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liirI?conomic Rcscurch

No. 9492

GENERAL EQUILIBRIUM IN ASSETMARKETS WITH OR WITHOUT SHORT-SELLINC

I;y Roso-Annc I)ana. ('uung I.e Vanand I'r~nFois Magnicn

November 1994

R2i

ISSN 0924-7815

D K.U.B.BIBLIOTHEEKTILBURG

August 1994

GENERAL EQUILIBRIUM IN ASSETMARKEiS YITH OR HITNOUT SHORT-SELLING~

Rose-Anne DANA(GRé1.fAQ, University of Toulouse-I)

Cuong LE VAN(CNRS, CEPREMAP, Paris)

FranGois MAGNIEN(DELTA, THF2.SA, University of Cergy-Pontoise)

~ A preltminary verslon of rheorem 1 ot the paper has been wrltten by

C.Le Van durlnq hls vlslt In Apr11 1994 at CentEA and the department of

Econometrlcs of Tllburq Unlverslty. The authors would ltke Co thank

Antoon ~an den Elzen for hls remarks and suqqestlons.

1

ABSIRACI - In this paper we extend the results of Cheng (1991), Brownand Werner (1993) on the exístence of equilibrium in infínitedímensional asset markets : we do not assume that each agent's preferred

sets have a uniform direction of improvement but only assume that thepreferred sets of attainable allocations have non-empty interiors. We

then deduce existence theorems for asset markets without short-sellingand for the CAPM.

Corresoondence to : Cuong LE VANCEPREI~IAP140, rue du Chevaleret75013 - PARIS (France)

zINTRODUCTION

1The CAPM model of Sharpe (1964) and Lintner (1965) has been the

first model of equilíbrium with consumption sets unbounded below. Whilethe implications of the model (the "mutual fund" result and the "betalaw") were widely used in finance, the problem of existence of anequilibrium itself was ignored. Existence results were obtained only afew years ago by Nielsen (1990a,b) and Allingham (1991).

The fírst finite dimension equilibrium existence result whenconsumption sets are unbounded below was proven by Hart (1974) under theassumption that agents' utility functions were Von-Neumann Morgensternand that theír directions of improvement were "positivelysemí-independent". Much later Werner (1987) and Nielsen (1989)reconsidered the problem. Werner gave an existence result based on ageneralisation of Gale-Nikaido-Debreu's lemma under the assumptfon thatthere was at least one price for which there was "absence of arbitrageopportunity" for all agents. Nielsen who makes fairly weak hypotheses onpreferences obtains a very general result under the assumption thatagents' directions of improvement were "positively semi-independent".

In the infinite dimension case, two existence results based onNegishi's method were given by Cheng (1991) and Brown and Werner (1993)and applied to subspaces of Lp and Von-Neumann Morgenstern utilíties.

In this paper, ue extend the results of Cheng, Brown and Werner inthe following sense : we do not assume that each agent's preferred setshave a uniform direction of improvement nor do we assume the continuityof utílity functions. We only assume that the preferred sets ofattainable allocations have non-empty interiors.

We first deduce from our result an existence theorem when theconsumptíon sets are the posítive orthant of a locally convex solidRiesz space. This result improves theorem 10.1 of Mas-Colell and 2ame(1991). We make a local non-satiation assumption instead of an uniformdirectíon of ímprovement assumption for the attainable allocations.

3

This assumption can be viewed as a weaker form of the F-propernesscondition in Mas-Colell and Zame.

41e then get existence theorems for the C.A.P.M with an infinitenumber of assets wíth or wíthout a riskless asset.

The paper is organised as follows. In section 1, we present themodel and its assumptions. In section 2, we give some críteria for theclosedness and boundedness assumptions. The main result and its proofare given ín section 3. Section 4 and 5 are devoted to applications,an equilibrium without short-selling and the C.A.P.M.

1. TE~ MODEL

we consider an exchange economy with commodity space E. The space Eís assumed to be a locally convex topological vector space. There are mconsumers. Each consumer 1 is descríbed by a consumption set Xi c E, aninitial endowment ei and a preference relation which Ss represented by a

mutility function ui : X1 -~ R. Let e- E eí be the total endowment. An

i-1allocation is a m-tuple x-(xl,...,xm) with xi e X1, Vi. It is

mattainable if E xi - e. It is indivldually ratlonal attainable if ít

i-1

is attainable and íf ui(xi) a ui(ei), tli. Let A denote the set of all

individually rational attainable allocations. We normalíze the utility

functions by requiring ui(eí) - 0, Wi.

An allocation x is weakly-optimal (N.O) if x e A and if there does

not exist another x'e A such that ui(xi) ~ ui(xi) for every i. We denote

by F the set of weakly-optimal allocations. x is Pareto-optimal (P.0) if

x e A and íf there exists no x' e A such that uí(x~) z ui(xí), Vi, and

u~(x~) ~ u3(xs) for some j. The utility set U is defined as follows :

U-((zl,...,zm) e Rm I 3 xe A s.t. ui(xi) a zi, di T .

4For xi E Xi, define the preferred-set of xi :

Pi(XS) -{ Xi E Xi I ui(X1) ~ ui(Xi)~ .

xi is a satiation-polnt of Xi if Pi(xi) -~.

Let E' denote the topological dual of E. A quasi-equilibrlum is acouple (x,p) such that :

1) xEA , pEE'`{p} ;

11) PXS - Pei. Vi :

iii) ui(xi) ~ ui(xi) -~ pxi a pei.

An equilibrlum is a couple (x,p) which satisfies í),ii) andSií bis) ui(xi) ~ ui(xi) -~ pxi ~ pei.

It is well-known that a quasi-equilibrium (x,p) is an equilibriumif Vi, pxi ~ inf pXi.

TLe assumotions

Ne make the folloxing assumptions :

H1 : X1 is closed, convex, non-empty for every i- 1,...,m.H2 : ei E Xi, Vl.H3 : ui is strictly quasi-concave, Vi.H4 : U is closed.HS : U ís bounded.H6 : If x-(xl,...,xm) E A then dí, int P1(xi) s~.

2. CRITERIA FOR CLOSEDNESS AND BOUNDEDNESS OF U

Let us recall that HS is verífied if, Vi, Xi is the positiveorthant of a topological lattíce.

m mAW -{ x E II Xi I xi E Wi, Vi and ï x1 - O}.

i-1 i-1

mIf A~ x{0} and if x E Am and x x 0, then pxi ~ 0, Vi wíth p E n Si,

i-1mcontradicting E xi - 0. Thus Am - {0} and A i s compact. Since ui is

i-1continuous for every i, U is compact.

ii) Conversely, assume that U is closed and bounded. Let a belong

to the unit-simplex of Rm with aci ~ 0, di. Let u be a solution to

max { E aizi ; z E U}. There exists x e A such that di, ui - ui(xi). Hence,i

„ m m „there does not exist x E II Wi with E xi - 0, and

i-1 i-1m mE- Wi is closed, contains no line and ( n-WO)0 - E- Wi .1-1 i-1 i-1

Therefore n- WD has a noni-1

Let us make some additional remarks :

empty interior. Equivalently n Si :~.i-1

a

a) We say that p is "viable" for agent i if his demand at p exists.

It may easily be proven that if E-Rl, p ís "viable" for agent i iff

p E Si.

Hm

Thus the hypothesis n Si x m is equivalent to the existence of ai-1

price which is viable for every agent.

b) We recall that the notion of absence of free lunch is more

restrictíve than the notion of arbitrage free. Indeed if p E RL admitsno free lunch for agent i then it is arbitrage free for this agent (seeBrown and Werner, 1993, proposition 1).

c) The hypothesis which is hard to verify is H4. We quote tworesults.

Proposltlon 5- (Chichilnisky-Heal, 1993). Assume that E is a reflexiveBanach space and Xi - E, Vi. Assume moreover :

a) A is norm bounded,b) ui is norm continuous and concave.

Then H4 and HS are fulfilled.

Proof - Obviously, HS is satisfied. We prove that H4 holds. Let {zn} e Uconverge to z. There exists xn e A such that ui(xi) ~ zi, Vi. Since A isconvex, norm closed and norm bounded, it is weakly compact. Since ui isnorm-continuous and concave, it is weakly upper semi-contínuous. Thus

there exists a converging subsequence x~ ~-~ x and ui(xi) a

1'im ui(x~ ) z zi, Vi. Thus z e U.

a

Proposition 6 - (Cheng, 1991). Assume :

a) Xi - Lpfp), Vi where ~a is a finite measure and 1 s p s m,

9

b) tli, ui(x) -J

Uí(x(s))dp(s) wlth Ui . R-~ R, strictly

increasing, strictly concave and

~ Ui(x(s))dp(s) e R, tlx e Lp(p)

Then U verifies H4 and HS.

Proof - It follows from proposition 2 and example 1 that HS holds. The

proof of H4 which is long and delicate can be found in Cheng (1991).

Under the above hypotheses Cheng (1991) also shous that if for at

least one agent i, lim 8Ui(x) ~ t m, then A is not p-norm bounded. Sox-~-m

that in that case the hypothesis a) of proposition 5 is not fulfilled.

3. TEIEOREM 1

Assume H1,..., H6. Then there exists a quasi-equilibrium.

Proof

This will be done in several steps. Throughout thís section we

assume that e- (el,...,em) is not weakly-optimal. This assumption is

not restrictive since if e is W.O. then there exists p e E'~ {0} such

that (e,p) is a quasi-equilibrium (see e.g. Cheng, 1991).

Lemma 1 - Assume H4, HS. Then U is compact, convex with non

empty-ínterior.

Proof - It is obviously convex, compact. Its interior is non-empty since

e is not W.O.

a

0

loLet A be the unit-simplex of Rm, i.e,

( m

A-{ SER~ I E S. - 1},l i-1 i

and let f: G-~ R4 be defined by

s E A-~ f(s) - max {a E Rf~as E U}.

Since U has a non-empty interior, f(s) ~ 0 for every s E A.

Lemma 2- Assume H1,...,HS. If x E A and if there exists an s E A such thatui(xi) e f(s)si, tli, then x is weakly-optimal.

Conversely, for every s E A, there exists an x E A such that ui(xi) '-f(s)si, dí.

Proof - Obvious.

By Lemma 2, one can define

s E A-~ H(s) -{x E A I ui(xi) e f(s)si, tli}.

Lemma 3 - Assume H4, H5, then f is continuous.

Proof - Let sn e ~-~ s and f(sn) -~ a. Since U is closed, as e U, andhence, a s f(s). Assume a ~ f(s). From Lemma 1, there exists W e U withWi ~ 0, tli.

Define : W~ -(1-~)f(s)s t~W, for ~ E 10,1[.

For ~ sufficiently close to 0, one has

asi ~ Wi, tli,

and therefore f(sn)si ~ Wi, Vi, for n sufficiently large. Since W~ e Uwe have a contradiction with the definition of f(sn).

0

0

11

Remark 1- The proof of Lemma 3 does not require the continuity of the

utílity functions ui as in Cheng (1991). One can also observe that Lemma

3 is true if U is compact, convex, comprehensive, i.e.,

V E U and 0 S V' S V-~ V' E U,

and if U contains an element N with Wi ~ 0, Vi.

Lemma 4 - There exist a convex symmetric open set V of E, an integer k,

m finite sets of k elements of E, (Ni,...,i~} vith i-1,...,m, k

elements xi, ..,xk of A and k open sets of A, U1, ..,Uk Which cover G,such that :

VS E A, s E U~ -~ V1, ui(Xi t i'~i t Z) ~ f(s)si, VZ E V.

Proof - Let s E A and xs E H(s). From H6, Vi, there exists vi E E such that :

ui(xi t vi) ~ ui(xi) z f(s)si.

Since f is continuous, Vi, there exists an open neigborhood Ui of s in

A such that :

ui(xi 4 vi) ~ f(s')si, Vs' E Ui .

Let s" E A, x"s E A be such that :

ui(x~s) a f(s")si - max (ffs')si~ s' E Ui }.

From H6, Vi, there exist a vï and a convex symmetric open set Vi such

that :

s sui(xis t vi t z) ~ ui(xïs) '- f(s')si, Vz E Vi, ds' E Ui

12

mDefine Us - n Ui

í-1

and di, vis by xi t vis - xis t vi .

Then we have :

Vi, ui(xi t vis t z) ~ f(s')si, tlz E Vi, ds' E Us.

Let {Usl, ..,Usk} be an open covering of A. One has :

tls E A, S E Usj -~ ui (xi`) t Visj t Z) ~ f(s)si , VZ E Vij .

Define for every i :

sWi - vi j

m k sV-n n V.jii-1 j-1

s.and note : xi - xi~

sUj - U j, Vi, Vj.

Then :

VS E 0, S E Uj -~ Vi, Ui(Xi t Wi t Z) ~ f(s)Si, VZ E V.

Next, define for s E A and for j- 1,...,k, such that s E U~ .

Wj(s) - S Wi, andí

0

P(s) -{p E E'~~pz~ ~ i, Vz E mV,pW~(s) ~ 1, `dj such that s e U~

13Lemma 5

i) Vs e A, P(s) is convex, c(E',E) - compact and non-empty.

ii) Vs e A, 3 p e P(s) such that :

tlx. ui(xí) ~ f(s) si. tli - 1,..., m-~ p E xi z pe.i

Proof - Let s e A. Define :

Ví, ai(s) -{x e Xí ~ ui(x) ~ f(s)si},

and G(s) - E ní(s) - e.i

From Lemma 4, ui(xi t Wi t z) ~ f(s)sí, Vz e V, and tlj such that

s e U~. Hence, for j such that s e U~, Wj(s) t m V c G(s).

From the very definition of f, 0 E G(s). Thus, there exists p e E'`{p}

such that :

p E zí ~ pe, tlz wfth ui(zí) z f(s)si,tli.i

Let x with ui(xi) z f(s)si, tli,and let I-{i Ixí is not a satiation-point}.

For i e I, from H3, there exists vi(xi) such that :

Va e 10,1(, ui(xí t a vi(xí)) ~ f(s)sí.

From Lemma 2, there exísts y e A verifying ui(yí) ~ f(s)si, Vi, and

from H6, yí is not a satiation-point for any i. Hence, for i e I,

ui(xi) ~ ui(Yi) ~ f(s)si.

Define xi - xi t a vi(xi) for i e I,

xí - xi for i E I.

Then E xi - e E G(s), and thereforei

14

p i xi t a i p. vi(xi) z pe.i ieI

Letting a-~ 0 we obtain p E xi z pe, and statement ii) has been1

proved.

Since, dj, Wj(s) t m V c G(s), we have :

tlj, pWj(s) ~ 0 and I pz ( s p Wj(s), Vz e m V.

Let j0 verify pWJD(s) - min p.Wj(s). We havej

I p . z I s 1 S p . Wj(s), Vz e m V.

pWjD(s) pW,D(s)

Thus, p'- pj belongs to P(s), i.e, P(s) is non-empty.It ispW ~(s)

compact by Alaoglu's theorem. Obviously P(s) is convex.

Defíne P(s) -{P e P(s) ~ uí(xi) a f(s) si, Vi -~ p E xi ~ pe}.i

Lemma 6

Vp E P(s), Vi, Vxi e Xi, ui(xi) ~ f(s)si -~ pxi ~ pxi, Vxs e H(s).

In particular : di, ui(xi) z ui(xí) with xs E H(s) -~ pxi ~ pxi.

0

15

Proof - Let xi e Xi with ui(xi) z f(s)sí, and let xs E H(s). Define :

x' - x~ , d~ x 1,

xi - xí .

Let p e P(s). Then p~ xs - pe á p~ x~ , and hence , px1 a pxi.

Let s e A, xs e H(s), and defíne :

~(s) -{(yí....,ym) e Rm I yi - P(ei - xi). di, with p e P(s) ).

Lemma 7

(1) m is convex uniformly twunded valued.

(ii) ~ has a closed graph.

Proof - (i) P(s) is convex since P(s) i s convex and therefore ~(s) is

convex.

Now, Vi, let x1 e X1 such that uí(xi) a max {uí I(u1,...,um) E U}.

From Lemma 6, pxi ~ pxi , V1, dp e P(s). Since pe - E p xi, we have :i

pxi -Pe- E px~ ape-p. E x~ zBjxi jxi

since p e P(s) which is Q(E',E)-compact.

sThen Ip(ei-xí1 I`- IP . ei~ t Ip . xi ~

5 max { ~ p.eí~ t Ip . xi~ } t B.

pEP(s)

16

(11) Let y- 1Lm yn, s- lím sn with yn E m(sn), Vn. We have

pn e P(sn) c P(sn). For n sufficiently large, sn E U~ for every j such

that s e U~.

In other words, pn E P(s) for n large enough. Hence pn P~ p e P(s).

From H6 and H3, there exists ví(xi) such that :

Va e)0,1[, ui(xi t avi(xí)) ~ ui(xi) '- f(s)si, Vi.

The function f being continuous, we derive that

ui(xi t aví(xi)) ~ f(sn)si, Vi, dn large enough.

sFrom lemma 6, pn(xs t avi(xi)) ~ pnxin - pnei -

and p(xí t a vi(xi)) ~ pei -

nyi '

Let a-~ 0. This gives : pxi ~ pei - yi . Since E xi - e and E yi - 0,i i

we have, Vi, pxi - pei - yi

We now prove that p e P(s).

Let x with ui(xi) '- f(s)si, di, and

let I-{i ~ xi is not a satiation-point }.

For i e I, from H3, there exists vi(xi) such that Va e)0,1(,ui(xi t avi(xi)) ~ f(s)si.

From Lemma 2, [here exists x' E A such that,

ui(xi) a f(s)si, Vi.

From H6, xi is not a satiation-point for any i. In particular,i E I-~ ul(xl) ~ ui(xi) z f(s)si.

Since f is continuous, for every n large enough,

ui(xi ~ a v(xi)) ~ f(sn)si, Vi E I,

and ui(xi) ~ f(sn)si, Ví E I.

Since p E P(sn) we haven

mpn E xi t a E pn.vi(xi) a pne,

i-1 1E1

mand p ï x 4 a E p.v (x ) 2 p.e, for every a e)0,1(.

1-1 1 SEI i í

mHence p E xi ~ pe .

i-1

Final step

Fírst we verify that m fulfills the boundary condítlons. Indeed,

si - 0-~ Cls)sí - 0- ul(ei).

Thus, Crom Lemma 6, pei z pxi -~ yi '- 0.

1 }i

'fherclore, Irum Ih~r yenera112ed Kakul,,rni Lheurr~m, t.herc exlsts s e Awlth O e m( s), l.e., Lhere exlsts

p e P(s), xs E A veritying : p xi - peí, Vi,

and uí(xí) z ui(xi) -~ p xi '- p xi.

In other words, ( x,p) is a quasi-equilibrium.

a

4. APPLICATION 1: EQUILIBRIl1M NITHOUT SHORT-SELLING

In thís section the commodity space E is a topological locallyconvex-solid Riesz space. Its positive orthant Et is closed and convex.

For x E Et, we define :

I(x) - (y E P ~ ~y~ s ax for some a~ 0}.

IC x ls strlctly posl[Ive then I(x) is dense in F Isee Aliprantis.Brown and Burkinshaw, 1989). We make the Collowing assumptions :

A1 : Vi, Xi - Et ;

A2 : ei E E;, di ;

e- E eí is strictly posítíve ;i

A3 : Vi, ui is strtctly quasi-concave and continuous from Et into R;

A4 : U is closed ;

AS : U is bounded ;

A6 : If x-(xl,...,xm) e A, then, tor every i, t.herc rzist an r,p~.n

neigborhood of 0, Ni, and a vector vi such that :

I(v.) - I(e) andidx E 10,11, ui(xi i L(vi . z)) ~ ui(x) if z e 41i and if xi t a(vi ~ z) E[:r.

Remark 2

i) In asswoption A6, the condition I(vi) - I(e) is equívalent to :

there exíst x ~ 0, {~ ~ 0 such that ice ~ vi ~ Aé. If int Et s m, and if

ví and e are strictly positive than this condition is satisfied since

I(e) - I(vi) ~ E.

ii) Consider the second statement of assumption A6. This condition

is weaker than the F-properness mentioned in Mas-Colell-2ame (1991). One

can also observe that, since ui is strictly quasi-concave, this condition

is the reformulation of assumption H6 (non-satiation) of Theorem 1 when

int E} is empty.

Theorem 2

Assume A1,...,A6. There exists a quasí-equllibrium.

Proof - It will be done in two steps.

Step 1- we shall prove there exists a quasi-equilibrium for the economy

with I(e) as commodity space. The consumptíon set for an agent i is

It(e) - E}ni(e). The set of indtvidually rational attainable allocations

of this economy is equal to A, and hence the utility set is U. Thus,

assumptions H1,...,HS of Theorem 1 are satisfied.

Let us check now assumptíon H6 of Theorem 1. we shall endow I(e)

with the following norm :

Ilxllé - ínf { A~ 0 ~ ~xl s xe ).

zo

Since E is solid, every neigborhood of 0(with the initial

topology) in E contains a neigborhood of 0 in I(e) (with the norm ~I II-).e

Consider the open set W. of assumption A6. From the remark aboveithere exists an open neighborhood of 0 in I(e), Vi, which is containedin Wi.Since I(vi) - I(e), one can choose Vi such that vi t Vi c(ae,{~e]for some x~ 0, {~ ~ 0, and therefore : xi t vi t z e E}, Vz e Vi. Hence,H6 of Theorem 1 is verified.

One concludes there exists a quasi-equilibrium (x`,p`) of the

economy with I(e) as commodity space and p' e I(e)' (the dual of I(e)with the norm ~I Ilé).

SLep 2 - We shall prove that p' is continuous in the initial topology

and has an extension p e E' such that ( x', p) is a quasi-equilibrium forthe initial economy.

Denote by Wi, vÍ the open sets and the vectors associated with x'by assumption A6.

Since, ui(xi t vi t z) a ui(x~), Vz e V1 (Vi is defined abovel, we~

have

p'(vi t z) ~ 0, Vz E Vi ,

and hence p` vi ~ 0.

Normalize p` by p' i vi - i.i

Without loss of generality, assume Wi symmetric and solid. Letm

W' - n Wi. We have just to prove that p' is bounded on W` n I}(e).i-1

Let y E W' n I}(e). There exísts x~ 1 such that 0~ y s A e.

Define z- ~ y s e- E xi.i

zl

One has : i xi - z }~ ~ vi n 0,

since vi is strictly positive (I(vl) - I(e), e strictly positive -~ vistrictly positive ; see Aliprantis, Brown and Burkinshaw, 1989).

By the Riesz deco~pasition, there exists (zl,...,zm) verifyi:~

Ezi -z.i

0 5 zi s xi t~ vi, Vi.

Since 0 5 zi s z-~ y e~ W' , and W' is solid, one has zi e~ N', Vi.

From A6 : ui(xi i~(vi - x zi)) ~ ui(xi), Fli ;

and 1- p' E vi z p" y.i

Ne have proved that p' is continuous on I(e) with the initialtopology.

Since I(e) is dense in E, p` has a unique extension p e E'.

Since E- I(e) -( x E E ~ ~x~ A n e-~ ~xl }, and ui is

continuous on E},if x' e E} and ui(x') ~ ui(xi), one has x' A n e-~ x'

and hence, for n large enough ui(x' A n e) ~ ui(xi). Therefore,

p(x' Ane) ~pxi andpxi~pxi-pei.

In other words, (x',p) is a quasi-equilibrium.a

z25. APPLICATION Z : C.A.P.M.

As we mentioned in the introduction, the CAPM played a veryimportant role in the finance literature although the problem ofequilibrium exístence was only discussed rather recently by Allingham(1991), Nielsen (1990a, 1990b).

In the next paragraph, we show that the "mutual fund" result andthe "beta law" are only necessary conditlons for equilibrium. They arehowever important relations from an equilibrium point of view becausethey may be used to bring down an infinite dimensional equilibriumproblem to a two dimensional one.

5.1. The model

There are S states of the world. A v-field .y models agents commoninformation on the set S of states of the world and P is either anobjective probability or agent's common subjective probability on (S,Y).

The economy E is described as follows. There is only one good takenas numéraire tradable at every state s.

There are n agents. Agent i is described by a consumption space X.iS L2(P) (we do not assume here that Xi is finite dimensional), anendowment el and a utility function uí : Xi -~ R assumed to be "meanvariance", in other words, there exists Ui : R x Rt-~ R such that ui(z)- Ui(E(z),var(z)), z e Xi where E(z) and var(z) denote the expectationand variance of z.

We make the following assumptions :

B1 X. - Z, b'i, where Z is a closed subspace of Lz(f'1in

B2 eí e Z, E(ei) ~ 0, Vi. e- E ei is not a constant, E(e) - 1,i-1

Var(e)-1 ,

23

B3 di, Ui is strictly concave, C2, Ui(.,y) is increasing dy e R},while Ui(x,.) i s decreasing dx e R;

B4 1 e Z (there exists a riskless asset).

As Z is a closed subspace of L2(P), an asset price iO being acontinuous linear form is identified by Riesz representation theoremwith aa element of 2. 11e denote by ~x,y~ the dot product of x and y in

Z. Given a price ~, the budget set of an agent i is defíned by

Bi(~p) -{ci E L2(P) s.t. 3z e Z ; ~~p,z~ 5 0, ci - eitzJ.

Equivalenltly,

gi(r~) -~ci E Z, ~ rp,ci~ s ~~p,ei~~.

Definttion - An equilibrium is a pair (c,~p) e Zn x 2 with

c - (cl,...,cn) such that

a) ci maximizes ui(ci) subject to ci e Bi(~p), for every í-1,...n,

b) En-1 ci - e.

4ie first remark that the mutual fund result and the beta law arenecessary conditions for equilibrium.

Let H denote the span of 1 and e.

Proposition 7- If (c,p) is an equilibrium, then ~p e H and c e Hn. More

precisely, there exist ( a,al,...,an) e R}}1 and (b,bl,...,bn) e Rntl

such that (') ~p --ae t b, and (" ) ci - aie t bi for every

i-1,...,n.

Proof - Let ci(~p) denote agent's i demand at price ~p, being the solution

to the problem :

24

max U1(E(zl,var(-r.)) sub~ect to

z E Z. ~~p,z~ s ~p,ei~.

It exists iff there exists a multiplíer pi e Rt such that

aii t 2 bi(ci(N) - E(ci(rp))1) - F~iN and

F~i(~N.ci(~p)~ - ~~p.ei~) - 0.

where ai-Ui1(E(cí(~p)),var(ci(~P))) and bi - Ui2(E(ci(W)),var(cí(~P))).

Thus ci(~p) - aí 1- ti ~p with ti E Rt.

At equílíbrium Ei-1 ci(~p) - e. Therefore summing up over agents,

(ïi-1 ti) W - - E } ~-1 Ai .

Under B2 since e is not a constant, we may write

~p--ae f b, aeRt, beR,

ci - ai e t bi, aie Rt,bi e R for every 1-1,...,n,

xhich proves the claim. As it is well known (~') is a mutual fund result

and (') is a version of the beta law.

It follows from proposition 7 that if (c,~p) is an equilibrium, then

~p e H and ci - ai e t bi for every i- 1,...,n with (ai, bi) solution of

max Ui(a t b, aZ)

~ W. a F t b~`- ~~p. Ei~

and Ei-1 ai - 1, Ei-1 bi-0.

zs5.2. Existence Qf eauilibrium when there exis ~ rískless asset

From the previous section it follows that we need only consider

prices in H and that we may substitute for the original economy, theeconomy ó which has the same equilibria, and is described by the list :

S s{X, ui : X c H-~ R, ei, 1-1,...n} where

X- { a e} b~ a e R, b e R}, ui(a e t b) - Ui(atb,a2) and ei is the

projection of ti on H, in other words ei - ai e } bi wíth

ai - cov(e,ei), b1- E(ei) - ai.

Let (p1,p2) denote the price of e and 1, i.e. pi -~~,c~,p2 - t ~p,l~. An equilibrium is now a quadruple (a', b', pi, p2) e R} x Rn

x R x Rt with a' -(ai,...,an), b' -(bi,...,bn) such that

a) (ai,bi) maximizes Uifatb,a2) subject to

pi a} p2 b s pi ai } p2 bi for every i-1,...,n,

b) 51-1 ai - 1, Ei-1 bi - 0.

Since Ui is increasing in b, p2 ~ 0 and we may normalize prices sothat p2-1. Obviously, it is easier to work in the space of expectations

and variances. Let us therefore make the change of coordinates

x- a t b, y- a, q--p1 t 1. Finally let Vi : R x R-~ R be defined

by Vi(x,y) - U1(x,y2). For every i-1,...,n, Vi is strictly concave and

C2 and Viy(x,0) - 0 for every x.

Lle may redefine an equilibrium as follows. Let xi - E(ei),

yi- cov(e,ei) for every i-1,...,n. M equilibrium is then a triple (x`.y'.q')

E Rnx Rn x R with x' -( x', .. x'), y' -(Y' y') such that~ 1 ' n 1' ' n

26a) (xí,yí) maximizes Vi(x,y) sub~ect to

x- q'y 5 Xi - q'yi for every i-1,...,n,

b) Ei-1 xi - 1. ~-1 yi - 1.

In other words (x',y',q`) is the equilibrium of a two dimensionaleconomy Y2 with consumption space R2, agent's i utility Vi(x,y) and

agent's i endowment xi - E(ei) and yi - cov(e,ei).

Next we define the set of individually rational allocations ad ofthe reduced economy ~ by :

( n n ld-{(x..Y-)n I E x. - 1. E y. - 1, V.(x..Y.) '- V.(x y ),Vi }l i i i-1 i-1 i i-1 i i i i i i' i 111

If (xi'yi)i-1 E ,1 we have

(i) 0~ Ui(xi.Yi) - Ui(xi.Yi) ~ Uil(xi.Yl)(xi-xi) t UiZ(xi,Yl) (yi -yi),Vi,

implying, as Uii(xi'yi) ~ 0, Vi

Uí2(Xi.Yí) -2(ii) xi - xi ~ YiUii(xi.Yi)

-2Ui2(xi.Yi) Y2

Vi.iUii(Xi.Yi)

n nAs E x - E x. - 1, we have

i-1 1 i-1 1

n Ui2(xi.Yi) -2 n UiZ(xi.y2) z~~ E y - E y

i-1 Uii(xi,Yi) i i-1 Uii(xi,yi) 1

Therefore yi 5 Ai, Vi for some Ai ~ 0.

From (ii), we also get :

-2xi L Xi 4 Uí2(xi,Y2) Yi di, since UiZ(xi,Yi) ~ 0.

Ui1(xi.Yi)

nVia E xi - 1, xi is also bounded above for every i.

1-1

Thus ,d is bounded. Since Vi is continuous, ,~ is compact and so is

the utility set of ~. Hence assumptions H4 and HS of theorem 1 are

fulfilled for the reduced economy. H6 is obviously satisfied since

there is no satiation point. We have therefore :

Theorem 3 - Under assumptions B1, B2, B3, B4, there exists an

equilibríum.

5.3. Existence of eauilibrium wit out ~ riskless asset

In this section we assume

64 Bis : 1 E Z(There doesn't exist a riskless asset).

Let n denote the projection of 1 on Z. Ne replace B2 by

B2 Bis . ei e Z, E(ei) ~ 0 Vi ;

ne- E ei is not proportional to ~. E(e) - i.

i-1

Let us first remark the following :

Proposition 8 - Agent's i utility has a satiation point si - tin for ti~ 0.

Proof - Uí(E(z),var(z)) being concave, has a maximum iff

aip f 2bi(si - E(si)n) - 0 with ai - Uil(E(si),var(si)) and

28

bi - Uíz(E(si),var(si)). Thus sí is a satiation point iff there exists at such that si - tn and the function t-~ Ui(tE(n),var(tn)) has a

maximum. Since Ui i s concave,

Ui(tE(n),var(tn)) 5 Ui(0,0) t tE(n)Uii(0,0) t t2 var(n)Ui2(0,0).

Thus, Ui(tE(n),var(tn)) -~ -m as t-~ tm, since U12(0,0) ~ 0. ThereforeU.(tE(n),var(tn)) has a maximum.i

Since its derivative at t-0 equals E(n)Uii(0,0) ~ 0, the maximumis reached at a tí ~ 0.

Let H' denote the span of n and e. Using the same proof as inproposition 7, while replacing ~l,cí~ by ~n,ci~, one gets a mutual fundresult and a beta law :

Proposition 9- If (c,~p) is an equilibrium, then ~p e H' and c e H'n. More

precisely, there exist (a,ai,...,an) E Rt}1 and (b,bl,...,bn) e Rntl

such that ( ') ~p -- ae t bn and (" ) ci - aie t bin, Vi.

It follows that if (c',~p') is an equilibrium, then ~p' e H'.Furthermore,

ci - aie i bin for every i- 1,...,n with (ai,bi) solution of

max Ui(atbE(n),var(aetbn)) such thatc~p',aetbn~ ~ c ~p~,ei~ - ~ ~p~,ei~

where ei is the projection of ei on H' and ii ai - 1, i~-1 bi - 0.Thus, we need only consider prices in H' and we may substitute for the

original economy the economy ó ', whích has the same equilibria, and is

described by the list :

~' -{X, ui : X c H' -~ R, ei, i-1,...,n} where

X-{aetbnl a E R, b e R}, ui(a,b) - Ui(atbÈ(n),var(aetbn)) and

e. - a.e t b.n-i i i

29

Let (pl,p2) denote the price of e and n, i.e. pl - ~~,e~,PZ - ~~,n~.

and

M equilibrium is a quadruple (a`,b',pi,p2) e R} x Rn x R x Rwith a'-(ai ,..,an), b'-(bi,...,bn) such that

a) (a?, bf) maximizes U1(a t bE(n),var(ae t bn)) subject to

pia ~ pZb s piai t pZbi for every 1-1,..., n,

~ rb) i-1 ai - 1, i-1 bi - 0.

Let us now show that the utility set of ó' is compact. Indeed, sinceUi is concave

Ui(0,0) - ui(a,b) ~-(a t bE(n))Uil(o,0) - var(ae t bn)Ui2(0,0).

Let M- Ui(0,0) - ui(ei). Then

{(a,b)~ ui(a,b) ~ ui(ei)} S{(a,b) I M a-(a t bE(n))Uil(0,0)

- var(ae t bn)Ui2(0,0)}.

Tedious computation of var(ae 4 bn) and the negativity of Uiz(0,0)

show that this last set is bounded. Let ~' denote the set ofindividually rational allocations :

n n~' -{(ai,bi)i-1 ~ E ai-1, E bi-0, ui(ai,bi) a ui(ei), tli }.

i-1 i-1

Thus ~' is compact and again it follows that the utility set of l;'

is compact which takes care of assumptions H4 and H5.

In the spirit of Nielsen [1990b), we add two more assumptions in

order to get H6 :

30

BS : ui(ei) ~ Ui(0,0)

( y Ui2(1'y) 1 1-E(n)B6 : max {-

Uil(1'y) }~ 2 with Y- E(t1) ~

Remark - Nielsen assumes that ui(ei) ~ Ui(0,0). Let us show that BS is aless strong assumption. Indeed by definition, ei - eif Ci where lí e H'l.Since ei is the projection of ei on H', E(ei) - ~l,ei~ -~n'ei~ -~p,ei~ - ~l,ei~ - E(ei). Thus E(fi) - 0 and cov (ei,li) - ~eí.li~ -E(Ci)E(ei) - 0. Therefore, var(ei) - var(ei) ; var(li) ~ var(ei), and

ui(ei) - Ui(E(ei),var(ei))- Ui(E(ei),var(ei)) ~ Ui(E(ei),var(ei)) - ui(ei).

Assume B5. If (ai'bi)i-1 E ,rl', then

Ui(aitbiE(~I),var(aie t bin)) a ui(ei) ~ Ui(0,0).

Therefore ai t bíE(n) ~ 0, Vi.

nSince E (ai t biE(n)) - 1, we have 0 ~ ai t biE(p) ~ 1.

i-1

Next, we show that B6 implies that E(tin) - tiE(n) ~ 1, with ti~ a

satiation-point.

Indeed, B6 is equivalent to

E(n) U(1, var(n))t Z var(n) U(1, var(n)) ~ 0, di,11 E(n)2 E(n) i2 E(n)2

since var(n) - E(n)(1-E(n)).

This implies that at t- E~j , the function Ui(tE(n), t2var(n)) is

increasing. Thus t. ~ 1 .i Rj

Consequently, if (ai,bl)n-1 e a4', then aie t bin ~ tin, di, which

implies that H6 is fulfilled.

Theorem 4 - Under B1, 62 bis, B3, B4 bis, BS and B6 there exists an

equilibrium.

31

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1. C.D.Aliprantis, D.J.Brown and O.Burkinshaw -"Existence and Optimalityof Competitive Equilíbria", Springer-Verlag, 1989.

2. M.Alligham - Existence Theorems in the Classical Asset PricingModel, Econometrica,1991, 59-4, 1169-74.

3. D.Brown, and J.Werner- Arbitrage and Existence of Equilibrium inInfiníte Asset Markets, Unpublished, Stanford University 1993.

4. H.Cheng - Asset Market Equilíbrium in Infinite Dimensional CompleteMarkets, Journal of Hathematical Economics, 1991, 20, 137-152.

5. G.Chichilnisky and G.M..Heal - Competitive Equilibrium in Sobolev

Spaces without Bounds on Short Sales, Journal of Economic Theory,

1993, 59, 364-384.

6. O.Hart - On the Existence of Equilibrium in a Securities Model,Journal of Economic Theory, 1974, 9, 293-311.

7. J.Líntner - The Valuation of Risk Assets and the Selection of RiskyInvestments in Stock Portfolios and Capital Budgets, Review

of Economics and Statistics, 1965, 47, 13-37.8. A.Mas-Colell, and W.2ame - Equílibrium Theory in Infinite

Dimensional Spaces, in W. Hildenbrand and H.Sonnenschein eds., TheHandbook of Mathematícal Economics, Vo1.IV, 1991. North Holland.

9. L.T.Nielsen - Asset Market Equilíbrium with Short-Selling, Review of

Economic Studies, 1989, 56, 467-474.30.L.T. Nielsen - Equilibrium in C.A.P.M without a Riskless Asset,

Review of Economic Studies, 1990a, 57, 315-324.11.L.T.Nielsen - Existence of Equilibrium in C.A.P.M, Journal of

Economic Theory, 1990b, 52,223-231.12.W.Sharpe - Capital Asset Prices : A Theory of Market Equilibrium

Under Conditions of Risk, Journal of Finance, 1964, 19, 425-442.

13.J.Werner - Arbitrage and the Existence of Competitive Equilibrium,Economeirica, 1987, 55, 1403-1418.

Discussion Paper 5eries, CentER, Tilburg University, The Netherlands:

(For previous papers please consult previous discussion papers.)

No. Author(s)

9409 P. Smit

9410 J. Eichberger andD. Kelsey

941 I N. Dagan, R. Sen-anoand O. Volij

9412 H. Bester andE. Petrakis

9413 G. Koop, J. Osiewalskiand M.F.J. Steel

9414 C. Kilby

9415 H. Bester

9416 J.J.G. Lemmen andS.C.W. Eijffinger

9417 J. de la Horra andC. I~crnandcz

9418 D. Talman and "L. Yang

9419 H.J. Bierens

9420 G. van der Laan,D. Talman and Z. Yang

9421 K. van dcn Rrink andR.P. (iilles

9422 A. van Soest

94?3 N. Dagan and O. Volij

94?4 R. ~an dcn Brink andP. Bomi

94?~ P.ILM. Riqs andR.P. Gilles

9.1?h I'. l'allan andA. van Sncst

Title

Amoldi Type Methods for Eigenvalue Calculation: Theory andExperiments

Non-additive Belíefs and Game Theory

A Non-cooperative View of Consistent Bankruptcy Rules

Coupons and Oligopolistic Price Discrimination

Bayesian Efficiency Analysis with a Flexible Form: The AIMCost Function

World Bank-Borrower Relations and Project Supervision

A Bargaining Model of Financial Intennediation

The Price Approach to Financial Integration: DecomposingEuropean Money Market Interest Rate Differentials

Sensitivity to Prior Independence via Farlie-Gumbel-Morgenstern Model

A Simplicial Algorithm for Computing Proper Nash Equilibriaof Finite Games

Nonparametric Cointegration Tests

lntersection Theorems on Polytopes

Ranking the Nodes in Directed and Weighted Diraaed Graphs

Youth Minimum Wage Rates: The Dutch Experience

Bilateral Comparisons and Consistent Fair Division Rules in theContext of Bankruptcy Problems

Digraph Competitions and Cooperative Games

I"he Interdependence between Production and Allocation

I amily I,abuur tiupply and " 1"axcs in Iroland

Nu. Authur(s)

9427 R.M.W.J. Beetsmaand F. van der Ploeg

9428 J.P.C. Kleijnen andW. van Grocncndaal

9429 M. Pradhan andA. van Soest

9430 P.J.J. Herings

9431 H.A. Keuzenkamp andJ.R. Magnus

9432 C. Dang, D. Talman andZ. Wang

9433 R. van den Brink

9d34 C. Veld

9435 V. Feltkamp, S. Tijs andS. Muto

9436 G.-J. Otten, P. Bomi,B. Peleg and S. Tijs

9437 S. Hurkens

9438 J.-J. Herings, D. Talman,and Z. Yang

9439 E. Schaling and D. Smyth

9440 J. Arin and V. Feltkamp

9441 P.-J. Jost

')dJ2 .I. Bcndor. D. Mookhcrjcc.and U. Ray

9443 G. van der Laan,D. Talman and Z. Yang

9444 G.J. Ahnekinders andS.C. W. Eijffinger

9445 A. De Waegenaere

Title

Macroeconomic Stabilisation and Intervention Policy under anExchange Rate Band

Two-stage versus Sequential Sarnple-size Determination inRegression Analysis of Simulation I:xpcrimcnts

flousehold Labour Supply in Urban Areas of a DevelopingCountry

Endogenously Determined Price Rigidities

On Tests and Significance in Econometrics

A Homotopy Approach to the Computation of EconomicF,quilibria on the Unit Simplec

An Axiomatization of the Disjunctive Permission Value forGames with a Perrnission Structure

Warrant Pricing: A Review of Empirical Research

Bird's Tree Allocations Revisited

The MC-value for Monotonic NTU-Games

Learning by Forgetful Players: From Pritnitive Formations toPersistent Retracts

The Computation of a Continuum of Constrained Equilibria

The Effects of Intlation on Growth and Fluctuations inDynamic Macroeconomic Models

The Nucleolus and Kernel of Veto-rich Transferable Utilit~Games J

On the Kole ofCommitment in a Class uf Signalling I'roblcmti

Aspirations, Adaptive Leaming and Cooperation in RepeatedGames

Modelling Cooperative Games in Permutational Structure

Accounting for Daily Bundesbank and Fcderal ReservcIntcrvention: A Priction Modcl with a tinltcl l npprc:uion

Equilibria in Incomplcte Financial Markcts with I'ortfi~liuConstraints and Transaction Costs

No. Author(s)

9446 E. Schaling and D. Smyth

9447 G. Koop, J. Osicwalskiand M.F.J. Steel

9448 H. Hamers, J. Suijs,S. Tijs and P. Bonn

9449 G.-J. Otten, H. Peters,and O. Volij

94511 A.I.. Bovcnbcrg :mdS.A. Smulders

9d~ I F. Verboven

9452 P.J.-J. Hcrings

9453 D. Diamantaras,R.P. Gilles andS. Scotchmer

9454 F. dc Jong, I'. Nijmanand A. Riicll

9455 F. Vella and M. Verbeek

9456 H.A. Keuzenkamp andM. McAleer

9457 K. Chatterjee andB. I)utta

9458 A. van den Nouweland,B. Pclcg and S. "I ijs

9459 T.ten Raa and E.N. Wolff

946o G.J. Almekinders

9461 J.P. Choi

946? J.P. Choi

9463 R.I L Gordon andA.1.. Bo~ cnberg

Titlc

The Effects of Inflation on Growth and Fluctuations inDynamic Macrceconomic Models

I lospital lil7icicncy Analysis ' I'hrough Individual Effects: ABayesian Approach

The Split Core for Sequencing Games

Two Characterizations of the Unifonn Rule for DivisionProblems with Single-Peaked Preferences

fransilional Impacts of Environmental Policy in an lindogcnousGrowth Model

International Príce Discrimination in the European Car Market:An Econometric Model of Oligopoly Behavior with ProductDifferentiation

A Globally and Universally Stable Price Adjustment Process

A Note on the Decentralization of Pareto Optima in Economieswith Public Projects and Nonessential Private Goods

Price Effects of "I'rading and Components of the Bid-ask Spreadon the Paris Bourse

Two-Step Estimation of Simultaneous Equation Panel DataModels with Censored Endogenous Variables

Simplicity, Scientific Inference and Econometric Modelling

Rubinstein Auctions: On Competition for Bargaining Partners

Axiomatic Characterizations of the Walras Correspondence forGeneralized Ecunomies

( tutsourcing of Services and Productivity Growth in GoodsIndustries

A Positive Theory of Central Bank Intervention

Standardization and Experimentation: Ex Ante Versus Ex PostStandardization

11erd Behavior, the "Penguin Effect", and the Suppression ofInformational Diffusion: An Analysis of InfortnationalExtemalities and Payoff Interdependency

Wlty is Capital so Immobile Intemationally'?: PossibleExplanations and Implications for Capital Income Taxation

No. Author(s)

9464 E. van Damme andS. Hurkcns

Title

Games with Imperfectly Ohservable Commitment

9465 W. Guth and E. van Damme Information, Stmtegic Behavior and Faimess in UltimatumBargaining - An I?xpcrimcntal Study -

9466 S.C.W. Eijffinger andJ.J.G. Lemmen

9467 W.B. van den Hout andJ.P.C. Blanc

9468 H. Webers

9469 P.W.J. De Bijl

9470 'I~. van dc Klundcrt andS. Smuldcrs

9471 A. Mountford

9472 A. Mountford

9473 L. Meijdam andM. Vcrhoeven

9474 I.. Mcijdam :mdM. Vcrhixvcn

9475 "1,. Yang

947G H. Hamers, P. Borm,R. van de Leensel andS. Tíjs

9477 R.M.W.J. Beetsma

9d78 R.M.W.J. Beetsma

9479 J.-J. Hcrings andD. Talman

9480 K. Aardal

9481 G.W.P. Charlier

9482 J. Bouckaert andI I. Degryse

The Catching Up of European Moncy Markets: The DegrecVersus the Speed of Integration

The Power-Series Algorithm fur Markovian Queueing Nehvorks

The Location Modcl with Twu Periods of Price Competition

Delegation of Responsibility in Organizations

Nurth-South Knowlcdgc Spilluvcrs and Cumpctilion.Convergencc Vcrsus Divcrgence

Trade Dynamics and Endogenous Growth - An OverlappingGenerations Model

Growth, History and Internaliunal Capital Flows

Comparative Dynamics in Perfect-Foresighi Models

('onslrainlx in Pcrlccl-Foresi}~hl M~~dclc I hc Cnsc of OId-AgcSaviugs and I'ublic I'cnsion

A Simplicial Algorithm for Tcsting the Integral Property of aPolytope

The Chínese Postman and Delivery Games

Servicing the Public DehC Canunent

Inflation Vcrsus la.eation: RcprescntaUvc Ucrm,c.racy and I':nt,;Nominations

Intersection Thcorcros with a('ontinuum uf Intcrscction I'oint,

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A Smoothed Maximum Scorc Iatimator liir Ihc Binary (' huiccPancl Data Modul wilh Individual I ixcd I.Ilcds ;uidApplicalion lo Labuur Pnrcc I';irlicipali~~u

Phonebanking

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9483 B. Allen, R. Deneckere, Capacity Precommihnent as a Barcier ro Entry: A BertrandT. Faith and D. Kovenock -Edgeworth Approach

9484 1.-J. Herings, Equilibrium Adjustment of Disequilibrium PricesG. van der Laan, D. Talman,and R. Venniker

948í V. Bhaskar Informational Constraints and the Overlapping GenerationsModel: Folk and Mti-Folk "fheorems

9486 K. Aardal, M. Labbé, On lhe 'fwo-levcl Uncapacitated Facility Lcx:ation ProblemJ. Leung, end M. Queyranne

9487 W.B. van den Hout and The Power-Series Algorithm for a Wide Class of MarkovJ.P.C Blanc Processes

9488 F.C. Drost,C.A.J. Klaassen andBJ.M. Werker

Adaptive Estimation in Time-Series Models

9489 "L. Yang A Simplicial Algorithm for Testing thc Intcgral Propcrty ofPolytopes: A Revision

9490 H. Huizinga Real Exchange Rate Misalignment and Redistribution

9491 A. Blwne. D.V. DeJong, Evolution of the Meaning of Messages in Scnder-ReceiverY.-G. Kim, and Games: An ExperimcntG.B. Sprinklc

9492 R.-A. Dana. C. Le Van, General Equilibrium in Asset Markets with or without Short-and F. Magnien Selling

P.O. BOX 90153. 5000 LE TILBURC~. THF NFTHFRLANDSBibliotheek K. U. BrabantN i ~ i~~~~u ~ ~ W ~ ~ ~w iu