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Financial Equilibrium with differential Information: a
Theorem of generic Existence
Lionel Boisdeffre
To cite this version:
Lionel Boisdeffre. Financial Equilibrium with differential Information: a Theorem of generic Existence. 2017. �hal-01871571�
Centre d’Analyse Théorique et de
Traitement des données économiques
CATT-UPPA
UFR Droit, Economie et Gestion Avenue du Doyen Poplawski - BP 1633 64016 PAU Cedex
CATT WP No. 12
August 2017
FINANCIAL EQUILIBRIUM WITH
DIFFERENTIAL INFORMATION:
A THEOREM
OF GENERIC EXISTENCE
Financial equilibrium with differential information: a theorem of generic existence
Lionel de Boisde¤re,1
(July 2017)
Abstract
We propose a proof of generic existence of equilibrium in a pure exchange econ-omy, where agents are typically asymmetrically informed, exchange commodities, on spot markets, and securities of all kinds, on incomplete …nancial markets. The proof does not use Grasmanians, nor di¤erential topology (except Sard’s theorem), but good algebraic properties of assets’payo¤s, whose spans, generically, never col-lapse. Then, we show that an economy, where the payo¤ span cannot fall, admits an equilibrium. As a corollary, we prove the full existence of …nancial equilibrium for numeraire assets, extending Geanakoplos-Polemarchakis (1986) to the asymmetric information setting. The paper, which still retains Radner’s (1972) standard perfect foresight assumption, is also a milestone to prove, in a companion article, the exis-tence of sequential equilibrium when the classical rational expectation assumptions, along Radner (1972, 1979), are dropped jointly, that is, when agents have private characteristics and beliefs and no model to forecast prices.
.
Key words: sequential equilibrium, temporary equilibrium, perfect foresight, exis-tence, rational expectations, …nancial markets, asymmetric information, arbitrage. JEL Classi…cation: D52
1 INSEE, Paris, and Catt-UPPA (Université de Pau et des Pays de l’Adour),
France. University of Paris 1-Panthéon-Sorbonne, 106-112 Bd. de l’Hôpital, 75013 Paris. Email address: lionel.de.boisde¤re@wanadoo.fr
1 Introduction
This paper proposes a non standard proof of the generic existence of equilib-rium in incomplete …nancial markets with di¤erential information. It presents a two-period pure exchange economy, with an ex ante uncertainty over the state of nature to be revealed at the second period. Asymmetric information is represented by private …nite subsets of states, which each agent is correctly informed to contain the realizable states. Finitely many consumers exchange consumption goods on spot markets, and, unrestrictively, assets of any kind on incomplete …nancial markets. Those permit limited transfers across periods and states. Agents have endowments in each state, preferences over consumptions, possibly non ordered, and no model to forecast prices. Generalizing Cass (1984) to asymmetric information, De Boisde¤re (2007) shows the existence of equilibrium on purely …nancial markets is character-ized, in this setting, by the absence of arbitrage opportunities on …nancial markets, a condition which can be achieved with no price model, along De Boisde¤re (2016), from simply observing available transfers on …nancial markets.
When assets pay o¤ in goods, equilibrium needs not exist, as shown by Hart (1975) in the symmetric information case. His example is based on the collapse of the span of assets’payo¤s, that occurs at clearing prices. In our model, an additional problem arises from di¤erential information. Financial markets may be arbitrage-free for some commodity prices, and not for others, in which case equilibrium cannot exist. We show this problem vanishes owing to a good property of payo¤ matrixes. Attempts to resurrect the existence of equilibrium with real assets noticed that the above "bad " prices could only occur exceptionally, as a consequence of Sard’s theorem. These attempts include Mc Manus (1984), Repullo (1984), Magill & Shafer
(1984, 1985), for potentially complete markets (i.e., complete for at least one price), and Du¢ e-Shafer (1985, 1986), for incomplete markets. These papers apply to sym-metric information, build on di¤erential topology arguments, and demonstrate the generic existence of equilibrium, namely, existence except for a closed set of measure zero of economies, parametrized by the assets’payo¤s and agents’endowments.
The current paper proposes to show generic existence di¤erently, under milder assumptions and for economies parametrized by assets’payo¤s only (in a restricted sense). This result applies to both potentially complete or incomplete markets, to ordered or non transitive preferences, and to symmetric or asymmetric information. The proof does not use Grassmanians, but properties of payo¤ matrixes and lower semicontinuous correspondences built upon them. It yields well-behaved nor-malized price anticipations at equilibrium, which serve to prove, in a companion paper, the full existence of sequential equilibrium in an economy where both Rad-ner’s (1972, 1979) rational expectations are dropped. That is, agents endowed with private beliefs and characteristics can no longer forecast equilibrium prices perfectly. So the paper be self-contained, we resume some techniques of De Boisde¤re (2007). Finally, we derive the full existence of equilibria for numeraire assets as a corollary, using a di¤erent and asymptotic technique. This latter result extends Geanakoplos-Polemarchakis’(1986) to the asymmetric information setting.
The paper is organized as follows: Section 2 presents the model; Section 3 states and proves the existence Theorem; Section 4 shows the existence of equilibria for numeraire assets; an Appendix proves a technical Lemma.
2 The model
We consider a pure-exchange economy with two periods,t 2 f0; 1g, and an uncer-tainty, at t = 0, on which state of nature will randomly prevail, att = 1. Consumers exchange consumption goods, on spots markets, and assets of all kinds, on typically incomplete …nancial markets. The sets, I, S, H and J, respectively, of consumers, states of nature, consumption goods and assets are all …nite. The non random state at the …rst period (t = 0) is denoted bys = 0 and we let 0 := f0g[ , for every subset,
, of S. Similarly, l = 0 denotes the unit of account and we let H0 := f0g [ H. 2.1 Markets and information
Agents consume or exchange the consumption goods, h 2 H, on both periods’ spot markets. Att = 0, each agent,i 2 I, receives privately some correct information signal, Si S (henceforth set as given), that tomorrow’s true state will be in Si. We assume costlessly that S = [i2ISi. Thus, the pooled information set, S :=
\i2ISi, always containing the true state, is non-empty, and S= S under symmetric information. A collection of #I subsets of S, whose intersection is non-empty, is called an (information) structure, which agents may possibly re…ne before trading. Since no state from the set SnS may prevail, we assume that each agent, i 2 I, forms an idiosyncratic anticipation, pi := (pis) 2 RS++inS of tomorrow’s commodity prices in such states, ifSi6= S. Yet, to alleviate subsequent de…nitions and notations, we will take pis= pjs= ps (henceforth given), for any two agents (i; j) 2 I2 such that
s 2 Si\ SjnS. This simpli…cation does not change generality. Thus, we may restrict tomorrow’s price set to P := fp := (ps) 2 (RH+)S : kpsk 6 1; 8s 2 S; ps = ps; 8s 2 SnSg, and we refer to any pair, ! := (s; ps) 2 S RH, as a forecast, whose set is denoted .
Agents may operate transfers across states inS0 by exchanging, att = 0, …nitely many assets, j 2 J, which pay o¤, at t = 1, conditionally on the realization of fore-casts,! 2 . We will assume that#J 6 #S, so as to cover incomplete markets. These conditional payo¤s may be nominal or real or a mix of both. They are identi…ed to the continuous map, V : ! RJ, relating forecasts,! := (s; p) 2 to rows,V (!) 2 RJ, of assets’cash payo¤s, delivered if state s and price p obtain. Thus, for every pair ((s; p); j) 2 J, the jth asset delivers the vector v
j(s) := (vhj(s)) 2 RH
0
of payo¤s if state sprevails, namely,v0
j(s) 2 Runits of account, and a quantity,vhj(s) 2 R, of each good h 2 H, whose cash value at price pisvj(!) := vj0(s) +
P
h2H phvjh(s). For allj 2 J, we let Vj := (vj(!))!2 be the column vector of the asset’s payo¤s across forecasts.
For p := (ps) 2 P, we let V (p) be the S J matrix, whose generic column is de-noted Vj(p), and whose generic row is V (p; s):=V (s; ps) (for s 2 S); we letVS(p) be its truncation to Sand < VS(p) > be the span of VS(p) inRS.
At asset price,q 2 RJ, agents may buy or sell unrestrictively portfolios,z = (z j) 2
RJ, for q z units of account at t = 0, against the promise of delivery of a ‡ow,
V (!) z, of conditional payo¤s across forecasts, ! 2 . The model is dispensed with the so-called "regularity condition" onV, otherwise used in generic existence proofs. For notational purposes, we let V be the set of (S H0) J payo¤ matrixes, as de…ned above, and M0 be the set of matrixes having zero payo¤s in any good and any state s 2 SnS(i.e., assets are nominal and pay o¤ in realizable states only). For every > 0, we let M := fM02 M0; kM0k 6 g and V := fM 2 V : kM-V k 6 g. The sets M0 and V are equiped with the same notations as above (de…ned for V 2 V). Non restrictively along De Boisde¤re (2016), we assume that, before trading, agents have always inferred from markets the information needed to preclude arbitrage.
We point out the good algebraic properties of payo¤ and …nancial structures, summarized in the following Claim 1, which later serve to circumvent the possible fall in rank problems a la Hart (1975). Claim 1 shows that, except for a closed negligible set of assets’payo¤s, no such fall in rank may occur. Then, we will show that an economy, where the payo¤ span can never collapse, admits an equilibrium. Claim 1Given ( ; M; p; ") 2 ]0; 1[ V P ]0; [, we let Vp:= f M 2 V : rank(MS(p)) =
#J g, := fM 2 V : M 2 Vp; 8p 2 P g and Bo(M; p; ") := f(M0; p0) 2 V P :
kM0 M k + kp0 pk < "g and, similarly, Bo(p; ") := fp0 2 P : kp0-pk < "g be given. The following Assertions hold:
(i)if M 2 Vp; 9 "0 > 0 : (M0; p0) 2 B0(M; p; "0) =) M02 Vp0 ; (ii)if M =2 Vp; 9 M"2 V"p; (iii) 9 M02 M0: 8(M0; p0) 2 V P; 9 2 R; (M0+ M0) 2 Vp 0 ; (iv)along Assertion (iii), 9 2 R : (V + M0) 2 ;
(v) 8M02 , @((z
i); p) 2 (RJ)Inf0g P :
P
i2I zi= 0 and M0(p; si) zi> 0, 8(i; si) 2 I Si. And we say that the payo¤ and information structure, [M0; (S
i)], is arbitrage-free;
(vi)the above set, := fM 2 V : M 2 Vp; 8p 2 P g, is non empty and open in V ;
(vii)the set, fMS(p) : M 2 V ; p 2 P; M =2 Vpg, of S J payo¤ matrixes, which fall in rank, is closed and negligible.
Proof
Assertions(i)-(ii)are well-known. Their proofs are obvious from the de…nitions and the continuity of the scalar product, and left to the reader.
Assertion(iii): letM02 M0, with full column rank, and(M0; p0) 2 V P be given. To simplify notations, we assume w.l.o.g. thatS= S. It is clear that, for > 0
small enough, either (M0+ M0) 2 Vp0 or
(M0 M0) 2 Vp0. From Assertion
the latter result holds, ifM02 Vp0
. If not, let (ek)16k6K be an orthonormal basis of A := fv 2 RJ : M0(p0)v = 0g and (ek)K<k6#J be an orthonormal basis of A?. By construction, the systemsfM0(p0)ekgK<k6#J,f(M0(p0)+ M0(p0))ekgK<k6#J and
f(M0(p0) M0(p0))ekgK<k6#J are all linearly independent, for > 0small enough, so that f(M0(p0) + M
0(p0))ekg16k6#J and f(M0(p0) M0(p0))ekg16k6#J are also linearly independent by construction. Assertion(iii)follows.
Assertion (iv) Assume, by contraposition, that Assertion (iv) fails, namely:
8n 2 N; 9 pn 2 P; (V + M0=n) =2 Vpn. Then, the sequence fpng may be assumed to converge in a compact set, say to p 2 P. From Assertions (i)-(iii), there exist > 0 and " > 0, small enough, such that (V + M0) 2 Vp
0
for every
p0 2 Bo(p ; " ), which contains an ending section of the sequence fpng, say
fpngn>N. We let the reader check, as tedious but straightforward from con-tinuity arguments, the fact that (V + M0=n)(pn) tends to V (p ) and has same rank for big enough integers, and from the arguments of the proof of Assertion
(iii), that the latter relations, (V + M0) 2 Vpn and (V + M0) 2 Vp , imply, for
n> N large enough,(V + M0=n) 2 Vpn, in contradiction with the above. Assertion(v): letM02 and((zi); p) 2 (RJ)I P be given, such thatP
i2I zi= 0 and M0(p; si) zi > 0 for every (i; si) 2 I Si. It follows from the fact that M0(p) has full rank, that the latter relations imply(zi) = 0, proving Assertion(v). Asssertion (vi): let M0 2 , a non-empty set from Assertion (iv), be given. Assume, by contraposition:8n 2 N; 9(Mn; pn) 2 V P; kM0 Mnk < 1=n; Mn2 V= pn. By the same token as above, we letlimn!1pn= p 2 P. From Assertion(i), there exists" > 0, small enough, such thatM " 2 Vp" for every(M "; p") 2 Bo(M0; p ; " ), which contains an ending section of f(Mn; pn)g, say f(Mn; pn)gn>N. The latter relations imply,Mn2 Vpn, forn> N, contradicting the former.
To simplify, we assume throughout that S = S and, at …rst, that#S = #J. We letf : V (RH)S RJ ! RJ be de…ned (with model’s notations) by f (V0; p0; ) :=
P
j2J jVj0(p0), for every(V0; p0; := ( j)) 2 V (RH)S RJ. From Sard’s theorem (see, e.g., Milnor, 1997, p. 16), let C be the set of critical points of f (i.e., such that rank(df(V0;p0; )) < #J) then, f (C), called the set of singular values,
has measure zero. If #J < #S, we apply the same arguments as above to any
subset T S of #J states and the corresponding truncated prices and J J
matrixes. The union of all singular values so obtained, hence, also the closed (from Assertion (vi)) set fM(p) : M 2 V ; p 2 P; M =2 Vpg, have measure zero.
2.2 The agent’s behaviour and the concept of equilibrium
Each agent, i 2 I, receives an endowment, ei := (eis), granting the conditional commodity bundles, ei02 RH+ att = 0, and eis2 RH+, in each expected state, s 2 Si, if it prevails. Given prices and expectations, $ := ((p0; q); p := (ps)) 2 RH+ RJ P, the generic ith agent’s consumption set is(RH
+)S
0
i and her budget set is:
Bi($; V ) := f (x := (xs); z) 2 (RH+)S 0 i RJ : 2 6 6 4 p0 (x0 ei0)6 q z ps (xs eis)6 V (p; s) z; 8s 2 Si g.
Each consumer, i 2 I, has preferences, i, represented, for each x 2 (RH+)S
0 i, by
the set, Pi(x) := fy 2 (RH+)S
0
i : x i yg, of consumptions which are strictly preferred
to x. In the above economy, denoted E = f(I; S; H; J); V; (Si)i2I; (ei)i2I; ( i)i2Ig, agents optimise their consumptions in the budget sets. So the concept of equilibrium:
De…nition 1 A collection of prices,$ := ((p0; q); p := (ps)) 2 RH+ RJ P, and strategies,
[(xi; zi)] 2 i2IBi($; V ), is an equilibrium of the economy, E, if the following holds:
(b) Pi2I(xis eis) = 0; 8s 2 S0;
(c) Pi2Izi= 0.
The economy, E, is called standard under the following conditions: Assumption A1 (monotonicity): 8(i; x; y) 2 I (RHSi0
+ )2; (x6 y; x 6= y) ) (x iy); Assumption A2 (strong survival): 8i 2 I; ei2 RHS
0 i
++ ;
Assumption A3: 8i 2 I, i is lower semicontinuous convex-open-valued and such that x ix + (y x), whenever (x; y; ) 2 R
HS0 i
+ Pi(x) ]0; 1].
3 The existence Theorem and proof
Theorem 1 Generically in the set of assets’ payo¤s, in cash value at market prices, and in realizable states only, a standard economy, E, admits an equilibrium. From Claim 1-(vii) and its proof, Theorem 1 will be demonstrated if we show that equilibrium exists if the economy’s …nancial structure is represented by an arbitrary element V 2e := fM 2 V : M 2 Vp
; 8p 2 P g 6= ?, for some > 0. Hereafter, we set as given such elements, > 0andV 2e , and show they yield an equilibrium.
3.1 Bounding the economy
For every (i; $ := ((p0; q); p)) 2 I P0 P, we let:
Bi($; eV ) := f (x; z) 2 (RH+)S 0 i RJ : 2 6 6 4 p0 (x0 ei0)6 q z + 1 ps(xs eis)6 eV(p; s) z + 1; 8s 2 Si g; A($; eV ) := f[(xi; zi)] 2 i2IBi($; eV ) : P i2I(xis-eis) = 0; 8s 2 S0; P i2Izi= 0g. Lemma 1 9r > 0 : 8$ 2 P0 P; 8[(xi; zi)] 2 A($; eV ); k[(x i; zi)]k < r Proof : see the Appendix.
Lemma 1 permits to bound the economy. Thus, we de…ne (along Lemma 1), for every$ := ((p0; q); p) 2 P0 P, the following convex compact sets:
Xi:= fx 2 (RH+)S
0
i: kxk 6 rg; Z := fz 2 RJ: kzk 6 rgandA($) := A($; eV )\( i
2IXi Z). 3.2 The existence proof
For every i 2 I and every$ := ((p0; q); p) 2 P0 P, we let:
B0
i($) := f(x; z) 2 Xi Z : p0(x0 ei0)6 q z + (p0;q) and ps(xs eis)6 eV(p; s) z + (s;ps); 8s 2 Sig;
B00
i($) := f(x; z) 2 Xi Z : p0(x0 ei0) < q z + (p0;q) and ps(xs eis) < eV (p; s) z + (s;ps); 8s 2 Sig,
where (p0;q):= 1 k(p0; q)k, (s;ps):= 1 kpsk, 8s 2 S and (s;ps):= 0, 8s 2 SnS.
Claim 2 For every (i; $ := ((p0; q); p)) 2 I P0 P, Bi00($) 6= ?.
Proof Let i 2 I and $ := ((p0; q); p) 2 P0 P be given. From Assumption A2, we may choose x 2 Xi, which meets all budget constraints, and with a strict inequality in each state s 2 Si, such that ps 6= 0. If p0 6= 0, or (p0; q) = 0, then, (x; 0) 2 Bi00($). Finally, ifp0= 0 andq 6= 0, then, forN 2 Nbig enough,(x; q=N ) 2 Bi00($).
Claim 3 For all (i; (p0; q); p) 2 I P0 P, B00i is lower semicontinuous.
Proof Let (i; $ := ((p0; q); p)) 2 I P0 P be given. The convexity of Bi00($) is obvious and implies, from Claim 2, B0
i($) = Bi00($). From the relation V 2 Ve p and Claim 1, B00
i is lower semicontinuous, as standard, for having a local open graph. Claim 4 For every (i; (p0; q); p)) 2 I P0 P, Bi0 is upper semicontinuous.
Proof Let (i; $ := ((p0; q); p)) 2 I P0 P be given. Bi0 is (as standard) upper semicontinuous at$, for having a closed graph in a compact set.
We introduce additional …ctious agents for markets and a reaction correspon-dence for each agent, de…ned on the convex compact set, := P0 P ( i2IXi Z). Thus, we let, for each i 2 I and every := ($ := ((p ; q); p); (x; z) := [(x; z)]) 2 :
0( ) := f[(p00; q0); p] 2 P0 P : P s2S0(p0s-ps) P i2I(xis-eis) + (q0-q)) P i2Izi> 0g; i( ) := 8 > > < > > : B0 i($) if (xi; zi) =2 Bi0($) Bi00($) \ Pi(xi) Z if (xi; zi) 2 Bi0($) 9 > > = > > ; ;
Claim 5 For each i 2 I [ f0g, i is lower semicontinuous.
Proof The correspondences 0is lower semicontinuous for having an open graph. We recall from Claim 1 that i( )will not yield any fall in rank problem in budget sets, since in all cases, V 2 Ve p, and that agents’anticipations will never vary along De Boisde¤re (2016), also from Claim 1, since markets always preclude arbitrage.
Assume that (xi; zi) =2 Bi0($). Then, i( ) = Bi0($).
Let V be an open set in Xi Z, such that V \ B0i($) 6= ?. It follows from the convexity ofB0
i($) and the non-emptyness of the open setBi00($)thatV \ Bi00($) 6= ?. From Claims 1 and 3, there exists a neighborhood U of $, such that V \ B0
i($0)
V \ Bi00($0) 6= ?, for every $02 U.
SinceBi0($)is nonempty, closed, convex in the compact setXi Z, there exist two open setsV1 andV2 inXi Z, such that(xi; zi) 2 V1,B0i($) V2 andV1\ V2= ?. From Claims 1 and 4, there exists a neighborhood U1 U of ($), such that Bi0($0) V2, for every $02 U
1. Let W = U1 ( j2IWj), where Wi:= V1 and Wj:= Xj Z, for every
j 2 Infig. Then,W is a neighborhood of , such that i( 0) = Bi0($0), and, from above,
V \ i( 0) 6= ?, for every 0:= ($0; (x0; z0)) 2 W. Thus, i is lower semicontinuous at . Assume that (xi; zi) 2 Bi0($), i.e., i( ) = Bi00($) \ Pi(x) Z.
Lower semicontinuity is immediate if i( ) = ?. Assume i( ) 6= ?. We recall that
a local open graph. As corollary & from Claim 1, the correspondence ($0; (x0; z0)) 2
! B00
i($0) \ Pi(x0i) Z Bi0($0) is lower semicontinuous at . Then, from Claim 1 and the latter inclusions, i is lower semicontinuous at .
Claim 6 There exists := ($ := ((p0; q ); p ); [(xi; zi)]) 2 , such that:
(i) 8((p0; q); p) 2 P0 P; P s2S0 (ps ps) P i2I (xis eis) + (q -q) P i2Izi > 0; (ii) 8i 2 I; (xi; zi) 2 B0i($ ) and Bi00($ ) \ Pi(xi) Z = ?.
Proof Quoting Gale-Mas-Colell, 1975-79 [9,10]: “Given X = m
i=1Xi, where Xi is a non-empty compact convex subset of Rn, let '
i: X ! Xi be m convex (possibly empty) valued correspondences, which are lower semicontinuous. Then there exists
x inX such that for each i eitherxi2 'i(x) or'i(x) = ?”.
The correspondences, 0: ! P0 P, i : ! Xi Z (for each i 2 I) satisfy the conditions of the above theorem and yield Claim 6.
Claim 7 Pm
i=1zi = 0.
Proof Assume, by contraposition, that Pm
i=1zi 6= 0. Then, from Claim 6-(i),
(p0 p0) Pi2I (xi0-ei0) + q Pmi=1zi 6 q
Pm
i=1zi, for every (p0; q) 2 P0, which implies
q Pmi=1zi > 0 and (p0;q)= 0. From Claim 6-(ii), the relations (xi; zi) 2 B0i($ ) hold,
for eachi 2 I, whose budget constraint in states = 0isp0 (xi0 ei0)6 q zi. Adding them up yields p0
P
i2I (xi0 ei0)6 q
Pm
i=1zi < 0, which contradicts Claim 6-(i). Claim 8 Pi2I (xis eis) = 0; 8s 2 S0 := \iSi0.
Proof Lets 2 S0be given, says 2 S, and assume thatPi2I (xis eis) 6= 0. Applying Claim 6 to good prices yields: ps Pi2I(xis-eis) > 0and (s;ps)= 0. Then, added budget
constraints in Claims 6-7 yield: 0 < ps Pi2I (xis-eis)6 eV(s; ps)
P
Claim 9 (x ; z ) := [(xi; zi)] 2 A($ ), hence, k(x ; z )k < r.
Proof Claim 9 follows immediately from Claims 6-7-8 and Lemma 1 above.
Claim 10 For each i 2 I, (xi; zi)is optimal in B0 i($ ).
Proof Let i 2 I be given. Assume, by contraposition, there exists (xi; zi) 2
B0
i($ ) \ Pi(xi) Z. From Claim 9, the relations kxik < r and kzik < r hold and, from Assumption A3, the relations kxik < r and kzik < r may be assumed.
From Claim 3, there exists (x0
i; zi0) 2 Bi00($ ) Bi0($ ). By construction, (xni; zin) :=
[n1(x0
i; zi0) + (1 n1)(xi; zi)] 2 Bi00($ ), for every n 2 N. From Assumption A3, (xNi ; ziN) 2
Pi(xi) Z holds, for N 2 N big enough. Hence, (xNi ; zNi ) 2 Bi00($ ) \ Pi(xi) Z, which contradicts Claim 6-(ii).
Claim 11 (p0;q)= 0, i.e., k(p0; q )k = 1, and (s;ps)= 0, i.e., kpsk = 1, 8s 2 S.
Hence, B0
i($ ) = Bi($ ; eV ), for every i 2 I, where:
Bi($ ; eV ) := f (x := (xs); z) 2 (RH+)S 0 i RJ: * p0 (x0 ei0)6 q z ps (xs eis)6 eV(p ; s) z; 8s 2 Si g.
Proof Let (i; s) 2 I S0 be given, say s = 0, the proof being the same for s 2 S.
From Claim 6, the relation p0(xi0 ei0) 6 q zi + (p0;q) holds. Assume, by
con-traposition, that p0(xi0 ei0) < q zi + (p0;q). From Claim 9, kxik < r, and, from
Assumptions A1-A3, there exists xi2 Pi(xi) (di¤ering from xi in xi0 only), close to
xi so that p0(xi0-ei0)6 q zi + (p0;q). This contradiction to Claim 10 insures that
p0(xi0 ei0) = q zi + (p0;q) holds for each i 2 I. Then, Claim 9 yields the relations:
0 = ps Pi2I(xis-eis) = q
P
i2Izi + #I (p0;q)= #I (p0;q).
Proof The collection(p0; q ; p ; eV ; [(xi; zi)])meets all Conditions of De…nition 1 of equilibrium of an economy, whose …nancial structure is Ve. Theorem 1 is proved.
Remark Theorem 1 may surprise. It reduces set of parameters to assets’ cash payo¤s (instead of payo¤s and endowments); it applies to asymmetric information and non-odered preferences; it yields normalized (instead of unknown) spot prices at equilibrium; it uses simple nominal asset techniques of proof. The reason for this is the good behaviour of payo¤ matrixes under Claim 1. In fact, the non-semicontinuity of demand correspondences, that may result from a fall in rank problem a la Hart (1975), is not binding. It can always be circumvented, owing to Claim 1-(vii).
4 The existence Theorem with numeraire assets
We consider an economy, where assets only pay in the same (bundle of) commodi-ties, e 2 RH
+ (we let kek = 1), in any state. These assets are referred to as numeraire. The economy is in anything alike that of Section 2, but the fact that there exists a given S J matrix, V, of payo¤s in numeraire, e, across states. This matrix, V, can also be written as an element of the set of (S H0) J payo¤ matrixes. Agents and markets have the same charateristics as above, and we resume all notations and assumptions of Section 2. Moreover, agents’preferences are now represented by continuous, strictly concave, strictly increasing functions, ui: Xi! R, for eachi 2 I. From Claim 12 and Theorem 1, above, for everyn 2 N, there exists an equilibrium,
Cn:= (pn
0; qn; pn:= (pns); eVn; (xn; zn) := [(xni; zin)]), for some payo¤ matrix Ven2 V1=n along Claim 1. From compactness, we may assume the price sequence,f(pn
0; qn; pn := (pns))g, converges to some (p0; q ; p := (ps)) 2 P0 P, such that k(p0; q )k = 1 and kpsk = 1, for each s 2 S, whilef eVng converges to V.
Without an additional assumption, nothing prevents the fall to zero of the value of the numeraire (ps e), in some state s 2 S, and a subsequent arbitrage problem. This is resolved by assuming that agents’ utilities are separable, that is, for each
i 2 I, there exist continuous utility indexes, us
i : R2+ ! R (for s 2 Si), such that
ui(x) = P s2Si u s i(x0; xs), for everyx 2 (RH+)S 0
i. Moreover, we assume, non restrictively,
that agents’signals,(Si), embed the information markets reveal, along De Boisde¤re (2016), therefore that the payo¤ and information structure,[V; (Si)], is arbitrage-free, along the latter paper’s de…nition. To shorten exposition, but w.l.o.g., we …nally assume that the S J payo¤ matrix,VS, has full column rank.2
The above equilibrium sequence,fCng, satis…es the following properties. Lemma 2The following Assertions hold :
(i) 8(n; i; s) 2 N I S0, xn is2 [0; E]H, where E := max(s;h)2S0 H X i2I eh is;
(ii)it may be assumed to exist (x ; z ) := [(xi; zi)] = limn!1[(xni; zin)];
(iii)for each s 2 S0, X i2I
(xis eis) = 0 and, moreover,
X
i2I
zi = 0.
Proof Assertion (i) is standard, from market clearance conditions of equilib-rium. Assertion (ii): the fact that the sequencef(xn; zn)g is bounded, hence may be assumed to converge, results from Lemma 1 (see the Appendix). And Assertion(iii)
results from the market clearance conditions on fCng, passing to the limit. And we show the following full existence Theorem.
Theorem 2 The above collection, C := (p0; q ; p ; (x ; z )), of prices, expectations & strategies is an equilibrium of the numeraire asset economy with payo¤ matrix V. Proof Let us de…ne C := (p0; q ; p ; (x ; z )) as above. From Lemma 2-(ii)-(iii), C meets Conditions(b)-(c) of De…nition 1 of equilibrium. Thus, it su¢ ces to show that
the relations, [(xi; zi)] 2 i2IBi($ ), of Section 2 (where $ := ((p0; q ); p ) 2 P0 P), and De…nition 1-(a)hold.
Let i 2 I be given. From the de…nition, the relations pn0 (xni0 ei0)6 qnzin hold, for alln 2 N, and, yieldp0(xi0-ei0)6 q zi, in the limit. Similarly, and from standard continuity arguments, the relationsxi 2 Xi andps(xis eis)6 V (p ; s) zi also hold for each s 2 Si. Hence, [(xi; zi)] 2 i2IBi($ ).
We now assume, by contraposition, that C fails to meet Condition (a)of
De…n-ition 1, that is, there exist i 2 I, (x; z) 2 Bi($ ) and " 2 R++, such that:
(I) " + ui(xi) < ui(x).
We may assume: (II) 9 ( ; M) 2 R2
++: xs2 [ ; M]H; 8s 2 Si.
Indeed, the upper bound,M, exists from the de…nition ofxand, for 2]0; 1]small enough, the strategy(x ; z ) := ((1 )x+ ei; (1 )z) 2 Bi($ )meets both relations(I) and(II), along Assumption A1 and the continuity of the mapping 2 [0; 1] 7! ui(x ). So, relation (II) may be assumed. Then, it is immediate from the relations (I)-(II)
and (x; z) 2 Bi($ ), from Lemma 2, Assumptions A2-A3 and continuity arguments, that we may also assume there exists 2 R++, such that:
(III) p0(x0 ei0)6 q z and ps(xs eis)6 + V (p ; s) z, 8s 2 Si.
From relations(I)-(II)-(III), we may also assume there exists 0 2]0; [, such that:
(IV ) p0(x0 ei0)6 0 q z and ps(xs eis)6 0+ V (p ; s) z, 8s 2 Si.
Indeed, the above assertion is obvious, from relations(III), ifp0(x0 ei0) < q z. Assume that p0(x0 ei0) = q z. If p0= 0, then, q 6= 0, and relations(IV ) hold if we replacez byz q =N, forN 2 Nbig enough. Ifp 6= 0and x 6= 0, the desired assertion
results from Assumption A1 and above. Otherwise, q z = p0 ei0< 0, and a slight change in portfolio insures relations(IV ). From relations(IV ), the continuity of the scalar product and Lemma 2, there exists N12 N, such that, for every n> N1:
(V ) pn0 (x0 ei0) < qnz and pns (xs eis) < eVn(pn; s) z; 8s 2 Si.
Along relations(V ), Assumption A1-A3, Lemma 2 and the de…nition of
equilib-rium, there exists N2> N1, such that: (V I) ui(x)6 ui(xni) < " + ui(xi), 8n > N2. Letn> N2 be given. Then, Conditions (I)-(V I)yield: ui(x) < " + ui(xi) < ui(x). This contradiction proves thatC is an equilibrium and Theorem 2 holds.
Appendix
Lemma 1 9r > 0 : 8$ 2 P0 P; 8[(xi; zi)] 2 A($; eV ); k[(x
i; zi)]k < r
We start with Lemma 1 in the general setting of of Section 3.
Proof Let $ := ((p0; q); p) 2 P0 P, and [(xi; zi)] 2 A($) := A($; eV )be given. As seen under Lemma 2, the relations,xis2 [0; E]H, whereE := max(s;h)2S0 H
X
i2I
eh is, hold, for every (i; s) 2 I S0, from market clearance conditions.
From above and the relation (ps) 2 (RH ++)SnS
0
, Lemma 1 will be proved if the portfolios,(zi), are bounded independently of$. We now prove that property. Let = 1 + (k(p)k + 1)k(ei)k. Assume, by contraposition, that, for every n 2 N, there exists [(xn
i; zin)] 2 A($n), for some $n := ((pn0; qn); pn) 2 P0 P, such that
kznk := k(zn
P
i2I zin= 0; and V (pe n; si) zin> ; 8(i; si; n) 2 I Si N. We may assume limn!1 $n= $ := ((p0; q ); p ) 2 P0 P. For every(i; n) 2 I N, we let x0n
i := xn i kznk+ (1 kz1nk)ei and z0ni := zn i kznk. Then, the relations[(x0n
i ; zi0n)] 2 A($n) andk(z0ni )k = 1hold and the sequencef[(x0ni ; z0ni )]gn2N has a cluster point,[(xi; zi)], such thatk(zi)k = 1, and satis…es the relations:
P
i2I zi0n= 0,V (pe n; si) zi0n> =n; 8(i; si; n) 2 I Si N, and, passing to the limit,
P
i2I zi= 0; eV (p ; si) zi> 0; 8(i; si) 2 I Si,
SinceV 2e , the latter relations imply(zi) = 0, from Claim 1-(v), and contradict the fact that k(zi)k = 1. This contradiction ends the proof.
We proceed with Lemma 1 for the numeraire asset economy. Proof
As above, we need only show portfolios are bounded, but, then, accross all economies, En
= f(I; S; H; J); eVn; (S
i)i2I; (ei)i2I; (ui)i2Ig. We let the reader check that all contraposition arguments above translate, mutatis mutandis, to double indexed sequences of prices,$(n;k):= ((p(n;k)
0 ; q(n;k)); p(n;k)) 2 P0 P, and strategies
[(x(n;k)i ; zi(n;k))] 2 A($(n;k); eVn), where (n; k) 2 N2 (n standing for the economy), whose …nal contraposition arguments are:
P
i2I zi0(n;k)= 0 , Ven(pn; si) zi0(n;k)> =k; 8(i; si; n; k) 2 I Si N2, and, in the limit,
P
i2I zi= 0; V (p ; si) zi> 0; 8(i; si) 2 I Si, withk(zi)k = 1.
Along the model’s speci…cation, the latter relations will imply (zi) = 0, from Claim 1-(v), hence, the same contradiction as above, whenever ps e > 0 holds for every s 2 S. Lemmata 1, below, shows this latter property indeed holds.
First, we introduce new notations and let, for all(i; s; x) 2 I S (RH +)S
0 i:
y i
sxdenote a consumption, such that ui(y) > ui(x) and ys0 = xs0,8s02 Si0nfsg;
A := f(xi) 2 i2I(RH+)S
0
i :P
i2I xis=Pi2I eis; 8s 2 S0g;
Ps:= fp 2 RH+; kpk = 1 : 9i 2 I; 9(xi) 2 A; such that (y isxi) ) (p ys> p xis> p eis)g.
Lemmata1 The following Assertions hold:
(i) 8s 2 S; Ps is a compact set;
(ii) 9 > 0 : 8(s; p) 2 S Ps; p e> ;
(iii) 8(n; s) 2 N S; pn
s 2 Ps, hence, ps e> > 0.
Proof Assertion (i) Let s 2 S and a converging sequence fpkg
k2N of elements of
Ps be given. Its limit, p = limk!1 pk, satis…es kpk = 1. We may assume there exist (a same) i 2 I and a sequence, fxk
gk2N:= f(xki)gk2N, of elements of A, converging to some x := (xi) in the closure of A in mi=1(R+[ f+1g)LS
0
i, such that, for each k 2 N,
(pk
s; i; xk)satis…es the conditions of the de…nition ofPs. From Lemma 2-(i) f(xkis0)gk2N, is bounded, hence, xs0 := (xis0) is …nite, for each s02 S0.
For every k 2 N, let exk := (xek
i) 2 A be de…ned by (xeki0) := (xi0), (xekis) := (xis) and
(xek
is0) := (xkis0), for eachs02 Sinfsg. Then, the relationspk (xkis eis)> 0, for everyk 2 N, yield, in the limit,p (exkis eis) := p (xis eis)> 0. We now show there existsk 2 N, such that (p; i;exk) satis…es the conditions of the de…nition of Ps (hence, p := lim pk 2 Ps). By contraposition, assume the contrary, i.e., for each k 2 N, there existsyk 2 (RL
+)S
0 i,
such thatyk
s0 =xekis0, for eachs0 2 Si0nfsg,ui(yk) > ui(xeki)andp (ysk xis) < 0. Then, given
k 2 N, there exists (from Assumption A3 and separability) K > k, such that, for every k0 > K, ui(yk) > u
i(xk
0
i ). The latter relations imply, by construction of each xk
0
(fork0> K),pk0
s (yks-xk
0
the inequality,p (yk
s xis) < 0, assumed above. This contradiction proves thatp 2 Ps, hence,Ps is closed, therefore, compact.
Assertion(ii)Lets 2 Sand p 2 Ps be given. We prove, …rst, that p e > 0. Indeed, let(p; i; x) 2 Ps I Ameet the conditions of the de…nition ofPs. From Assumption A2, there exists ai2 (RL+)S
0
i such that, a
is0 = 0, for eachs02 S0infsg, andp ais< p eis6
p xis. Then, for every n > 1, we let xni := (n1ai+ (1 1 n)xi) 2 (RL+)S 0 i, which satis…es ps (xnis xis) < 0by construction. Let Eis2 (RL+)S 0 i be de…ned by Es is= e, Esis0 = 0, for
s0 6= s. Along Assumptions A1-A3, there exists n > 1, such that y := (xn
i + (1 n1)Eis) satis…esu(y) > u(xi), which implies, p xis6 p ys= p (xnis+ (1 1n)e) < p xis+ (1
1 n)p e. Hence, p e > 0. The mapping 's: Ps! R++, de…ned by's(p) := p eis continuous and attains its minimum for some element p on the compact set Ps, say s > 0. Then, Assertion(ii) hods for := min s, fors 2 S.
Assertion (iii) is immediate from the de…nition of equilibria, of the sets Ps, for each s 2 S, and of Assertion (ii).
End of the proof: Lemmata 1 insures the desired contradiction with Claim 1-(v)
(or the fact that the payo¤ and information structure, [V; (Si)], is arbitrage-free).
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