A topological invariant for competitive markets

ELSEVIER

Journal

of Mathematical

Economics

28 ( I9971 445469

A topologica invariant for competitive

Thih paper summarizes the concepts Chichitnisky ( Eccrnmzic I’~zcwry. 1995. ing that limited arbitrage is necessary equilibrium and for the compactness ( Anzrtkmz Em~zorpzir: t&dew, 1992,

of globrti cones and limited arbitrage introduced in and ihe corresponding results establishand sufficient for the existence of of Pareto frontier (announced in Chic 84, 427-434. and Chichilnisky ( Bzzlietirr Americ*uzz Mutherzzrzticd Society. 1993, 29, 3 89-207). Using the same global cones I my earlier results to encompass ‘mixed economies’ based on Chichilnisky ( sion Paper No. 9527. 1995). I introduce a topofo@wl in~czrimzt for competitive markets which deepens the concept of limited arbitrage. This invariant encodes exact informati~ on the equilibria and on the social diversity of the economy and all its s&economies, predicts a failure of effective demand. K~p~rd.~:

Arbitrage;

Topology;

Markets;

5. 79-108),

Sociul

diversity;

Limited

Itrbitrage

1. Intrductim Limited arbitrage is central to resource allocation. It is simultaneously necessary and sufficient for the existertce of a competitive equilibrium, * for non-emptiness af the core ’ and for the existence of satisfactory social choice * UNESCO

Professor of Mathematics and Economics and Director. Program on ln~o~ti~ Columbia University. E-mail: gc9Gholumbiu.edu. ’ See Chichilnisky (1991. 1992b. 1993~. 6994, l995a, b and Chichiln&y and Heal (1992). ’ Set Chichilnisky (l993b. 1994. l995b, 1996).

R~SOUUXS.

0304~4048/97/$17.00 Q 1997 Elsevier PII S0304-4066f97~00798-7

Scicmx

S.A. All rights

reserved,

a

rules, ’ recently, it was shown to be related to the uniqueness of market equilibrium (Chichilnisky, 1996c). lJsing the original concepts of global cones and limited arbitrage introduced in Chichilnisky ( 1992b. 1995a), I expand the scope of the theory by showing that limited arbitrage is necessary and sufficient for the existence of a competitive equilibrium in economies where different traders may have different types of preferences. ’ What if limited arbitrage fails? To study this question I introduce a new concept: a topological inv,ariant for competitive markets, denoted CH. This invariant defines various types of social diversity. ’ Ilt deepens limited arbitrage, encoding exact information about the existence of equilibrium, the core and social choice for every subeconomy (Chichilnisky, 199Sb). CH predicts a failure of *effective demand’ in Arrow-Debreu economies (Section 5). Based on my results on necessary and sufficient conditions for the nonempty intersection of families of sets in Chichilnisky (1993a), I show that the topological invariant CM gauges precisely which market cones intersect and which do not. A special case is when the invariant C’H is zero: then ail the market cones intersect and the economy has limited arbirrage. Both limited arbitrage and the’ invariant CH are defined on the basis of a concept of global cone 1 introduced in 199 1 (Chichilnisky, 1991, 1992b) and in Chichilnisky (1995a). My global cones, and the condition of limited arbitrage they imply, are always the same throughout my work including this paper: their notation is adapted to the context. Global cones consist of those directions along which the utility function does not achieve a maximum. They are generally smaller than the recession cones which are used in other literature on arbitrage and equilibrium: the latter cones consist of those directions along which the utility does not decrease. ” This makes limited arbitrage unique in that it only bounds the feasible utility levels which the economy achieves. Other no-arbitrage conditions bound, instead, the set of individually rational and :Pasible allocations. Limited arbitrage is therefore weaker than the rest. which ix why it can be simultaneously necessary and sufficient for the existence of eqtitiibrium. It has been shown that limited arbitrage is equivalent to the compactness of the Pareto frontier in utility space, ’ and this controls the existence of equilibrium, the core and social choice rules, a crucial relationship which is extended here to mixed economies.

’ See Chichilnisky # 1991. 199%. 199-l. 199Sh). ’ Following Chichilnisky ( I995h”. ’ SW ~1st Chichilnisky ( 1992n. : 99Sh). ” The difference is significant. 1Jhen a utility function is conau~. the recession cone is the whole space. while my glohal cone is empty. Reccxsion cones are very different from gtohal cones. ’ The Pareto frontier is the WI of undominuted and individually rational utiliry values. This equivalence was established in Chic MnisLy ( 199%. 1994. 199%. h. I99ba) and Chichilnisky itnd Heal (1985. I99Ib. 1993). where it was used to prove the e)tiacnce of equilibrium. the core and social choice rule\ . .

CH is a topological invariant because it does not vary with continuous deformation of the commodity space. In particular it does not depend 01 of measurements of the economy. indeed, CH is only sensitive to t properties ol: the preferences of the traders, as measured by their global cones. This paper identifies an important role for social diversity. Markets need diversity to be useful, but as defined here too much diversity can be a hindrance. This t-&es questions about the role of diversity in resource allocation. Diversity is generally a positive force in the adaptation of a group to its environment. Hs it possible that markts require less diversity to function that what would be ideal for society’s success:l?lladaptation? In more general terms: Are our economic institutions sustainable? The paper is cqanized as foilows. Sections 2-4 summarize my originat results on the equivalence of limited arbitrage, the compactness of the Pareto frontier and the existence of an equilibrium given in Chichilnisky (1992b, 1994, 1995a) extending them on the basis of Chichilnisky (1995b) to encompass a larger class of mixed economies. Sections 5 an : 6 cover social diversity and the failure of effective demand, and Section 7 introduces the topological invariant CH and develops its properties.

2.

Definitions

This section addresses ‘mixed economies’ on the basis of Chichilnisky ( 1995bj. This is a class of economies where different traders may have different types of preferences, and is larger than the class of economies I covered earlier in Chichilnisky (1995~1). The definitions and results provided here are idenrical to those in Chichilnisky ( 1995a, i 994a) when restricted to the domain of ‘homogeneous’ economies covered by Chichilnisky ( l99Sa, l996a), but are correspo ingly more general than those of Chichilnisky (1995a, t996a) within mixed economies. ’ In all my work the global cones are the same and limited arbitrage is the same condition. The notation is adapted to the context.

’ The results on limited arbitrage. the Pareto frontier and equilibrium in Chichilnisky ( 1995*, I996a) hold in ‘homogeneous’ economies where either all indifferences have half lines or none do, ati where either all traders hnvc indifference surfilces bounded below. or none do; see p. 103. !&xtion 7.0.1. lines 1-5 of Chichilnisky (19951); mixed cases were covered in Chichilnisky (1995b). These results originated from a thcorcm in Chichilnisky and Heal (1984). a paper which was submitted for publication in February 1984 and WBS published in February 1993 fChichilnisky and Heal, 1993). Chichilnisky and Heal (1993) introduced a non-arbitrage condition C which is identirof IQ lirn~~~d arbitrage for preferences without half lines, and proved shut it is sufficient for the Pareto frontier (Lemmas 4 without half lines. ;Ind proved that it is sufficient for the Pare10 frontier (Lemmas 4 and 5) and for the existence of an equitibrium Whcorem half lines, with or without short sales, und in finite or infinite dimensions. Chichi Section 4 below discuss the iitcrzture further.

An Arrow-Debreu markei =(X, f2,1, ufl, h = I ,..., H} has H2 2 traders, all with the same trading space X, ’ X = Rf or X’ = RN, N 2 2. The results presented here also hold in infinite dimensions (Chichilnisky and Neal, 1992). Traders may have zero endowments of some goods. L!,, E RT denotes trader h’s property rights. When X = R y, J-J,,# 0, VIZ, and C,, & = J2 z+ 0. Assumptiorz 0. Trader h has a preference represented by a continuous, quasiconcave and increasing function ZI,,: X + R. lo All the assumptions and all the results in this paper are ordinal, ‘I therefore without loss of generality we choose a particular utility representation for the traders’ preferences which satisfies: for all It, M,,(O)= 0 and sup{ c:\ E ,YI~f,,= x. This condition can be removed without changing any result at the cost of more notation. Assumptions I. When X = R y, VI+ > 0, any boundary ray f Il,

‘(I-).

is trausversal to

‘I

Remurk 1. Assumption 1 means that either a boundary ray r does not intersect u/T ‘(t-1 or, if it does, the utility clh does not achieve a maximum on r. l Lkfinitiol2 1. A preference on RN is unifkdy non-satic~trd if it has a utility representation u with a uniformly bounded rate of increase; e.g. when II is smooth 3K, E > 0, such that K > 11Dla(x) 11> C, I/x, y E RN. I4 The following Assumption 2 restricts the rate of increase of the representing utility, and allows indifferences with or without ‘flats’. Geometrically, this assumption includes two types of preferences: (a) those preferences where on every

449

indifference surface of a trader’s utility the map x -+ Drr(x)/ 11D4x) 11from allocations to normaiized gradients is closed, and (b) those preferences where this map is open. The two types of preferences are quite different, indeed the two cases are exclusive: IIn case (a) every indifference surface contains half lines in every direction while in case (b) no indifference surface contains half lines in any direction, Case (b) was covered previously in Chichilnisky (l!EGa, 19%& Formally: 2. When X = RN, preferences are uniformly non-satiated, and they satisfy one of the following two mutually exclusive conditions: (a) the normalized gradient to any closed set of indifferent vectors define a closed set I5 or (b) no indifference surface contains half lines.

Assrmption

Remwk 2. There is one situation when the case X = RN is formally identical to the case where X = Ry and where all traders have interior endowments: this is when X = RN and all traders’ indifference surfaces are bounded below. I6 In this case short trades are allowed but no trader wilt trade below the utility level of the initial endowment so short trades are bounded below. It is worth observing that uniform non-satiation ensures that if one trader has one indifference surface bounded below, then all this trader’s indifference surfaces are bounded below. 2. A mixed economy is one which has more than one typo of preferences: some traders have preferences of type (a) and others of type (bB, and/or some traders have indifferences bounded below, and others not.

Definition

3. Sections 2-7 of this paper and Chichilnisky (1995b3) cover mixed economies; in Chichilnisky (1995a) either all traders had indifferences bounded below or none did, and either all traders were of type (a), or all traders were of type (b); see Chichilnisky (1995a, p. 103, Section 7.0. I ., lines f-5). The definitions and results are however identical to those in Chichilnisky (1995a) for the cases which that paper covered. Remark

This section defines global cones, a concept which hs the same throughout my work and appeared first in its most general form in Chichilnisky (1995aI. The Is I.e. the map x + D;r,(.r)/ 11Du,~ ix [I maps closed sets of an indifference in the unit sphere. This mcanh fi?at ihe gradient directions to any indifference set. and conresponds to cae (a) in Chichilnisky (iPEaS. encompassing indifferences arc banda! below or not. I’ A set S is said to lx ~.WNI&V~ b&~ when 3~ E R” :Vx E S. s 2 .v.

surface into closed sets surface defk a cloxed here’ preferences whae

cones presented here are identical to those in Chichilnisky (1995a), for the homogeneous economies considered in that paper. The notation is adapted to fit the context. Two cases, X = RN and X = RF, are considered separately. Consider first X = RN. Re$%tion 3. For trader h define the cone of directions A,(f2,1) along which utility increases without bound: The rays of this cone intersect all indifference surfaces preferred to J2/,. In case (a) the cone A,(fl,,) is open (Proposition 1 of Chichilnisky (1995a)). When augmented by the part of its boundary along which utility never ceases to increase A,,(fi,,) det?nes the gZ&aZ cone G,S(L?,l)-both cones are new in the literature: ” G,,(O,,) = {XEX

and

It is important to observe that this cone is identical to the global cone G&f],,) which was used in Chichilnisky (1995a) to define limited arbitrage; see Chichilnisky (i995a, p. 85, (4)). The alternative notation used here is simplifying in that the cone G,$L$,,) treats all convex preferences in a unified way. In all cases under Assumption 2, G,,(O,,) is shown in Proposition 3 of the Appendix to be identical to the global cone in Chichilnisky (199Sa), namely in case (a) when preferences have half lines G,j Q,#) equals A,( a,,>, and it equals its closure when indifferences contain no half lines, i.e. in case (b); G,,(O,,) is independent of the initial endowment a,,. ” The marker cone of trader /2 is u,(Cn,,) =(z~X:Vym&(fJ,,),

(xv v>=)

D,, is the convex con2 of prices assigning strictly positive value to all directions in GA 4 1. I7 The cone

A,JRa)

has points

in common

with

Debreu’s

(1959)

‘asymptotic

cone’

corresponding

to

the preferred set of I{/, at the initial endowment R,,, in that along any of the rays of A,,(&) utility increases. Under Assumption I, its closure 3 a,, 1. equals the ‘recession’ cone used by Rockafelier in the 1940s. hltr not generally: when preferences have ‘fans’ as defined in Chichilnisky ( 1995b). then the recession cone differs from A,$fl,). Along the rays in A,(fL,) not only does utility increase forever. but it increases beyond the utitity level of any other vector in the space. This condition need not be satisfied by asymptotic cones or by recession cones. For exampte. for Leontief preferences the recession cone through the endowment in the closure of the upper contour. which includes the indifference curve itself. By contrast. the cone A,,(&) is the interior of the upper contour set. Related concepts appeared in Chichilnisky (1976, 1986. 1991. 1995a. 1996b); otherwise there is no precedent in the literature for global cones A,(fI,,) or A;$ fl,,,>. ” The cone G,( &I is identical to that defined in Chichilnisky (1995a. p. 18 (4)): because it is the same at every allocntion under Assumption 2. it is called ‘uniform’ and denoted also G,,,

G. Chichilnish~

/ Joumd

of MarhemnGcal

Economics

28 I /997)

445469

Consider next the case: X = Rt. The market cone of trader h is: Dl(f&)

= D,,( 0,) = D,,( 0,),

I-I S(E),

if S(E) c

otherwise.

where S(E) is the set of sqmrts defined as I’)

to rational uJt?wdable eficient

(3) allocations,

S(E) = (I: E RN : if ( xi . . . xH) E f T (L’. x/t - In,) = 0, and u,,( z,,) 2 lt,$ xlr) then (c, zA - -rr,> 2 0, Wh),

(4)

and where N is the set of prices orthogonal to the endowments. defined as ” = (I* E R:-(O)

:31z with (L:, 12,j) =O).

(5)

The market cones Dl(J2,) typically vary with the initial endowments. When < VI&,, E Rr,, D\f ( 0,) = D& st,). 21 2.2. Limited rrrbitrage This section defines my concept of limited arbitrage, the same concept that I have used throughout my work. Limited arbitrage was defined in Chichilnisky (1995a) is based on the global cones introduced there and defined above. Definition 4. If X = R N, then E satisfies limited rrrbitrage when W;14+%. h= I When global cones are independent of the initial endowments, this condition is satisfied simultaneously at every set of endowments. When preferences are in case (b) (LA) is identical to the no-arbitrage condition C introduced in Chichilnisky and Heal ( 1984, p. 374), and used there to prove the compactness of the Pareto frontier and the existence of an equilibrium with short sales. This definition of limited arbitrage is also identical to that in Chichilnisky (1995a, p. 89, Section 3.1) in the cases which that paper covers.

I” r = ((A-#. . . . , xl, 1: V. IJ,,(x,,) a u,,(JI,,)). The e.~prtxsiotz ‘II*. sh - &,I = 0’ was missing in the definition of S( El in Chichilnisky (1995a) due to a typographical error, which was corrected in the revised version of Chichilnislcy (199%). 2’ N is empty when Vh. S2, a 0. is the interior of RT. The market cone DL contains Ry+ when S( El has it vector assigning ” Rf, strictly positive income to all individuals. If some trader has zero income. then this trader must have a boundary endowment.

Fig. I. Limited arbirrage is satisfied. The two global cones Iic in the halfspxe defined by P. There are no feasible trades that increase utilities without limit: these would consist of pairs of points symmetrically placed about the common initial endowment. and such pairs of points lead to utility values below these of the endowments at a bounded distance from the initial endowments.

This definition is identical to that in Chichilnisky (1995a, (8), p. 91, Section 3.3).

Fig. 2. Limited arbitrage are sequence of feasible

does not hold. The allocations such as (

gIobalCO?lCS are not contained w,. w;.w,. Wi 1 which produce

in ;I half space. and there unbounded utilities.

The set of feasible indiuidwdiy XH : vk tt&xlr) 2 tth(gtlr)r and cf= rational Latility alkocatims is

) = (u=

(U,,...,U”)

rational I xh s 0).

tmdes is T = ((xl, . . . , x,) E e set of feasible in raally

ERH:u~~=lli,(xn)

2 tt,i(d2,) where (x ,,.. ., xH) E Tj. The Pm-et0.fronikr

P(E) is the set of ~~domi~ated vectors in U( - 3VE

U(E)

with V> U).

A competitive equilibrium is a price vector p * E R$’ and an all~atio~ cx; . . . X;;)EXH such that C;=, x; - 9‘r,t = 0 and xi maximizes uk over t budget set B,(~*)={xEX:(X-a,?, p*> =Oj. 2.3. A fmancinl

interpretation

rzf limited

arbitrage

It is useful to show the connection between limited arbitrage alad the lotion of ‘no-arbitrage’ used in finance. The concepts are generally different, but in ce cases they coincide. In financial markets an arbitrage opportunity exists gains can be made at no cost or, equivalently, by taking no risks. illustration of the link between limited imrbitrage and no-arbitrage is an where the traders’ initial endowments are zero, J2, = 0, and the

In brief: in this exam bolrnded or limited gains from trade at zero cost. In case (b) limited arbitrage has a stricter meaning: it implies that t condition C introduced in Chichilnisky and Heal ~1984, p. 3741. When the traders’ utilities are linear functions, the twa concepts coinci re is limited arbitrage if and only if there is no-arbitrage as defined in finance. Iin brief: in linear economies, limited arbitrage ‘collapses’ into no-ah&age.

The Pareto frontier P(E) is the set of feasible. individually rational rrndominated utility allocations. The compactness (and non-emptiness) of Pareto frontier is a crucial step in establishing the existence of a c equilibrium.

It is useful to clarify the difference between limited arbitrage and conditions used elsewhere. ” Other conditions used in the literature require a compact set of feasible individually rational allocations T. 23 For example, the no-arbitrage condition C of Chichilnisky and Heal (1993) ensures a compact set of individually rational and feasible trades T, and thus the compactness of P (El. ‘-I By contrast, limited arbitrage ensures a compact utility possibility set, witizmlt requiring the compactness of the underlying set of trades r: see Chichilnisky (I 991, 1992a, b, 1994, 1995a, b, 1996b) and Chichiinisky and Heal ( 199 1b). It has been established within homogeneous economies that limited arbitrage is necessary and sufficient for a compact utility possibility set U(E), and hence the compactness of its boundary P(E); ” for details see Chichilnisky (1992b, 1995a) and Chichilnisky and Heal (1991a, 1993). The following theorem extends this result to mixed economies following Chichilnisky ( I995b).

-” Case (b). when indiffercnces contain no half lines. is particularly simple: under limited arbitrage feasible and rational trades always define a compact set. exactly as in economies without short sales, Chichilnisky (l996a). ” Without short sales the set ‘I’ is always compact. In economies with short sales.Chichilnisky and Heal (1984. 1993). introduced the ‘no-arbitrage condition C’. and similar conditions were used in Werner (19871. Nielsen (1989). Koutsougeras (199.5) (among others). the latter three based on recession cones. Any no-arbitrage condition based on recession cones amounts to ;t compact set of feasible and desirable allocations ‘T m commodity space. set’ e.g. Lemma 2 of Chichilnisky (199%. 1996a secdon 4.2) and Lemma 10.1. Section IO, p. 20 of Koutsougerirs ( 1995). Werner’s huft?cient conditions for existence is somewhat weaker because he removes some constnnt directions from the recession cone, but like all the others it is strictly stronger than limited arbitrage. He requires S = n S, # @ (see Theorem I on p. 1410 of Werner. 1987). where S, = ( p: ( p. a> > 01 for all s E W,} (bottom of p. 1409). and where Wj is the rcccssion cone minus the directions along which the utility is constant. see 1408. p. 6. last two lines. Observe that \V, contains many directions which are not included in G,, such as those directions along which the preference is eventually constant without being constant. Such directions are not part of the global cone G,, defined here and in Chichilnisky ( 1995ak Therefore the cone Wi is strictly larger than the global cone G,,. and the condition S z fi is strictly stronger than limited arbitrage as defined here and in Chichilnisky (1995a). Werner’s condition is therefore not necessary for existence of an equilibrium unless indifferences contain no half lines. see also Proposition 2. (ii). p. 1410 of Werner (1987). In this latter case. Werrrer’s condition bounds the set of all feasible individually rational allocations r as do all other non-arbitrage conditions exctpt for limited arbitrage. ‘-I Lemmas 4 and 5 of Chichilnisky and Heal (1984) prove the compactness ‘no-arbitrage condition C’ which is identical to limited arbitrage when indiffercnccs see Chichilnisky and Heal (1984) and Chichilnisky (199%. b). ” Lemma 2 of Chichilnisky (199Sa) established that limited arbitrage is sufficient F’(E) and Theorem I of Chichilnisky (199%). by the second welfare theorem. arbitrage is necessary for compactness of P(E). because it is necessary for equiiibl itim.

of P(E) using its have no half lines: for compactness of proved that limited the existence of an

cient for the compactness and non-emptiness of the Pareto frmtier P( allocations and randominated utility allocutions may however define wbormded sets. ProoJ See the appendix.

Cl

Proposition 1. When X = RN, limited arbitrage implies that the Pareto frontier P(E) c Ry is homeomorphic to a simplex. 26 Proof This follows from Theorem 1 and the convexity of preferences, cf. Arrow and Hahn (1971). 0 3. I. Interpretation

qf limited arbitrage as bounded gains from trade ,t*lren X = R”

Limited arbitrage has a simple interpretation in terms of gains from trade when X = RN. Gains from trade are defined as follows: C(E) =

supf ,,T,,~,4~h) ( s, . . . . . ,{

-4u4J:

T was defined above as the set of feasibIe and individually rational allocations. Proposition 2. The economy E satisfies limited arbitruple if and on/y if it bus bounded gains -from trade which are attainable, i.e. 3x * E T:

ProoJ Proposition 3 in the Appendix shows that under the conditions global cones are independent of the initial endowments, so we n;ay assume without loss of generality that Cn,!= 0 for all tz. Sufficiency first. The sum Ef=, uh defines a continuous function of ( !l,, . . . , 1~~)E Ry; this continuous function always attains a maximum when the vector II,, . . . , 14~~ varies over the set U(E) C I?:. which is compact by Theorem 1 when limited arbitrage is satisfied. The converse is immediate from the proof of necessity in Theorem t (see the Appendix), which establishes that when limited arbitrage fails, there exist no undominated individually rational utility allocations, i.e. P(E) is empty. q ” A topological space X is homcomorphic to another Y when there exists a on-to-one onto map .f: X --, Y which is continuous and has a continuous inverse. ” In the following equation l z’ must be replaced by ~up(~. , c xl (E,“, Iu,(.r)u,j11,,)13 w&n Vh, sup , E x u,j .t I< 3~. The difference between Proposition 2 2nd Corollary 1 is that b.xmded gains from trade need not be uttainable.

PrtmfI Recall that idscase (a), G,, = A,r by Proposition 3 in the Appendix. In this case limited arbitrage is also necessary for bounded gains from trade.

4. Gomgetitiv~

equiIibrium

0

and Bimited arbitrage

It was proved in Chichilnisky ( 1995a) that limited arbitrage is simultaneously necessary and sufficient for existence of a competitive equilibrium in economies with or without short sales. ” This result was established in Chichilnisky (1995a) for economies where prefererlces were homogeneous: either all preferences are of type (a) or all of type (b), and either all have indifferences bounded below or none do. Below I show that this is true for mixed economies, where some preferences may be of type (a) and others of type (b), where some indifferences are bounded below and others are not. This follows Chichilnisky (1995b). This result includes the classic economy of Arrow and Debreu without short sales, which was neglected previously in the literature on no-arbitrage. The equivalence between limited arbitrage and equilibrium extends to economies with infinitely many markets, see Chichilnisky and Neal (1992). Recall that limited arbitrage implies a compact set of feasible and individually rational utility allocations U(E), without reqcriri/lg that the set of underlying trades T be compact. Therefore under limited arbitrage the set of equilibria, the set of undominated utility allocations and the core may all be unbounded when X = R*‘. ” Chichilnisky and Heal (1984). Hitrt (1974) Hammond (1983) and Werner (1987) among nthtrs. have defined various no-arbitrage conditions which they prove. under certain conditions on prcfercnccs;. to be sufficient for existence of equilibrium. Hart and Hammond study asset market models which arc incomplete economies because they lack forward markets. and therefore have typically inefficient equilibria, None of these no-arbitrage conditions is generally necessary for existence. For the speciitl c;1sc (b). within economies with short sales (which exclude the classic Arrow-Debreu’s market), and where recession cones are assumed to be uniform, Werner ( 1987) remarks correctly (p. 14 10. last para. I that another related condition (p- 1410, line -31 is necessary for existence, without however providing a complete proof of the cquivalcncc between the condition which is ncccssary scfficient. The two conditions in Werner (1987) arc different and are detincd on the sufficient condition is defined on cones S, ip. 1410. line -14) while the defined on other cones, D, (p. 1410. -3). The equivalence between the two cones of vet another family of cones W, (see p. 1410. lines 13-4). The definition of W, shows :Cat W, is different from the recession cone R,. (which ure uniform therefore tbl? cone W, need not be uniform even when the recession cones are. Proposition

2.

and that which is different sets of cones: necessary condition is depends on properties on page 1408. line -15 by assumption) and its needed in Werner’s

G. Chichilttisky

/ Jorrmd

qf Mu~hetttatid

Economics

28 llW7)

445-460

Theorem 2. Consider an economy E = {X, uh, 52,. h = 1,. . . , H). where H 2 2, with X=RN or X=R$! and N 2 1. Then the f<~iiowing two properties ore eyuicalent : (8 The economy E has limited arbitrage. (ii) The economy E has a competitifie equilibrium. PM$

See the appendix.

IJ

Remark 5. When X = RN the set of competitive equilibria need not be bounded. 4. I. Subeconomies with competitice equilibria As shown in Chichilnisky ( I995a) the condition of limited arbitrage need not be tested on all traders simultaneously: in the case of RN. it needs only be satisfied on subeconomies with no more traders than the number of commodities in the economy, N. plus one. Definition 6. A k-trader sub-economy of E is an economy F consisting of a subset of k I H traders in E, each with the endowments and preferences as in E: H), cardinality CJ) = k I If}. F = Ix, uk. J-J,,, h EJC{l,..., Theorem 3. The f&llowing few properties of m econom? E with trading space R fV are equirxlient: (i) E has a competitice equilibrium. (ii) Erery sub-economy of E tc*ith at most N + 1 traders has a :*ompetitiice equilibrium. (iii) E has limited arbitrage. (iv) E has limited arbitrage&r any subset of traders rvith no more that N + 1 members. Prooj Theorem 2 implies (i) CJ (iii) and (ii) * (iv). That (iii) = (iv) follows from Helly’s theorem. which is a corollary in Chichilnisky (1993): Consider a family WJj= 1. . . H of convex sets in RN, H, N 2 I. Then H n U,. + 4 if and only if n Ui # 6. i=l

jEJ

for any subset of indices .I C{ 1 . . . H} having at most N + 1 elements. In particular, an economy E satisfies limited arbitrage, if and only if it satisfies limited arbitrage for any subset of k = N + 1 traders, where N is the number of commodities in the economy E. CI 5. A failure OFeffective demand when X = R”, In economies without short sales X = Ry a compensated equilibrium or quasiequilibrium always exists, Arrow and Hahn ( 197 1) and Negishi ( 1960).

However, when limited arbitrage fails, quasiequilibrium are ill-behaved; I show below that every quasiequilibrium has a failure of ‘effective’ demand, in the sense of Arrow a;;d Hahn (1971, p. 345). 7. A compensated eiiuilibrium or quasiequilibrium is a price p * and an allocation x * at which every trader minimizes the cost of achieving the give:1 utility level, and Cf= 1(xi - fi,,) = 0.

Definition

Although a quasiequilibrium allocation x + satisfies Zf= ,(xi - R,,) = 0, x * can exhibit ‘excess demand’, if demand is computed from utility maximization under a budget constraint. This is because at the quasiequilibrium allocation traders minimize costs, but may not maximize utility. If at a quasiequilibrium allocation every trader maximizes utility, then the quasiequilibrium is also a competitive equilibrium. 8. An allocation and a price exhibit a failure of effective demand in the sense of Arrow and Hahn (1971) when in some markets there exists excess demand (computed from utility maximization subject to a budget constraint), but the value of this excess demand is zero.

bJ?izitkm

Theorem 4. When X = R$! a failure of limited crrbitrage implies that there is a failure of effecm’ivedemand: every qtlasieqlrilibriutn has excess demand, but the market value of this demand is zero. Proof A quasiequilibrium price p * = ( p; , . . . , pI; > E Ry always exists because X = RT. As seen in Theorem 2 if limited arbitrage fails there is no competitive equilibrium. This implies that at any quasiequilibrium some trader has zero income (for otherwise a quasiequilibrium is always a competitive equilibrium). Therefore for at least one market i, pi = 0, and the value of excess demand in the ith market is therefore zero. For some trader h, p * GZ0; (since otherwise the quasiequilibrium would be a competitive equilibrium) so that there is no maximum for if,, on the budget set defined by p * . Therefore there is excess demand at p * in the ith market, but the value of excess demand in this market is zero. Cl 6. Social diversity and limikd

arbitrage

If the economy does not have limited arbitrage, it is called

soci~dly

dirIerse:

Definition 9. When X = RN, the economy is socially diverse if n r= , D,, ==4. This concept is independent of the units of measurement or choice of numeraire. Social diversity admits as many different ‘shades’ as traders: Definitk~n 10. The economy E has index elf diversity I(E) = H - K if K + 1 is the smallest number such that 3J c (1 . . . H) with cardillality of J = K + I, and

n lrEJ4 = (8. The index i(E) ranges between 0 and H - 1: the larger the i the larger the social diversity. The index is smallest (= 0) when all t cones intersect: then all social diversity disappears, and the economy has limited arbitrage. Theorem 5. The index of social diversity is I(E) if and c&y ij H - I(E) is the maximum number of traders fk which euery sabecorromy has a competitive equilibriiam. P~oo~TThis is immediate from Theorem 3. 7. A topd~gical

invariant

for competitive

0 markets

This section introduces a new concept, a tapokgical incariant fbr competitive markets. This is an algebraic object which describes important properties of the market - its social diversity arid the existence of an equilibrium for it or for its subeconomies - in a way that is robust, namely independent of the uni!s of measurement. The topological invariant CH contains exact information about t resource allocation properties of the economy, as shown below. Furthermore it is computabk by simple algorithms from th2 initial data of the problem: its endowments and preferences. Important properties of an economy E with short sales can be describe terms of the properties of a family of cohomology rings denoted CH(E). cohomology rings ” of a space Y contain informatinn about the topological structure of Y, namely those properties of the space which remain invariant when the space is deformed as if it was made of rubber. For formal definitions of the algebraic topology concepts used here a standard textbook is Spanier (19791. The following concept, called a mrve, defines a combinatorial o simpMcial complex, from any family of sets. Starting from any arbitrary family of sets in RN, one defines from it a triangulated set, obtained by ‘pasting up’ simplices in a well ordered fashion (see Spanier, 1979) called a simpkial complex. The simplicial complex created from the family of sets is usually called th.- ‘nerve of the family of sets’, The procedure for transforming any family of sets into a simplicial complex is as follows: each set is represented by a point. which is a vertex in the triangulated space; the intersection of any two sets in the family is represented as a one-simplex, the intersection of any three sets as a Z-simplex, etc. It is simple to see that putting all this together one obtains a simpliciai csrnptex (Spanier, 1979). Formally: 3 A ring is 8 set ~2 endowed with two operations. dcnoted + and X ; tk operation + must define a group structure for Q (every element has an inverse under +I and the operatior x defines a semi group structure for Q; both ogerutions together satisfy a distributive relurioa. A ty&al example of a ring is the set of the integers. another is the rational numbels. both with addit;sa e~:cl mu~tip~~c~i~.

Dt$ttitim

Il. The W~~LVof a family of subsets {V,), _ , . .., I1 in R”, denoted nerve(vi),, *. ...* L is a simplicial compIex defined as follows: each subfamily of k + I sets in w= l.....L with non-empty intersection is a k-simpIex of the nerve (V,)i, I..... L. Now that the nerve of a family of >ets is defined, one looks at the global topology of this simplicial complex, in words, how many ‘holes’ it has, and of what type. This is measure1.i by the coilomology rings of the space, which are groups with an additional operation, wtlich measure precisely the number and types of holes which the space has. The cohomology rings are computable by standard algorithms once the market cones of the economy are known. See Spanier (1979) for definitions. Now we are ready to define our topological invariant. A subfamily of the family of sets (D,),, = ,, . . . H is a family consisting of some of the sets in (D,), = 1v.... H, and is indicated (D,r)h E Q, where Q C { 1,. . . , Y}. The topoiogicul irzcwiant CH(E) of the economy E with X = R” is the family of reduced cohomofogy rings ” of the simplicial complexes defined by all subfamilies W,,), E o of the family (DB)h _ t-z.. . “, i.e. the cohomology rings of neru~ II,,),, c Q forevery Qc{l,...,H): = (H * ( nerve(&),,,Q, VQc{ L..., H)). In the following result I consider continuous deformations of the economy which preserve all the Assumptions in Section 2. Any such deformation will preserve our topological invariant. The following result shows how much information is encoded in CH(E). It shows that the existence of an equilibrium for the economy and its subeconomies are purely topological properties, and they are predicted exactly by the number and the type of ‘ho!es’ in the nerve of the family of market cones of the economy, i.e. by the cohomology rings of this nerve. This result depends on a new theorem in Chichilnisky (1993a) which established that a family of finitely many convex sets has non empty intersection if and only if each subfamily has a contractible union CH(E)

i.e. VQC(l,...,

H}, Id * (nerve{ D,,},,, Q) = 0.

H * (nerve{D,,},,, c,) - 0,

rind (ii) there exists T c (I,. . . , H) with cardinal@ T = K + 6 and

Prmfi This follows from Theorem 5 and Corollary 2 to Theorem 6, p. Chichilnisky (6993ad, which proves that an acyclic family of sets has non intersection if and only if every subfamily has acyclic union, 3’ see Chichilnisky (1993a) for details. 0

8. Conclusions I extended prior results on necessary and sufficient conditions for the existelnce of a competitive equilibrium in Chichilnisky (199Sa) to mixed economies which treat all preferences in a unified way, following Chichilnisky (69955). H defined social diversity and a topological invariant which contains exact information about the existence of a competitive equilibrium for the economy and its subeconomies, and showed that when limited arbitrage fails every quasiequilibrium has a fai6ure of ‘effective’ demand.

Acknowlegements Comments from Kenneth Arrow, Truman Bewley. Duncan Foley, Frank Hahn, Geoff Heal, Wa6t Weller, Ted Groves, Morris Hirsch, Paul Miigrom, Wayne Shafer and two anonymous references are gratefully acknowledged.

Propositim 3. Under Assutnption 2: (8 The cone G,( 0,) equa1.s Ah( aA) when ind&Kerences contain half lines (case (a)) and its closm-e when they do not, (case (b)), and thus it is identical tu the global cone defined in ChkhiM~ky ( 6995ak (ii) the cones Ah( Jr,) = c& fl,) are independent rzf initial endowments 0, E RN; (iii) when preferences are in cases (aI and (b) the gk&al ones G,( a,) are independent of initial endowments, but they are not mxessarily independent in general; and (iv) nmte of these results need hold when Assumption 2 is not satisfied. ” This

result

applies

also to more

general

I’amiiies.

462

ProqJ See also Chichilnisky (1995b). Define the sets: 4, t f&t ) =

1

x

E X : lit,{ O,, +

AX)

2 u,,( f& + TX) when A 2 772 0.

lim uh( R,, + A, X) = 14’)< X and ~j : 111,(0, + Aj.y) = ,,I’}, A,-+-r

and

lim I’~~(&?~~ + Aj-~) = u0 < m and - 3j: u~,(0,, + Ai”) - IC’) . A,+ -I> The three sets A,$ L!,,), B,,( In,,) and C,,,(0,) are disjoint pairwise and A,,( fW ” Bt,( a,) u Cd w u fu QJ = RN* (7) where H,,( a,,) is the complement of the set Ah( a,,) U B,,( f2,t) U Cn( a,,). Observe that by convexity of preferences I#,,(&?,,> is the set of directions along which the utility achieves a maximum value and decreases thereafter. The first step is to observe that if z f B,,(fl,,) U C,,(fI,,), then for all s E RN s a ; * s E A,,( f2,J

(8)

and S-e~.SEHI,(L2,,).

(9) This follows from monotonicity and Assumption 2, which implies that the rate of increase of the utility is uniformly bounded below above zero along the direction defined by any strictly positive vector. Since the utility u,, is non-satiated (8) and (9) together imply that for every E > 0, 3h > 0 s.t. an E neighborhood of a vector AZ E B,(f2,j> U C&IR,) contains a vector s in the set A,,(fi!,,) and another vector I in the set I#,*( iI,*). This implies that the set B,,(fIZ,,) U C,,(fi,,) is in the boundary of the set A,( a,,). The relation between G,(fI,,) and A,,( 0,) stated in (i) is now immediate, cf. Chichilnisky (1995a, p. 85 (4)). The next step is to show that if two different half-Iines I = (L!,; + AL’), 2 U and 171=-{/i,, +A&) are paraHe1 translates of each other, and I c A,(fi,,), then rrzc A,$ A,,), !!A,# E m. This is immediate from Assumption 2, which ensures that the rate of increase of the utility is uniformly bounded above. Therefore the cone A,, is independent of the initial endowments. Observe that for a general convex preference represented by a utility ~1,~the set G,(R,,) itself may vary as the vector J2{, varies, since the set B,$J2,> itself may vary with 0,: at some “n,# a direction z E aG, may be in Bl,(f2,1) and at others Bh(fJh) may be empty and z E C,,(fl,,) instead. This occurs when along a ray defined by a vector z from one endowment the utility levels asymptote to a finite limit but do not reach their limiting value, while at other endowments, along the

same direction z, they achieve this limit. This cannot happen when the preferences are in cases (a) or (b), but is otherwise consistent with Assumption 2. example, and similar reasonings for A,(Q,#) completes the proof of the tion. Cl Protlf t$ Tkr~em I (Limited arbitrage is equivalent to the compactness of U(E))). This result always holds when the consumption set is bounded below by some vector in the space, and in that case it is proved using standard arguments; see, for example, Arrow and Hahn (197 1). Therefore in what follows I concentrate in t case where X is unbounded below. Sufficiency first. Assume E has limited arbitrage. Since global cones are independent of initial endowments by Proposition 3, we may assume without loss of generality that 0, = 0 for all 11.If U(E) was tctjt bounded there would exist a sequence of individually rational net trades ( z{ . . . zb )j= !,?. . . and therefore cormspondingly positive utility levels, such that: Vj, C,zi = 0, and for s~tne ta, limj-+xuh(z~) = 30. Therefore for some Cp, limj 11L’X11= z, and there exists a subsequence of its normalized vectors, denoted also by {&/ 11zi II}, which is convergent. I will now prove that zh = limj ;//II zi’,11is in Gh, the closure of G,. The proof is by contradiction. If z,, E c,t then by quasiconcavity of uh and by the proof of Proposition 3, along the ray defined by zk the utility uln achieves a maximum level ~4’ at A0 z, for some A, 2 0. and decreases thereafter: A > A, a K,$ AZ,) < 11’. Define a function 8 : R, -+ R, by M& AZ,, + 0( A)e) = u*. where e=(l,..., I). I will show that 8 is a convex function. By the convexity of preferences,

= rr,,((aA + (1 -a)i)zl,+(*6(A)

+(1

-a)(e(X)j&?).

(l!)

Thus by monotonicity, 6( CPA+ (1 - cu)% 5 cuti(h) + (! - CX)~(&, which proves convexity. Assumption 2 together with monotonicity implies that the rate of increase of uk along the direction defined by e (or by any strictly positive vector) is uniformly bounded below: % > 0 : 1u,(x + 8e) - u,(x) 1 2 I8 1&, ve E R,, vx E RN* Therefore n,(Azh + tY(A)e) = U* z uh(Azh) + 0(A)&, so that ss,(AzA) s ~8’6( A)&. Note that 8(A,) = 0 and 8(A) > 0 for A > A,. Cince as seen above 8 is convex, lim, + x 8(A) = Q;, and U& Azh) 5 U* - 4%A)6 implies u,( AZ,) + - a. It follows that zh E C,, for otherwise, since limj II zi !l = m and the vectors zi wander arbitrarily close to the direction defined by +, lirrrj 4 r Al& z/j < 0 contradicting individually rationality of the allocations ( zi . . . zk )j E ,,lt.. . Now recall that for some 6 limj,, M ( zJ) = X. By Assumption 2, I u#(x) u,oQl 5ax - y IIV,x, y E RN, so for \ny n and j 1u,(z,“) - IJ#( 2: -je)l s

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K 11je If ; therefore for every j there exists an “j such that M~(2;) - je) > j. Take the sequence {zi, 1) and relabel it (zi}. Now consider the new sequence of allocations {z{ + je/(H - I), . . . , zi - je, . . . , zft + je/(H - I)} and call it also * this defines a feasible allocation for all j and by Assumption 2, k&= 1.2..*...HT V/z, u,~(z$ + P In particular Vh, 11;i 11---)m+ Define now C as the set of all strictly positive convex combinations of the vectors zI, for all tz. Tnen either C is strictly contained in a half space, or it defines a subspace of RN. Since c, zi = 0, C cannot be strictly contained in a half space. Therefore, C defines a subspace; in particular for any 6, 3h,, 10, Vh + s. such that

When one trader s has indifferences without half lines, case (b), then G,e= cX and therefore z,,~G,.=z,=(,. which by (*) contradicts limited arbitrage because h zip&?, z,,) 20 for z,, E c,, all h, and ( p, z, > > Oz, E GK. When for all h the normalized gradients to any closed set of indifferent vectors detine a closed set, case (a), the global cone G,, is open (Chichilnisky, l995a) so that its complement G;; is closed, and the set of directions in 6;; is compact. On each direction of G;,’ the utility cchachieves a maximum; therefore there exists for each h a maximum utility level for LI/, over all directions in G;;. Since along the scquerrce { z$ every trader’s utility increases without bound, Vh3j,! : i > j,, - zd E G,,. However Vj, Z,#zi = 0, again contradicting limited arbitrage. In all cases a contradiction arises from assuming that U(E) is not bounded, so U(E) must be bounded. The next step i:, to prove that the set U(E) is closed when limited arbitmge is satisfied. Consider a sequence of ahocations { zi)j= ,,z , h = I, 2,. . . , H, satisfying y$ IL1: , :i = 0. Assume tha; ‘$j, u,(;i), . . . , lfH(z&) C Ry and converges as j 4 m to a utility allocation c = ( LJ,,. . . , IV,) E Ry, which is undominated by any other feasible utility allocation. Observe that the vector 1’ may or may not be the utility vector of a feasible allocation: when limited arbitrage is satisfied, 1 will prove that it is. Tht* result is immediate if the sequence of allocations (z/}~= ,,? ..., tz = I, 2,. . . , H is bounded: thereeo‘o. rpy I concentrate in the case where the set of feasible allocations is not bounded. Let M be the set of all traders ir E {I, 2.,,,N), which may be empty, for whom the corresponding sequence of allocations I*?.. . is bounded, i.e. 12 E M - 3X,, : Vj, 11zd I[ < K,# < 30; let J be its complement, J = { 1, 2,. . . , H} -- M, which i assume to be non empty. There exists a subsequence of the original sequence of allocations, which for simplicity is denoted also { z8}j, rsz..., lz = I, 2.. . . , H, along which Vh F M, lim+$~, ,,l.+, = z/, exists, and by construction C,#E M zl, + limj ~ 3 C,, E J ti = 0. Consider now the sequence zi/ II zi 11for h E J; it has a convergent subsequence, denoted also zi/ II zd Il. Define zh = l&nj zi/ II zi 11,t2 E J. Then as seen in the first part of this proof, Wh f J, zI, E G,, because II zi Ii -+ x and I(lr(& 2 0. If Vh E J, z,, E G,#, then the uti!ity values of the traders attain their limit for all h { Z , J I } j =

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and the utility vector L: is achieved by a feasible allocation. Therefore the proof is complete. It only remains to consider the case where for some trader g E J, z, f Gh’. As above. let C be the convex cone of all strictly positive linear combinations of the vectors (i& E J. Then either C is contained strictly in a half-space of R’v, or C spans a subspace of RN. Since C,] E M zh -t+““;i ~ r C, E J zi = 0, C cannut be strictly contained in a half space, for otherwise hmi --+%C, E J zi would not be bounded as Vh E J, 11z{ II --, 0~.Therefore there exists a subspace S c RAYspannsrd by (zA),, E J. In particular, -z,ES, i.e. VhEJ, 3h,lQ such that (“J-z,== c 8~ J A,Zp Since in this last case Vh E J, h f g, zh E ch and for h = g, z, E Gg ., by limited arbitrage 3p E n ,, D,, s. t. ( p, :, > > 0, and Vh, ( p, tk) 2 0, which contradicts ( * 1. Since the contradiction arises from assuming that the set U(E) is not closed, U(E) must be closed. Thus U(E) is compact. In particular, since P(E) is the boundary of U(E), limited arbitrage implies a compact Pareto frontier P(E). I establish necessity next. If limited arbitrage fails, there is no vector y f RN such that ( y, z/,) > 0 for all (zh) E G,. Equivalently, there exist a set J consisting of at least two traders and, for each h E J, a vector z/, E G, such that &, e j zh = 0. Then either for some h, q, f A,, so that the Pareto frontier is unbounded and therefore not compact, or else for some h, z/, E aG, n G,, and therefore the Pareto frontier is not closed, and U(E) is not compact. In either case, the Pareto frontier is not compact when limited arbitrage fails. Therefore compactness of I..JtE] is necessary for limited arbitrage. LI Proof of Them-em 2 (Limited arbitrage is equivalent to the existence of a competitive equilibrium). Necessity first. Consider first the case X = RN and assume without loss of generality that 0!, = 0 for all h. The proof is by contradiction. Assume that limited arbitrage fails and let p’; be an equilibri;lm price and x* =(n-l,... xi ) the corresponding equilibrium allocation. A failure of limited arbitrage means that 3 ( z,, . . . , z,) : Vh, zh E G,z, and cf=z Lzp,= 0. By construction crj,(hzh) never ceases to increase with A, because z,~E G,. Since by Proposition 3 global cones are uniform x; + zl, E G,jxi 1, so that by the same reasoning x * cannot be Pareto efficient, contradicting the fact that x * is a competitive equilibrium. Consider next X = Ry. Assume that MB E S(E)3 h f (1,. . . , H} such that (4, 52,) = 0. Then if limited arbitrage is not satisfied f’7 f= , DL{&2,1) = B), which implies that Vp E S(E), 3 h and U( p) E G,(J2,): (p, AL+(p))10,

VA>O.

w I will show that this implies that a competitive equilibrium price cannot exist: the proof is by contradiction. Let p* be an equilibrium price and x * E XN be the corresponding equilibrium allocation. Then p * E S( El. so that by (12) xi -t Au( p) is affordable and strictly preferred to x * for some A > 6, contradicting the assumption that x * is an equilibrium allocation. Therefore limited arbitrage is also necessary for the existence of a competitive equilibrium in this case.

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It remains to consider the case where 3p E S(E) such that V’h E H}, ( 11. 0,) + 0. But in this case by definition n F= I Dl< 0,) # fl since L..., VhE(l..* H), I?:+, c DC< a,,), so that limited arbitrage is always satisfied when an equilibrium exists. This completes the proof of necessity when X = Ry. Sufficiency next. The proof uses the fact that the Pareto frontier is homeomorphic to a simplex. When X = Ry the Pareto frontier of the economy P(E) is always homeomorphic to a simplex, see Arrow and Hahn (1971). In the case X = RN this may fail: for example U(E) may be unbounded. However, by Theorem 1 above, if the economy satisfies limited arbitrage then the utility possibility set U(E) c Ry and the Pareto frontier P(E) c Ry are compact; under the assumptions on preferences, P(E) is then also homeomorphic to a simpIex (Arrow and Hahn, 197 I). Therefore in both cases, P(E) is homeomorphic to a simplex and I can apply a method due to Chichilnisky and Heal (isl84) which extends Negishi’s method of using a fixed point argument on the Pareto frontier 32 to establish the existence of a pseudoequilibrium with or without short sales. 33 It remains, however, to prove that the pseudoequilibrium is also a competitive equilibrium. To complete the proof of existence of a competitive equilibrium consider first X=RN. Then V/z= l,..., H there exists an allocation in X of strictly lower value than the pseudoequilibrium xc at the price p*. Therefore by Lemma 3, Chapter 4, page 8 1 of Arrow and Hahn ( 19711, the quasiequilibrium ( p * , x * ) is also a competitive equilibrium, completing the proof of existence when X = RN. Next consider X = Ry+ and a quasi-equilibrium ( p * , x * ) whose existence was already established. If every individual has a positive income at p*, i.e. V/z, ( p * , &?,I>> 0, then by Lemma 3, Chapter 4 of Arrow and Hahn (1971) the quasiequilibrium ( p * , x* ) is also a competitive equilibrium, completing the proof. Furthermore, observe that in any case the pseudoequilibrium price p* E S(E), so that S(E) is not empty. To prove existence we consider therefore two cases: first the case where 3q * E S(E): Vh, (q * , a,,) > 0. In this case, by the above remarks from Arrow and Hahn (197 I), (y * , s * ) is a competitive equilibrium. The second case is when ‘ifq E S(E), 3h E {I,. . . , H) such that (4, n,,) = 0. Limited arbitrage then implies: 3q” ES(E):Vh, Let x* =x1*,...,

(q**

C) > @ for all 18E G,J on,).

(13)

xi E XH be a feasible allocation in T supported by the vector

7 * rt’egisr.:‘j method for proving the existence of a pseudoequilibrium (Negishi. 1969) applies only to the case where the economy has no short sales. This result was extended by Chichilnisky and Heal (1984. 1993) to economies with short sales. Cere feasibte desirable allocutions may be unbounded; see also Chichilnisky and Heal (1991b) and Chichilnisky ( 19950). 33 A p~adwijuiIibrium. ofso called quasiequilibrium. is an alloccation and a price at which traders minimize cost and markets clear.

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q * defined in (13): by definition, Wh, u,( xi ) 2 u,$J~,> and q + sup Note that any ;t minimizes costs at xi because 9 * is a support. Furthermore x,* is affordable under 4 * . Therefore, (4 * , x * ) can fail to be a competitive equi%ibrium only when for some h, (9 * , xl > = 0, for otherwise the cost minimizing ahocation is always also utility maximizing in the budget set B&q * ) = {w E X:(g’, w} = (y*, L?,>). It remains therefore to prove existence wken (4 * , x,3:> = 0 for some k. Since by the definition of S(E), xc is individuahy rational, i.e. Wh, ttk(xi ) > ra,( fl,), then {q*, xl > =U implies {q*, J-2,) = 0, because by definition q * is a supporting price for the equilibrium allocation x*. If Vh, u,+(x~ I= 0 then xi E aRy, and by the monotonicity and quasiconcavity of uh, any vector y in the budget set defined by the price p*, B&q * 1, must also satisfy M,(Y) = 0, so that xl maximiZea utriity in B,,(y * ), which implies that (q*, x * ) is a competitive equilibrium. Therefore (q * , x *) is a competitive equilibrium unless for some h, Lfh(X;: I> 0. Assume therefore that the quasiequilibrium (4 * , x * 1 is not a competitive equilibrium, and that for some 11with (q * , a,,> = 0, uk( x,TI> 0. Since tlli( xi ) >O and xi E 8RT then an indifference surface of a commodity bundle of positive utility u,(x,T 1 intersects c%?: at xi E a@!. Let r be the ray in aRy containing xi. If WE r then (q*, w) =O, because (q*, xi> =O, Since H,(x~) > 0, by Assumption I llh does not reach a maximum along r. so that w E Gh(-~i 1. But this contradicts the choice of q* as a supporting price satisfying limited arbitrage ( 13) since 3/tand

~‘G,(ln,)

such that (q*,

cc?)=O.

(14)

The contradiction between (14) and (13) arose from the assumption that (Q * , x * ) is not a competitive equilibrium, so that (4 * , x ’ ) must be a competitive equilibrium, and the proof is complete. q

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