zir = {z ∈ Zi | W (¯ q )z ∈ BD (0, r)} and Pˆir (x) = Pˆi (x) ∩ intBL (0, r).
The existence of a weak equilibrium of (Eˆr , F r ) is then a consequence of Section 5.1, that is, Theorem 4.2 with the additional Assumption K. We just have to check that Assumption K and all the assumptions of Theorem 4.2 are satisfied by (Eˆr , F r ). In view of Proposition 5.1, this is clearly the case for all the assumptions but the Survival Assumption C (vi) that is proved via a standard argument (that we recall hereafter). Indeed we first notice that (ei , 0)i∈I belongs to K(q), hence, for every i ∈ I, ei ∈ ˆ Xi (q) ⊂ intBL (0, r). Recalling that ei ∈ intXi (from the Survival Assumption), we deduce that ei ∈ intXi ∩ intBL (0, r) ⊂ int[Xi ∩ BL (0, r)] = intXir .
Proposition 5.2 Given q¯ ∈ RJ , if (¯ x, z¯, p¯, q¯) is a weak-equilibrium of (Eˆr , F r ) then it is also a weak-equilibrium of (E, F).
Proof of proposition 5.2. Let (¯ x, z¯, p¯, q¯) be a weak-equilibrium of the economy (Eˆr , F r ). In view of the definition of a weak-equilibrium, to prove that it is also a weak-equilibrium of (E, F) we only have to check, for every i ∈ I, [Pi (¯ x) × Zi ] ∩ Bi (¯ p, q¯) = ∅, where Bi (¯ p, q¯) denotes the budget set of agent i in the economy (E, F). Assume, on the contrary, that, for some i ∈ I the set [Pi (¯ x)×Zi ]∩Bi (¯ p, q¯) is nonempty, hence contains a couple (xi , zi ). Clearly the allocation (¯ x, W (¯ q )¯ z ) belongs to the set K(¯ q ), ˆ i (¯ ˆ i (¯ hence for every i ∈ I, x ¯i ∈ X q ) ⊂ int BL (0, r) and w ¯i = W (¯ q )¯ zi ∈ W q ) ⊂ int BD (0, r). Thus, for t ∈]0, 1] sufficiently small, xi (t) := x ¯i + t(xi − x ¯i ) ∈ int BL (0, r) and wi (t) := w ¯i + t(wi − w ¯i ) ∈ intBD (0, r). Clearly (xi (t), wi (t)) is such that wi (t) = W (¯ q )zi (t) where zi (t) = (¯ zi +t(zi −¯ zi )) ∈ Zi and (xi (t), zi (t)) belongs to the budget set Bi (¯ p, q¯) of agent i (for the economy (E, F)) and since xi (t) ∈ Xir := Xi ∩BL (0, r), zi (t) ∈ Zir := {z ∈ Zi | W (¯ q )z ∈ BD (0, r)}, the couple (xi (t), zi (t)) belongs also to the budget set Bir (¯ p, q¯) of agent i (in the economy (Eˆr , F r )). From the definition of Pˆi , we deduce that xi (t) ∈ Pˆi (¯ x) (since from above xi (t) := x ¯i + t(xi − x ¯i ) and xi ∈ Pi (¯ x)), hence xi (t) ∈ Pˆir (¯ x) := Pˆi (¯ x) ∩ intBL (0, r). p, q¯). We have thus shown that, for t ∈]0, 1] small enough, (xi (t), zi (t)) ∈ [Pˆ r (¯ x)×Z r ]∩B r (¯ i
i
i
This contradicts the fact that this set is empty, since (¯ x, y¯, p¯, q¯) is a weak-equilibrium of r the economy (Eˆ , F). 25
6
Appendix: Proof of lemma 5.2
Proof of Lemma 5.2: Part (i): We recall that, from Assumption C, there exists r > 0 such that, for all i ∈ I, BL (ei , r) ⊂ Xi , and we define the correspondence Γ from BL to ∆ by Γ(p) = {γ ∈ λ⊥ ∩ W o ∩ BD (0, ε) | ∃ u ∈ BL (0, r), ∃ w ∈ W, p
u
