Existence of Equilibria in Financial Markets With Restricted

Participation¹

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 [27]). In the presence of such portfolio restrictions we need to confine our attention to aggregate arbitrage-free asset prices. We also describe a considerably weak condition on the space of income transfers(the union of the agents’ payoff sets is linear), that ensure these asset prices to be part of a financial equilibrium. We also show that weakning this condition further to require the union of the payoff set to be convex results only in a quasi-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 [8], 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: October 2008 2

Paris School of Economics, Paris France and University of Kansas, Lawrence, KS 3

University of Texas at Dallas, Richardson, TX (ramu.gopalan@utdallas.edu)

1

Contents

1. Introduction           3
2. The T-period financial exchange economy   6
   1. Time and uncertainty in a multiperiod model . . . . . . . . . . . . . . . . .   6
   2. The stochastic exchange economy . . . . . . . . . . . . . . . . . . . . . . . .      7
   3. The financial structure      . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        7
   4. Financial equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  9
   5. Arbitrage and equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . .            10
3. Existence of equilibrium      11
   1. The main existence result . . . . . . . . . . . . . . . . . . . . . . . . . . . .           11
   2. Other definitions of quasi-equilibrium . . . . . . . . . . . . . . . . . . . . .      13
4. Proof of existence under additional assumptions.    14
   1. Additional assumptions     . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          14
   2. Preliminary lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   15
      1. The fixed point argument          . . . . . . . . . . . . . . . . . . . . . . . .      15
      2. Properties of (¯x,w,¯ p,¯ q¯). . . . . . . . . . . . . . . . . . . . . . . . . . 18
      3. Limit argument when  converges to zero and quasi-weak-

                          equilibrium.                    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                      21

4. 3. Proof in the general case (without additional assumptions) . . . . . . . . .        22
      1. Enlarging the preferences as in Gale and Mas-Colell . . . . . . . . .    23
      2. Truncating the economy . . . . . . . . . . . . . . . . . . . . . . . . .    24
5. Appendix: Proof of lemma 4.2         26

1. Introduction

In financial markets investors face several restrictions on what assets they can trade and the extent to which they can trade in these assets. For instance in September 2008, in the midst of the current financial crisis, short sales on several stocks were banned in UK. Similar actions were taken by the SEC in US a week later, when short sales in almost 800 stocks were banned for a couple of weeks. Such constraints on the investor’s portfolio choices are not exceptional cases. Even during peroids of normalcy there are several restrictions that investors face with respect to their asset market participation, apart from their own wealth constraints.

In general there are two reasons why such restrictions exist in financial markets - institutional or behavioral reasons. Some of the well known institutional restrictions are transactions costs, short sales constraints, margin requirements, frictions due to bid-ask spreads and taxes, collateral requirements, capital adequacy ratios and target rations. Elsinger and Summer [12] give a good discussion of these institutional constraints and how to model them in a general financial model. On the other hand investors may be restricted due to some behavioral reasons. For instance following Radner and Rothschild [25] we can suppose that agents have limits on the how much information they can process. This may cause each investor to concentrate on only a subset of assets to begin with.

Given that investors can face such restrictions on their portfolio choices, there are two ways in which such restrictions can be incorporated into a general financial model. The first is to assume that these restrictions are institutional and hence exogenously given. Then we can take them as primitives of the model. Alternatively, we can model them as arising engodenously. It is worth notiong that in a truly general model these constraints should be determined endogenously. Villanacci et al. [28] sumarize some of their work in this direction.

In this paper we adopt the view that these restrictions are institutional following the approach of Balasko, Cass and Siconolfi [2]. Moreover, we consider very general restrictions on portfolio sets as in Siconolfi [27], where each agent’s portfolio set is assumed to be closed, convex and contains zero. Such general portfolio set are able to capture all the institutional restrictions listed earlier (see [12]).

Investors primarily use financial assets to enable income transfers in order to achieve a consumption bundle that would maximize their preferences. With such investors, we need to address two important ideas that is relevant to economists - that of arbitrage and equilibrium. Such preference maximizing agents would be ready to seize any arbitrage opportunity that may arise in financial markets. However, if each investor faces different constraints on portfolios (there is good reason to believe that this is the case), then the arbitrage opportunities that open up to individuals will be different from one another and different from what may be available to the market as a whole. Hens et al. [18] show this distinction in a 2-date model with linear portfolio sets. Cornet and Gopalan [7] extend this result to a model with more than two dates. Also, since all agents will be choosing optimal consumptions and portfolios in an equilibrium, it is obvious that, with non-satiation, none of the agents will by themself find an arbitrage opportunity at equilibrium.

This paper primarily examines the existence of a financial equilibrium in a multiperiod model when investors face such general portfolio restrictions. In such models there are two possible ways to approach existence issue. One would be to directly show the existence of commodity and asset prices and commmodity and asset allocations which are optimal for all agents and all markets clear. The other approach, motivated by Cass ([4] and [5]), is to look for arbitrage-free asset prices that are also equilibrium asset prices.

In 2-date (one period) models without restrictions on portfolio sets, the existence issue has been extensively studied. Cass ([4]) and Werner ([29], [30]) showed existence with nominal assets. Duffie and Shafer ([11]) showed a generic existence result with real assets. This second approach has been extensively used. Magill and Shafer [19] 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. [28] and Magill and Quinzii [20]).

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 [9] pioneering model we consider an event-tree to represent the evolution of time and uncertainty. Magill and Quinzii ([20]) 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 nodes, 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 portfolio constraints, Cass ([4]), Duffie ([10]) and Florenzano and Gourdel ([13]) show the characterization of equilibrium and arbitrage free asset prices. In the presence of such constraints Cass ([4] and [5]) approached the existence problem by assuming that there is some agent who is unconstrainted. Then this agent would be able to accommodate the excess demands of all the other agents facilitating the existence proof. Around the same time Werner [29], Duffie and Shafer [11] and Magill and Shafer [19] provided similar approaches to show existence. This approach has been extensively used to show existence ever since, Florenzano and Gourdel ([13]), Magill and Quinzii ([20]), Angeloni and Cornet ([1]) among others.

This approach of assuming one agent to have unconstrained porfolio sets, breaks the symmetry of the problem and hence it is not possible to give a symmetric existence result (symmetric with respect to the agents’ problem). More recently in a working paper, with such general portfolio restrictions, Da-Rocha and Triki ([8]) 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 aggregate arbitrage-free asset price can be supported as an equilibrium asset price. The approach here is similar to that in Da-Rocha and Triki ([8]), however the notion of absence of arbitrage and the assumptions on the set of income transfers are weaker. In this paper we show two kinds of results. If the cone formed by the union of all the agents portfolio sets is convex then any aggregate arbitrage-free asset price will also be a quasi-equilibrium asset price. However if this cone is linear then we can show that any market arbitrage-free asset price will be an equilibrium asset price.

Section 2 describes the T-period model and the notion of a financial equilibrium. Section 3, states the main result and discusses the different notions of equilibrium. Section 4 gives a detailed proof of the central result in this paper.

2. The T-period financial exchange economy
   1. Time and uncertainty in a multiperiod model

We^[1]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 = {0,1,2,...,D} of length T and we shall essentially use the same model as Angeloni and Cornet [1], (we refer to [20] for an equivalent presentation with information partitions). The set D_t denotes the nodes (also called date-events) that could occur at date t and the family (D_t)_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 ∈ D_t.

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 ∈ D_t,t 6= 0 has a unique immediate predecessor in D_t−1, denoted pr(ξ). The predecessor mapping pr : D \ {ξ₀} −→ D is assumed to satisfy pr(D_t) = D_t−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 \ D_T .

Moreover, for τ ∈ T \ {0} and we define, by induction, pr^τ(ξ) = pr(pr^τ−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) = {ξ⁰ ∈ D : ∃τ ∈ T \ {0} | ξ = pr^τ(ξ⁰)},

D⁻(ξ) = {ξ⁰ ∈ D : ∃τ ∈ T \ {0} | ξ⁰ = pr^τ(ξ)}.

If ξ⁰ ∈ D⁺(ξ) [resp. ξ⁰ ∈ D⁺(ξ) ∪ {ξ}], we shall also use the notation ξ⁰ > ξ [resp. ξ⁰ ≥ ξ].

We notice that D⁺(ξ) is nonempty if and only if ξ 6∈ D_T and D⁻(ξ) is nonempty if and only if ξ 6= ξ₀. Moreover, one has ξ⁰ ∈ D⁺(ξ) if and only if ξ ∈ D⁻(ξ⁰) and similarly ξ⁰ ∈ ξ⁺ if and only if ξ = (ξ⁰)⁻.

1. 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 R^L, where L = H×D. An element x in R^L is called a consumption, that is x = (x(ξ))_ξ∈D ∈ R^L, where x(ξ) = (x(h,ξ))_h∈H ∈ R^H, for every ξ ∈ D.

We denote by p = (p(ξ))_ξ∈D ∈ R^L the vector of spot prices and p(ξ) = (p(h,ξ))_h∈H ∈ R^H is called the spot price at node ξ ∈ D. 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

p(ξ) •H x(ξ) = ^X p(h,ξ)x(h,ξ).

h∈H

Each agent i ∈ I is endowed with a consumption set X_i ⊂ R^L which is the set of her possible consumptions. An allocation is an element x = (x_i)_i∈I ∈ ^Q_i∈I X_i, where x_i ∈ X_i denotes the consumption of agent i ∈ I.

The tastes of each consumer i ∈ I are represented by a strict preference correspondence P_i : ^Q_j∈I X^j −→ X_i, where P_i(x) defines the set of consumptions that are strictly preferred by i to x_i, that is, given the consumptions x^j for the other consumers j 6= i. Thus P_i 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 u_i : X_i −→ R, for every i ∈ I, the strict preference correspondence is defined by P_i(x) = {x¯_i ∈ X_i | u_i(¯x_i)> u_i(x_i)}.

Finally, at each node ξ ∈ D, every consumer i ∈ I has a node-endowment e_i(ξ) ∈ RH (contingent to the fact that ξ prevails) and we denote by e_i = (e_i(ξ))_ξ∈D ∈ R^L her endowment vector across the different nodes. The exchange economy E can thus be summarized by

E = [D;H;I;(X_i,P_i,e_i)_i∈I].

1. 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 ξ⁰ of ξ(j), that is, for ξ⁰ > ξ(j). We denote by v(ξ,j) the payoff of asset j at node ξ. Since we consider only 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) is defined for all nodes ξ ∈ D and that v(ξ,j) = 0 if ξ ∈ D6 ⁺(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) ∈ D_T , v(ξ,j) = 0 for every ξ ∈ D.

For every consumer i ∈ I, if z_i^j > 0 [resp. z_i^j < 0], then |z_i^j| will denote the quantity of asset j ∈ J bought [resp. sold] by agent i at the emission node ξ(j). The vector z_i = (^z_i^j)_j∈J ∈ R^J is called the portfolio of agent i.

The price of asset j is denoted by q_j and we recall that it is paid at its emission node ξ(j). We let q = (q_j)_j∈J ∈ R^J be the asset price (vector).

We assume that each consumer i ∈ I is endowed with a portfolio set Z_i ⊂ R^J, which represents the set of portfolios that are admissible for agent i. If some agent i ∈ I has no constraints on her portfolio choices then Z_i ⁼ R^J. In this paper we consider portfolio sets that closed, convex and contain zero for every agent.

To summarize, the financial asset structure F = (J,(ξ(j),V ^j)_j∈J ,(Z_i)_i∈I) consists of

• A finite set of assets J , and each asset j ∈ J is defined by its node of issue ξ(j) ∈ D and its payoff vector V ^j = (v(ξ,j))_ξ∈D ∈ R^D, with v(ξ,j) = 0 if ξ /∈ D⁺.
• The portfolio set Z_i ⊂ R^J for every agent i ∈ I.
• The payoff matrix of F is the D × J matrix V = (v(ξ,j))_ξ∈D,j∈J • The full payoff matrix W_F(q) is the (D × J)−matrix with entries

wF(q)(ξ,j) := v(ξ,j) − δ_ξ,ξ(j)q_j,

where δ_ξ,ξ0 = 1 if ξ = ξ⁰ and δ_ξ,ξ0 = 0 otherwise.

So for a given portfolio z ∈ R^J (and asset price q) the full flow of payoffs across all nodes is W_F(q)z, a vector in R^D whose ξ-th component is the (full) financial payoff at node ξ, that is

[W_F(q)z](ξ) := W_F(q,ξ) •J z = ^X v(ξ,j)z_j − ^X δ_ξ,ξ(j)q_jz_j

                                                                           j∈J                               j∈J

                                            =              ^X                v(ξ,j)z_j −                 ^X               q_jz_j.

                                                  {j∈J| ξ(j)<ξ}                                   {j∈J| ξ(j)=ξ}

We shall extensively use the fact that, for λ ∈ R^D, and j ∈ J , one has:

[^tW_F(q)λ](j) = ^X λ(ξ)v(ξ,j) − ^X λ(ξ)δ_ξ,ξ(j)

                                                            ξ∈D                                  ξ∈D

(2.1)                                                       = ^X λ(ξ)v(ξ,j) − λ(ξ(j))q_j.

ξ>ξ(j)

In the following, when the financial structure F remains fixed, while only prices vary, we shall simply denote by W(q)the full payoff matrix. In the case of unconstrained portfolios, namely Z_i = R^J, for every i ∈ I, the financial asset structure will be simply denoted by F = (J,(ξ(j),V ^j)_j∈J ).

1. 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,(X_i,P_i,e_i)_i∈I;J,(ξ(j),V ^j)_j∈J ,(Z_i)_i∈I].

Given the price (p,q) ∈ RL × R^J, the budget set (resp. quasi-budget set) of consumer i ∈ I is⁵

B_i(p,q) = {(x_i,z_i) ∈ X_i × Z_i : ∀ξ ∈ D, p(ξ) •H [x_i(ξ) − e_i(ξ)] ≤ [W_F(q)z_i](ξ)}

= {(x_i,z_i) ∈ X_i × Z_i : p  (x_i − e_i) ≤ W_F(q)z_i}.

resp. B^˘_i(p,q) = {(x_i,z_i) ∈ X_i × Z_i : p  (x_i − e_i)  W_F(q)z_i}.

When F is fixed we can drop the subscript F from the budget set. We now introduce the equilibrium notion.

Definition 2.1 (i) A financial accounts clearing equilibrium is a list of strategies and prices x, of the financial exchange economy (E,F) if p¯ 6= 0 and

(with x(ξ),p(ξ) in R^H) we let p  x = (p(ξ)•_Hx(ξ))_ξ∈D ∈

1. for every i ∈ I,(¯x_i,z¯_i) ∈ B_i(¯p,q¯) and [P_i(¯x) × Z_i] ∩ B_i(¯p,q¯) = ∅;
2. ^Xx¯_i = ^Xe_i and ^XW^¯ (¯q)z_i = 0.

i∈I

(ii) The listis a financial accounts clearing quasi-

equilibrium of the financial exchange economy (E,F) if p¯ 6= 0, (b) holds and

(ai) for every i ∈ I,(¯x_i,z¯_i) ∈ B_i(¯p,q¯);

(aii) B˘_i(¯p,q¯) = ∅ and/or [P_i(¯x) × Z_i] ∩ B˘_i(¯p,q¯) = ∅;

(a iii) for every ξ ∈ D \ D₀,(x_i(−s),x_i(s)) ∈ P_i(¯x) implies p¯(s) •H (x_i(s)) ≥ p¯(s) •H (¯x_i(s));

Remark 2.1 The above definition requires the balancing of accounts (or payoffs), that is ^P W(¯q)¯z_i = 0 instead of the balancing of portfolios, ^P_i∈Iz¯_i = 0 which is needed in _i∈I

the standard definition of equilibrium (see Magill and Quinzii).

Remark 2.2 If in addition we assume the following condition due to Martins da-Rocha and Triki [8],

                      (WEQ)                          − ^XZ_i ∩ Ker W(¯q) ⊂ ^X(A(Z_i) ∩ Ker W(¯q)),

                                              i∈I                                                i∈I

then existence of the above defined accounts clearing equilibrium will imply the existence of an equilibrium in the usual sense where we have asset market clearing in portfolios that is ^P_i∈Iz¯_i = 0.

1. 5. 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 2.2 Given the financial structure F = (J,(ξ(j),V ^j)_j∈J ,(Z_i)_i∈I), a portfolio z¯_i ∈ Z_i is said to have no arbitrage opportunities or to be arbitrage-free for agent i ∈ I at the price q ∈ R^J if there is no portfolio z_i ∈ Z_i such that W_F(q)z_i > W_F(q)¯z_i, that is, [W_F(q)z_i](ξ) ≥ [W_F(q)¯z_i](ξ), for every ξ ∈ D, with at least one strict inequality, or, equivalently, if

.

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 arbitrage-free asset prices that are given by the following definition.

Definition 2.3 We say that the asset price q is aggregate arbitrage-free if one of the following equivalent conditions hold ^[2]

.

(ii) There exists such that λ •D w = 0 for all.

Consider the following non-satiation assumption:

Assumption NS (i) For every x¯ ∈ ^Q_i∈I X_i such that ^P_i∈I x¯_i = ^P_i∈I e_i,

(Non-Satiation at Every Node) for every ξ_i ∈ D, there exists x ∈ ^Q_i∈I X_i such that, for each ξ 6= ξ_i, x_i(ξ) = ¯x_i(ξ) and x_i ∈ P_i(¯x); (ii) if x_i ∈ P_i(¯x), then [x_i,x¯_i[⊂ P_i(¯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
   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,(X_i,P_i,e_i)_i∈I;J,(ξ(j),V ^j)_j∈J ,(Z_i)_i∈I].

Define the set of attainable consumptions by

and for each be the projection of X_b on X_i.

We introduce the following assumptions.

Assumption (C) (Consumption Side) For all i ∈ I and all x¯ ∈ ^Q_i∈I X_i,

1. X_i ⊂ R^L is closed, convex and bounded below by x_i.
2. [Continuity] the preference correspondence P_i : ^Q_i∈I X_i → X_i, is lower semicontinuous^[3]and P_i(¯x) is open in X_i (for its relative topology);
3. [Convexity] P_i(¯x) is convex;
4. (Irreflexivity) x¯_i 6∈ P_i(¯x);
5. (Non-Satiation of Preferences at Every Node) if ^P_i∈I x¯_i = ^P_i∈I e_i, for every ξ ∈ D there exists x ∈ ^Q_i∈I X_i such that, for each ξ⁰ 6= ξ, x_i(ξ⁰) = ¯x_i(ξ⁰) and x_i ∈ P_i(¯x);
6. (Survival Assumption) For all i ∈ I, e_i ∈ X_i.

Note that these assumptions on P_i 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, Z_i is closed, convex and contains zero;

(iii) For every asset price q ∈ R^J, for every i ∈ I, W(q)Z_i is a closed subset of R^D;

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 condition holds

(i) Closed Cone  is convex.

Then there exists (¯x,z,¯ p¯) with p¯ 6= 0, such that (¯x,z,¯ p,¯ q¯) is a financial accounts clearing quasiequilibrium.

(b) If we additionally assume that

(ii) Closed Cone  is linear,

then there exists (¯x,z,¯ p¯) with p¯(ξ) 6= 0, for all ξ ∈ D such that (¯x,z,¯ p,¯ q¯) is a financial accounts clearing equilibrium.

It is worth noticing that (WEQ) and (i) in the Theorem are satisfied in the following examples.

Example 3.1 For every i ∈ I,Z_i is closed, convex and contains zero and ^S_i∈IZ_i is a vector space. Consider for example, I = I₁ ∪ I₂ and

                                       ∀i ∈ I₁,                Z_i = {z | z · e₁ ≥ 0,e₁ = (1,0)}

                                      ∀i ∈ I₂,                Z_i = {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.

Example 3.2 (Cass Condition) For every i ∈ I,Z_i is closed, convex and contains zero and for some i₀ ∈ I,Z_i ⊂ Z_i0 and Z_i0 is a vector space. Then ^S_i∈IZ_i = Z_i0 is a vector space and we are in the structure of Example 3.1.

It is also worth noticing that in Condition (i) in Theorem 3.1, taking ^S_i∈IW(¯q)Z_i 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}

1. 2. Other definitions of quasi-equilibrium

Gottardi and Hens consider a two date incomplete markets model without consumption in the first date. Their definition of a quasi-equilibrium, suitably modified by Seghir et al. to include consumption in first date is presented below.

Definition 3.1 A list of strategies and prices x, is a quasiequilibrium of the financial exchange economy (E,F), if p¯ 6= 0,

(a i) for every i ∈ I,x_i ∈ P_i(¯x) and p¯ ₁(x_i −e_i) ≤ V z_i ⇒ p¯(0)•H(x_i(0)−e_i(0))+¯q•Jz_i ≥ 0;

(a ii) for every i ∈ I,p¯ (¯x_i − e_i) = W(¯q)¯z_i;

(a iii) for every ξ ∈ D \ D₀,(x_i(−s),x_i(s)) ∈ P_i(¯x) implies p¯(s) •H (x_i(s)) ≥ p¯(s) •H (¯x_i(s));

(b) ^Xx¯_i = ^Xe_i and ^Xz¯_i = 0.

      i∈I                i∈I                   i∈I

Let

B_ei(p,q) = {(x_i,z_i) ∈ X_i × Z_i | p(0) •H (x_i(0) − e_i(0)) < −q •J z_i;p  ₁(x_i − e_i) ≤ V z_i}

Notice that (a i) of this definition is true if an only if .

Comparing Definitions 2.1 and 3.1 we see that the quasi-equilibrium conditions a( ii), (a iii) and (b) are the same. 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. In particular we show how (a i) in Definition 2.1 is related to (a i) of Definition 3.1.

Lemma 3.1 Under the assumption^[4](F0): ∃ζ_i ∈ A (Z_i) such that V ζ_i >> 0,

(P_i(¯x) × Z_i) ∩ B^˘_i(¯p,q¯) = ∅ ⇒ (P_i(¯x) × Z_i) ∩ B_ei(¯p,q¯) = ∅.

Proof of Lemma 3.1. Suppose on the contrary there is some (x_i,z_i) ∈ (P_i(¯x) × Z_i) ∩

B_ei(¯p,q¯) 6= ∅. Then

p¯(0) •H (x_i(0) − e_i(0)) < −q¯•J z_i

p  ₁(x_i − e_i) ≤ V z_i.

Take ζ_i ∈ AZ_i with V ζ_i >> 0. Then for all t > 0 small enough

p¯(0) •H (x_i(0) − e_i(0)) < −q¯•J z_i − q¯•J (tζ_i) = −q¯(z + tζ_i)

p  ₁(x_i − e_i) << V (z_i + tζ_i)

and clearly z_i + tζ_i ∈ Z_i since ζ_i ∈ AZ_i. This contradicts (P_i(¯x) × Z_i) ∩ B^˘_i(¯p,q¯) = ∅.

4. Proof of existence under additional assumptions.
   1. Additional assumptions

We will first provide a proof of Theorem 3.1 under the following additional assumptions (together with those already made in Theorem 3.1).

Assumption K:

(i) The sets X_i and W(¯q)Z_i for a given q¯∈ R^J, are bounded;

(ii)[Local Non-Satiation] for every x¯ ∈ ^Q_i∈I X_i, for every x_i ∈ P_i(¯x), [x_i,x¯_i) ⊂ P_i(¯x).

(iii) [Strong survival assumption] For every i ∈ I e_i ∈ int X_i.

Then in the next section we will give the proof of Theorem 3.1 in the general case, that is without assuming Assumption K.

1. 2. Preliminary lemma

Before entering the proof of Theorem 3.1 we will state and prove the following:

Lemma 4.1 Let q¯∈ R^J, let W be a the closed convex cone then the following are equivalent:

.

(ii) There exists such that:

hWi ⊂ λ^⊥ := {t ∈ R^D | λ •D t = 0}.

The proof of Lemma 4.1 is standard (see Magill-Quinzii). In the following we let

W = closed convex cone (W(¯q)(∪_I∈IZ_i)).

Let ⁹ BL = {p ∈ RL | ||λ  p|| ≤ 1}. Given as in Lemma 4.1, for each p ∈ BL let

ρ(p) = (ρ(p,ξ))_ξ∈D ∈ R^D with ρ(p,ξ) = 1 − ||λ  p|| for all ξ ∈ D.

Given p ∈ BL and a function γ : BL → R^D, for all i ∈ I define

B_i^γρ(p) = ⁿ(x_i,w_i) ∈ X_i × WZ_i : ∃τ_i ∈ [0,1],p(x_i − e_i) ≤ w_i + τ_iγ(p) + ρ(p)^o,

B^˘_i^γρ(p) = ⁿ(x_i,w_i) ∈ X_i × WZ_i : ∃τ_i ∈ [0,1],p(x_i − e_i)  w_i + τ_iγ(p) + ρ(p)^o.

We state a lemma, which extends a previous result by Da-Rocha and Triki [8], the proof of which is given in the Appendix.

Lemma 4.2        (i) For every ε > 0, there exists a continuous mapping γ : BL → R^D 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∈IZ_i) is convex, then ∀ p ∈ BL,∃ i ∈ I, such that B˘_i^γρ(p) 6= ∅ .

1. 1. 1. The fixed point argument

Let γ : BL → R^D be the continuous mapping associated by Lemma 4.2 to some ε. For

(x,w,p) ∈ ^Q_i∈I X_i × ^Q_i∈I WZ_i × BL, we define the correspondences Φ_i for i ∈ I₀ := {0} ∪ I as follows:

Φ⁰(x,w,p) = ⁿp⁰ ∈ BL | λ  (p⁰ − p)•_L ^X(x_i − e_i) > 0^o,

i∈I

denotes the euclidean norm.

and for every i ∈ I,

                                                                             if (x_i,w_i) ∈ B/  _i^γρ(p) and B^˘_i^γρ(p) = ∅,



            Φ_i(x,w,p) =         B_i^γρ(p)                                                if (x_i,w_i) ∈ B/ _i^γρ(p) and B^˘_i^γρ(p) 6= ∅,

                                )                            if (x_i,w_i) ∈ B_i^γρ(p).

The existence proof relies on the following fixed-point-type theorem due to Gale and MasCollel ([15]).

Theorem 4.1 Let I₀ be a finite set, let C_i (i ∈ I₀) be a nonempty, compact, convex subset of some Euclidean space, let C = ^Q_i∈I0 C_i and let Φ_i (i ∈ I₀) be a correspondence from C to C_i, which is lower semicontinuous and convex-valued. Then, there exists c¯ ∈ C such that, for every i ∈ I₀ either c¯_i ∈ Φ_i(¯c) or Φ_i(¯c) = ∅.

We now show the sets C₀ = ^BL, C_i = X_i ×WZ_i (i ∈ I) and the above defined correspondences Φ_i (i ∈ I₀) satisfy the assumptions of Theorem 4.1.

Claim 4.1 For every c¯:= (¯x,w,¯ p,¯ ) ∈ ^Q_i∈I X_i × ^Q_i∈I WZ_i × BL,

1. Φ_i(¯c) is convex (possibly empty);
2. p¯ 6∈ Φ₀(¯c), and for all i ∈ I,(¯x_i,w¯_i) ∈6 Φ_i(¯c);
3. for every i ∈ I₀, the correspondence Φ_i is lower semicontinuous at c¯.

Proof of Claim 4.1: Let c¯:= (¯x,w,¯ p¯) ∈ ^Q_i∈I X_i × ^Q_i∈I WZ_i × BL be given.

Proof of (i): Clearly Φ₀(¯c) is convex. For every i ∈ I, recalling that P_i(¯x) and WZ_i are convex sets, by Assumption C and F, we have Φ_i(¯c) is a convex set.

Proof of (ii): Clearly, p¯ 6∈ Φ₀(¯c) and (¯x_i,w¯_i) 6∈ Φ_i(¯c) follows from the d efinitions of these sets and the fact that x¯_i ∈6 P_i(¯x) (from Assumption C).

Proof of (iii): We need to show that Φ_i is lower semicontinuous for all i ∈ I₀. Since Φ₀ has an open graph, clearly it is lower semicontinuous. To show lower semicontinuity of Φ_i for i ∈ I, let U be an open subset of X_i × WZ_i such that Φ(¯c) ∩ U =6 ∅ and we will distinguish three cases:

Case(1): (¯x_i,w¯_i) ∈ B/ _i^γρ(¯p) and B˘_i^γρ(¯p) = ∅. Then Φ_i(¯c) = {(e_i,0)} ⊂ U. The set Ω_i = {(x_i,w_i,p) | (x_i,w_i) ∈ B/ _i^γρ(p)} is an open subset of X_i × WZ_i × BL (by Assumptions C and F). To see this, let {(x_i,w_i,p)} be such that (x_i,w_i) ∈ B_i^γρ(p) and (x_i,w_i,p) → (x,w,p). Since for all n,(x_i,w_i) ∈ B_i^γρ(p), there exists τ_i ∈ [0,1] such that

p  (x_i − e_i) ≤ w_i + τ_iγ(p) + ρ(p)

In the limit we have

p  (x − e_i) ≤ w + τγ(p) + ρ(p)

Where τ = lim_n→∞τ_i ∈ [0,1]. Thus (x,w) ∈ B_i^γρ(p).

Thus Ω_i contains an open neighborhood O of c¯. Now, let c = (x,w,p) ∈ O. If B^˘_i^γρ(p) = ∅ then Φ_i(c) = {(e_i,0)} ⊂ U and so Φ_i(c)∩U is nonempty. If B˘_i^γρ(p) 6= ∅ then Φ_i(c) = B_i^γρ(p).

But Assumptions C and F imply that (e_i,0) ∈ X_i×WZ_i, hence  (with τ_i = 0 and noticing that ρ(p) ≥ 0). So {(e_i,0)} ⊂ Φ_i(c) ∩ U which is also nonempty.

Case (2): c¯ = (¯x_i,w¯_i,p¯) ∈ Ω_i := {c = (x_i,w_i,p) : (x_i,w_i) ∈ B/ _i^γρ(p) and B˘_i^γρ(p) =6 ∅}. Then the set Ω_i is clearly open (since its complement is closed).

On the set Ω_i one has Φ_i(c) = B_i^γρ(p). We recall that ∅ 6= Φ_i(¯c)∩U = B_i^γρ(¯p)∩U. We notice that B_i^γρ(¯p) = cl B^˘_i^γρ(¯p) since B^˘_i^γρ(¯p) 6= ∅. Consequently, B^˘_i^γρ(¯p) ∩ U 6= ∅ and we chose a point (x_i,w_i) ∈ B˘_i^γρ(¯p) ∩ U, that is, (x_i,w_i) ∈ [X_i × WZ_i] ∩ U and for some τ_i ∈ [0,1],

p¯ (x_i − e_i)  w_i + τ_iγ(¯p) + ρ(¯p).

Clearly the above inequality is also satisfied for the same point (x_i,w_i) and the same τ_i 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˘_i^γρ(p) ∩ U ⊂ B_i^γρ(p) ∩ U = Φ(c) ∩ U.

Case (3): (¯x_i,w¯_i) ∈ B_i^γρ(¯p). By assumption we have

∅ 6= Φ_i(¯c) ∩ U = B^˘_i^γρ(¯p) ∩ [P_i(¯x) × WZ_i] ∩ 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^˘_i^γρ(p)∩U. Since P_i is lower semicontinuous at c¯ (by Assumption C, there exists an open neighborhood Ω of c¯such that, for every c ∈ Ω, ∅ 6= [P_i(x) × WZ_i] ∩ M, hence

 for every c ∈ Ω.

Consequently, from the definition of Φ_i, we get ∅ 6= Φ_i(c) ∩ U, for every c ∈ Ω.

The correspondence Ψ_i := B^˘_i^γρ ∩ (P_i × WZ_i) is lower semicontinuous on the whole set, 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 ∩ Ψ_i(x,w,p) 6= ∅ hence ∅ 6= U ∩ Φ_i(x,w,p) (since we always have Ψ_i(x,w,p) ⊂ Φ_i(x,w,p)).

Given ε > 0, in view of Claim 4.1, we can apply the fixed-point Theorem 4.1. Thus there exists c¯= (¯x,w,¯ p¯) ∈ ^Q_i∈I X_i ×^Q_i∈I WZ_i ×BL such that, for every i ∈ I₀,Φ_i(¯x,w,¯ p¯) = ∅.

Written coordinatewise, this is equivalent to saying that:

(4.1)                           ∀ p ∈ BL, (λ  p) •_D ^X(¯x_i − e_i) ≤ (λ  p¯) •_D ^X(¯x_i − e_i)

                                                           i∈I                                                       i∈I

(4.2)                     ∀i ∈ I,(¯x_i,w¯_i) ∈ B_i^γρ(¯p) and B˘_i^γρ(¯p) ∩ (P_i(¯x) × WZ_i) = ∅.

1. 1. 2. Properties of (¯x,w,¯ p,¯ q¯).

We first prove that x¯ = (¯x_i)_i∈I satifies the market clearing condition.

Claim 4.2 Pi∈Ix¯i = Pi∈Iei.

Proof of Claim 4.2: Suppose ^P_i∈I(¯x_i − e_i) 6= 0. From the fixed-point assertion (4.1) we

P deduce that (λ                 and ||λ  p¯|| = 1. So

(4.3)p¯) •_L ^X(¯x_i − e_i) > 0.

i∈I

Recalling that (¯x_i,w¯_i) ∈ B_i^γρ(¯p), for all i ∈ I, by the fixed-point assertion (4.2), hence there exists τ¯_i ∈ [0,1] such that

p¯ (¯x_i − e_i) ≤ w¯_i + ¯τ_iγ(¯p) + ρ(¯p).

Summing up over i we get:

p¯^X(¯x_i − e_i) ≤ ^Xw¯_i + (^Xτ¯_i)γ(¯p) + (#I)ρ(¯p).

                                      i∈I                              i∈I                  i∈I

Taking the scalar product with λ we get,

(λ  p¯) •_L ^X(¯x_i − e_i) ≤ λ •_D ^Xw¯_i + (^Xτ¯_i)λ •D γ(¯p) + (#I)λ •D ρ(¯p).

                               i∈I                                          i∈I                  i∈I

On the right hand side, we have λ •D ^P_i∈Iw¯_i = 0 (by Lemma 4.1), λ •D γ(¯p) = 0 (by Lemma 4.2), and ρ(¯p) = 0 (since ||λ  p¯|| = 1). Thus (λ  p¯) •L ^P_i∈I(¯x_i − e_i) ≤ 0, which contradicts (4.3).

Claim 4.3 The following conditions hold:

1. If for some i ∈ I, B˘_i^γρ(¯p) 6= ∅ then (¯x_i,w¯_i) ∈ B_i^γρ(¯p) and B_i^γρ(¯p) ∩ (P_i(¯x) × WZ_i) = ∅;
2. For all;
3. For all i ∈ I,(¯x_i,w¯_i) ∈ B_i^γρ(¯p) and B_i^γρ(¯p) ∩ (P_i(¯x_i) × WZ_i) = ∅.

Proof of Claim 4.3: Proof of (i): The first assertion that (¯x_i,w¯_i) ∈ B_i^γρ(¯p) is a consequence of the Fixed-Point Assertion (4.2). We now show that B_i^γρ(¯p)∩(P_i(¯x_i)×WZ_i) = ∅. Indeed suppose that contains an element (x_i,w_i). Since, we

let (¯x_i,w¯_i) ∈ B^˘_i^γρ(¯p).

Suppose first that x¯_i = x_i, then, from above (x_i,w¯_i) ∈ [P_i(¯x) × WZ_i] ∩ B^˘_i^γρ(¯p), which contradicts the fact that this set is empty by Assertion (4.2). Suppose now that x¯_i 6= x_i, from Assumption C (iii), (recalling that x_i ∈ P_i(¯x)) the set [¯x_i,x_i[∩P_i(¯x) is nonempty, hence contains a point x_i(λ) := (1 − λ)¯x_i + λx_i for some λ ∈ [0,1[. We let w_i(λ) := (1 − λ) ¯w_i+ λw_i and we check that (x_i(λ),w_i(λ)) ∈ B˘_i^γρ(¯p) (since (x_i,w_i) ∈ B_i^γρ(¯p) and (¯x_i,w¯_i) ∈ B^˘_i^γρ(¯p)). Consequently, B^˘_i^γρ(¯p) ∩ (P_i(¯x) × WZ_i) =6 ∅, which contradicts again Assertion (4.2).

Thus

(4.4)                           (¯x_i,w¯_i) ∈ B_i^γρ(¯p) and B_i^γρ(¯p) ∩ (P_i(¯x) × WZ_i) = ∅

Proof of (ii): From Lemma 4.2 (ii), there exists i₀ ∈ I such that. Suppose there exists ξ ∈ D such that p(ξ) = 0. From Claim 4.2, ^P_i∈Ix¯_i = ^P_i∈Ie_i, and from the Non-Satiation Assumption at node ξ (for Consumer i₀) there exists x_i0 ∈ P_i0(¯x) such that

        0                                            0                                                            0

x_i0(ξ ) = ¯x_i0(ξ ) for every ξ 6= ξ; from Assertion (4.2),and, recalling that p¯(ξ) = 0, one deduces that . Consequently,

,

which contradicts Condition 4.4.

Proof of (iii): From Part (ii) of this Claim we have, for all, hence

p¯(ξ) · p¯(ξ) > 0 and (¯p  p >>¯ 0. Taking w¯_i = 0,τ¯_i = 0 and x¯_i = e_i − tp¯, for t > 0 small enough, we get x¯_i ∈ BL(e_i,r) ⊂ X_i (from the Survival Assumption). Then p¯ (¯x_i − e_i) = −t(¯p  p¯)  0 ≤ 0 + ρ(¯p). This shows that (x_i,0) ∈ B˘_i^γρ(¯p) =6 ∅.

Claim 4.4 The following conditions hold:

1. ρ(¯p) = 0
2. If closed cone  is linear then for all i ∈ I,τ¯_iγ(¯p) = 0 and ^P_i∈Iw¯_i = 0.
3. If closed cone  is convex then ||^P_i∈Iw¯ⁱ|| ≤ (#I)||γ(¯p)||.

Proof of Claim 4.4: Proof of (i): We first prove that the modified budget constraints are binding, that is

(4.5)                               p¯ (¯x_i − e_i) = ¯w_i + ¯τ_iγ(¯p) + ρ(¯p),        ∀ i ∈ I

Indeed, suppose that there exists i ∈ I such that

p¯ (¯x_i − e_i) < w¯_i + ¯τ_iγ(¯p) + ρ(¯p)

That is there exist ξ ∈ D such that

p¯(ξ) •H (¯x_i(ξ) − e_i(ξ)) < w¯_i(ξ) + ¯τ_iγ(¯p)(ξ) + ρ(¯p)

But by Claim 4.2, ^P_i∈Ix¯_i = ^P_i∈Ie_i and by the nonsatiation assumption C (v) for consumer i, there exists x_i ∈P_i(¯x) such that x_i(ξ⁰) = ¯x_i(ξ⁰) for every ξ⁰ 6= ξ. Consequently, we can choose x ∈ [x_i,x¯_i) close enough to x¯_i so that(x,w¯_i) ∈ B_i^γρ(¯p). But, from the local non-satiation (Assumption K (ii)), [x_i,x¯_i) ⊂ P_i(¯x). Consequently, B_i^γρ(¯p) ∩ (P_i(¯x) × WZ_i) 6= ∅ which contradicts Claim 4.3. This ends the proof of Equation (4.5).

Summing up over i ∈ I in the equalities (4.5) and using the market clearing condition (Claim 4.2) we get:

(4.6)                                        0 = ^Xw¯_i + (^Xτ¯_i)γ(¯p) + (#I)ρ(¯p)

                                                  i∈I                  i∈I

Taking above the scalar product with λ yields:

0 = λ •D (^Xw¯_i) + (^Xτ¯_i)λ •D γ(¯p) + (#I)λ •L ρ(¯p)

                                               i∈I                    i∈I

But λ •D (^P_i∈Iw¯_i) = 0 since ^P_i∈Iw¯_i ∈ W ⊂ < W >⊂ λ^⊥ by Lemma 4.1. Moreover 0 = λ •D γ(¯p) by Lemma 4.2. Consequently ρ(p) = 0.

Proof of (ii): From Assertion (4.6) and the fact that ρ(p) = 0 we get

(4.7)                                                  ^Xw¯_i = −(^Xτ¯_i)γ(¯p).

                                                      i∈I                      i∈I

Taking the scalar product with ^P_i∈Iw¯_i on both sides and using the fact the closed cone  is linear in Lemma 4.2 we get ^P_i∈Iw¯_i = 0. Again from Assertion

4.7 we get

(^Xτ¯_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¯_iγ(¯p) = 0.

Proof of (iii): From Assertion 4.7, taking scalar product on both sides with ^P_i∈Iw¯_i on both sides and recalling that τ_i ∈ [0,1] we get ^P_i∈Iτ¯_i ≤ #I hence

||^Xw¯_i|| ≤ (#I)||γ(¯p)||          i∈I

If closed cone  is a linear space and e_i ∈ int X_i for all i ∈ I then (¯x,z,¯ p,¯ q¯) is an accounts clearing equilibrium as in Theorem 3.1 as consequence of Claim 4.2, Claim 4.3 and Claim 4.4 (i and ii).

If however, closed cone  is a convex set then we can go to the limit as ε goes to zero and verify that the limit of the sequence will be quasi-weak-equilibrium.

1. 1. 3. Limit argument when converges to zero and quasi-weak-equilibrium.

Since e_i ∈ int X_i, let ε be such that B(e_i,ε) ⊂ X_i. Hence there exists a sequence e_iⁿ∈

 and eⁿ_i converges to e_i. 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 , to make the dependence on n explicit. Since BL and each of X_i and WZ_i are compact, without any loss of generality we can assume that the sequence converges to some element (x,¯¯ w,¯¯ p¯¯). Let z¯¯_i be such that w¯¯_i = W(¯q)z¯¯_i.

We first recall the following sets:

B_i(p,q) = {(x_i,z_i) ∈ X_i × Z_i : p(x_i − e_i) ≤ W(q)z_i}.

B^˘_i(p,q) = {(x_i,z_i) ∈ X_i × Z_i : p(x_i − e_i) << W(q)z_i}.

Note that we have

Claim 4.5 (i) γ(¯pⁿ) → 0.

2. ρ(p¯¯) = 0, that is kλ  p¯¯k = 1.
3. Pi∈Ix¯¯i = Pi∈Iei.
4. ^P_i∈IWz¯¯_i = 0.