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Arbitrage, duality and asset equilibria
Journal of Mathematical Economics 34 Ž2000. 397–413 www.elsevier.comrlocaterjmateco
Arbitrage, duality and asset equilibria Rose-Anne Dana a , Cuong Le Van b,) a
Ceremade, Place du marechal De Lattre de Tassigny, 75775 Paris, France ´ b CNRS, Cermsem, 106-112 bouleÕard de l’Hopital, 75013 Paris, France
Received 27 April 1998; received in revised form 2 August 1999; accepted 17 April 2000
Abstract In finite dimensional economies, it was proven by Werner wWerner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55, 1403–1418.x, that if there exists a no-arbitrage price Žequivalently, under standard assumptions on agents’ utilities, if aggregate demand exists for some price., then there exists an equilibrium. This result does not generalize to the infinite dimension. The purpose of this paper is to propose a ‘‘utility weight’’ interpretation of the notion of ‘‘of no-arbitrage price’’. We define ‘‘fair utility weight vectors’’ as utility weight vectors for which the representative agent problem has a unique solution. They correspond to no-arbitrage prices. The assumption that there exists a Pareto-optimum, can be viewed as the equivalent of the assumption of existence of aggregate demand. We may then define in the space of utility weight vector, the excess utility correspondence, which has the properties of an excess demand correspondence. We use a generalized version of Gale–Nikaido–Debreu’s lemma to prove the existence of an equilibrium. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Arbitrage; Duality; Asset
1. Introduction The problem of existence of an Arrow–Debreu equilibrium in economies with consumption sets unbounded below, has appeared in different economic settings, in particular in the theory of temporary equilibrium Žsee Green, 1973 and )
Corresponding author. Tel.: q33-1-40-77-8400; fax: q33-1-44-24-3857. E-mail address: [email protected] ŽC. Le Van..
0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 0 0 . 0 0 0 4 9 - 5
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Grandmont 1977, 1982. and in the asset market equilibrium theory Žsee Hart, 1974.. It was later considered as an abstract equilibrium theory problem Žfor a survey of the literature, see Page, 1996.. In finite dimensional economies, various conditions have been given for the existence of an equilibrium, in particular Ø the assumption of existence of a no-arbitrage price Žwith different concepts of absence of arbitrage., Ø the assumption of absence of unbounded and utility increasing trades, Žsee e.g. Page and Wooders, 1996. and Ø the assumption that the individually rational utility set is compact. Under standard assumptions on agents’ utilities, Žin particular that indifference curves contain no-half lines and that preferred sets have same asymptotic cone and that there is no satiation., these assumptions are equivalent to each other and also to the assumption that the individually rational attainable set is compact or to the existence of a Pareto-optimum Žfor a comparison of hypotheses and methods, see Dana et al., 1999.. In particular, Grandmont Ž1977, 1982. and later Werner Ž1987. gave an existence proof based on the demand approach. Werner Ž1987. assumes that there exists a ‘‘no-arbitrage’’ price or equivalently Žunder standard assumptions on agents’ utilities. that aggregate demand exists at some price. The problem of existence of an Arrow–Debreu equilibrium in infinite dimensional economies with consumption sets unbounded below has been discussed by Cheng Ž1991., Brown and Werner Ž1995., Chichilnisky and Heal Ž1993., Dana et al. Ž1997. and Aliprantis et al. Ž1998.. As it is well known, the assumptions stated above are not equivalent. The assumption of absence of free lunch is too weak, the assumption that individually rational attainable set is compact is generally not fulfilled. The standard assumption has been to assume that individually rational utility set is compact. A first attempt to understand the link between arbitrage theory and the existence of equilibrium has been made by Brown and Werner Ž1995. who introduced ‘‘arbitrage free prices’’ and proved that the assumption of existence of an arbitrage free price implies that the individually rational utility set is bounded. They also proved that if the utility set is closed and under standard assumptions on the economy, the existence of an equilibrium is equivalent to the existence of arbitrage free prices. In proving the existence of equilibrium in infinite dimensional economies with consumption sets unbounded below, all authors cited above have used Negishi’s approach in its topological version making no reference to arbitrage concepts in their proof. The purpose of this paper is to propose a ‘‘dual’’ Žin the sense of utility weight. interpretation of the notion of ‘‘no-arbitrage’’. In finite dimensional economies, a no-arbitrage price for the economy is a price such that pwi ) 0, ; wi g Wi _ 04 where Wi denotes the asymptotic cone of agents’ preferred sets. The existence of a no-arbitrage price is known to be equivalent to intÝ iWi8 / f.
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Similarly Žif the utility set is closed., a ‘‘fair utility weight vector’’ is a vector l such that l z - 0, ;z g U` _ 04 where U` denotes the asymptotic cone of agents’ utility set. The existence of a fair utility weight vector is equivalent to intU`8 / 0. In finite dimensional economies, this similarity is even more striking, a no-arbitrage price is a price for which aggregate demand exists. When there exists no-arbitrage price, one can define the excess demand correspondence and use a generalized version of Gale–Nikaido–Debreu’s lemma to prove the existence of an equilibrium. Fair utility weight vectors are utility weights for which the representative agent problem has a unique solution Žwhich is Pareto-optimal.. We may then similarly define in the space of utility weight vectors, the excess utility correspondence, which has the properties of an excess demand correspondence and use a generalized version of Gale–Nikaido–Debreu’s lemma to prove the existence of an equilibrium. The paper is organized as follows. In Section 2, we set the model and introduce some notation. In Section 3, we prove our main result which is the equivalence between existence of equilibrium and existence of a fair utility weight vector. This is done in several steps. We first prove the existence of Pareto-optima, we then define and characterize the excess utility correspondence. Lastly, we apply a generalized version of Gale–Nikaido–Debreu’s lemma to prove the existence of an equilibrium. Comments on our assumptions and links with the literature are given in Section 4. Section 5 is devoted to finite dimensional economies. We show the equivalence between the existence of a no-arbitrage price, existence of a fair utility weight vector, and compactness of the set of individually rational attainable allocations.
2. The model and notation We shall use the following notation. Given a subset C of R n , intC, EC, and C denote its interior, its boundary, and its closure. For a convex subset C : R n, int r C denotes its relative interior, when C is regarded as a subset of its affine hull. We consider a pure exchange economy with a commodity space F assumed to be a locally convex, topological space with dual F X . There are m agents. Agent i is described by a consumption set X i : F and an initial endowment wi g X i . The preferences of agent i are represented by a utility function u i : X i R. Let w s Ý mis1wi denote aggregate endowment. For the sake of completeness, we recall two definitions.
™
A pair Ž x, p . g P i X i = F X _ 04 is an quasi-equilibrium if Ži.; i, u i Ž x i . ) u i Ž x i ., implies px i G pwi , and Žii. Ý i x i s w.
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A pair Ž x, p . g P i X i = F X _ 04 is an equilibrium if Ži. ; i, u i Žx i . G u i Ž x i ., for every x i such that px i F pwi , and Žii. Ý i x i s w. We shall make the following assumptions about the agent’s characteristics. H1. X i is closed and convex, ; i. H2. ; i, u i : X i
™ R is strictly concave and u Ž w . s 0. i
i
H3. Ž w1 , . . . , wm . is not weakly Pareto-optimal and furthermore, there exists a symmetric neighborhood W : F of zero such that ;´ g W , there exists wX s Ž wX1 , . . . , wmX . g P i X i with Ý mis1wXi s w q ´ such that u i Ž wXi . G u i Ž wi ., ; i. For ´ g W , let m
½
AŽ ´ . s Ž x 1 , . . . , x m . gŁXi < Ý x i s w q ´ i
is1
5
be the set of attainable allocations when aggregate endowment is w q ´ ; U Ž ´ . s Õ g R m < Õi F u i Ž x i . , ; i ,
for some x g A Ž ´ . 4
be the utility set, V Ž 0 . s z g R m < u i Ž wi . F z i F u i Ž x i . , ; i ,
for some x g A Ž 0 . 4
be the set of individually rational utilities. H4. UŽ0. is closed. H5. V Ž0. is bounded. H6. ; x g AŽ0., 'Ž k 1 , . . . , k n . g F m , such that u i Ž x i q k i . ) u i Ž x i ., ; i. We postpone the comments on our assumptions to Section 4. Let us already remark that H1 is standard. As far as H3 is concerned, it is usual to assume that Ž w1 , . . . , wm . is not weakly Pareto-optimal, in other words that there exists wX g AŽ0. such that u i Ž wiX . ) u i Ž wi ., ; i. If not, it is well known that there exists a price p such that ŽŽ w 1 , . . . , wm ., p . is a quasi-equilibrium. Assumption H3 is stronger. H6 means that if x s Ž x 1 , x 2 , . . . . g AŽ0., then agent i is not satiated at x i. Let V be a closed convex set of R m , then V` s x g R m < x s lim n l n x n , l n G 0, l n 0, x n g V 4 is its asymptotic cone and V8 s y g R m < Õy F 0, ; Õ g V 4 its polar cone.
™
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Since we do not assume UŽ ´ . closed, we shall need to consider U`Ž ´ . the asymptotic cone of UŽ ´ . and U`8Ž ´ . the polar of the asymptotic cone U`Ž ´ .. For further use, let us remark the following: for a g R m , let V Ž a, 0 . s Õ g R m < a i F Õi F u i Ž x i . , ; i , x g A Ž 0 . 4 . The purpose of next proposition is to show that any subset of the utility set which is bounded below is bounded above. Proposition 2.1. Assume H1-H2-H4-H5, then V(a, 0) is compact, ;a. Proof. Since UŽ0. is closed, V Ž a, 0. is closed, ;a. Hence, it is bounded iff its m asymptotic cone equals 04 . V Ž a, 0.` s UŽ0.` l Rq s V Ž0.` s 04 by H5. B
3. Fair utility weight vectors and existence of equilibria In finite dimensional economies, one of the methods to prove the existence of an equilibrium when consumption sets are unbounded below consists first in defining prices for which aggregate demand exists Žno-arbitrage prices.. One applies subsequently a generalized version of Gale–Nikaido–Debreu’s lemma to prove the existence of an equilibrium. As it is well known, this method does not generalize to infinite dimensional economies even if consumption sets are bounded below. To prove the existence of an equilibrium in infinite dimensional economies where consumption sets are bounded below, Negishi’s method has often been used either in its topological version with a separation argument to find supporting prices or in its differential version. Supporting prices are then elements of the subdifferential of the support function of the utility set also called ‘‘the aggregate utility’’. In this paper, we follow the second approach. Our aim is to define the excess utility correspondence. We shall show that this correspondence is defined on the set of strictly positive weights for which there exists Pareto-optima. This is why we call these weights ‘‘fair utility weight vectors’’. We shall prove that it has all the properties of an excess demand correspondence in finite dimensional economies. While the excess demand correspondence is defined on the set of ‘‘no-arbitrage prices’’, the excess utility correspondence is defined on the set of ‘‘fair utility weight vectors’’. As for the excess demand correspondence, we apply a generalized version of Gale–Nikaido–Debreu’s lemma to prove the existence of an equilibrium. In order to define the excess utility correspondence, let us introduce some notation.For every l g R m , ´ g F, let h Ž l , ´ . s sup
Ýl i z i
zgU Ž ´ . i
be the support function of UŽ ´ .. Trivially, hŽ l , P. is concave.
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m Let DŽ ´ . s dom hŽ l , ´ . s l g R m N hŽ l , ´ . - `4 . Since U`Ž ´ . > Ry , it has non-empty interior. It follows from Aubin Ž1982. p. 34, Proposition 9 that m. m DŽ ´ . s U`8Ž ´ . ; ŽRy 8 s Rq . We shall prove that DŽ ´ . is independent of ´ and hence, denote it by D. For l g int D, we shall prove that there exists a unique x Ž l . Pareto-optimal such that hŽ l , 0. s Ý i l i u i Ž x i Ž l ... More precisely, For l g int D, let Õ Ž l . s Argmax l u N u g UŽ0.4 and x i Ž l . be such that Õi Ž l . s u i Ž x i Ž l ... We call int D the set of fair utility weight vectors.
For ´ g W ,
let E´ h Ž l , ´ .
s z g F X N h Ž l , ´ . y h Ž l , ´ y u . G zu, ; u g F 4 . We shall prove that E´ hŽ l , 0. is not empty, that it is convex, weakly compact, and u.h.c. We may now define the excess utility correspondence. For l g int D, ; i s 1, . . . , m, let Ei Ž l . s
½
z P Ž x i Ž l . . y wi
li
5
, z g E´ h Ž l , 0 . .
Our main result is the following. Theorem 3.1. Assume H1-H2-H3-H4-H6. (a) If there exists a fair utility weight Õector, then there exists a quasi-equilibrium (x, z) with z g E´ h(l , 0) and l g D. Assume furthermore ;l g D, ; p g E´ h( l ,0) _ {0}, ; i, pwi ) infpX i , then, (b) the existence of a fair utility weight Õector is equiÕalent to the existence of an equilibrium.
The Proof of Theorem 3.1 is done in four steps, ŽProposition 3.1, 3.2 and 3.4.. In particular in Proposition 3.4, we give a sufficient condition for the existence of fair utility weight vectors. Proposition 3.1. (a) Assume H1-H2. Then ;´ g W , h(P, ´ ) is conÕex and lower semi-continuous. (b) Assume H1-H2. Then ;´ g W , ´ X g W , D(´ ) s D(´ X ). (c) Assume H1-H2-H3. Then ;l g D(´ ), h( l , P) is continuous in W .
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Proof. To prove Ža. the function hŽP, ´ . is convex, and lower semi-continuous as a support function Žsee Rockafellar, 1970, Theorem 13.2.. To prove Žb. assume that hŽ l , ´ . - ` and there exists ´ X g W such that hŽ l , ´ X . s `. Let ´ Y g W be such that ´ s t ´ X q Ž1 y t .´Y with t gx0, 1w. We then have: q` ) h Ž l , ´ . G th Ž l , ´ X . q Ž 1 y t . h Ž l , ´ Y . s q`, since hŽ l , ´ Y . ) y`, a contradiction. Hence, hŽ l , ´ Y . - q` and DŽ ´ X . s DŽ ´ .. Let us lastly prove that hŽ l , P. is continuous in W . Let ´ g W . It follows from H3 that there exists wX g AŽ ´ . such that u i Ž wiX . G u i Ž wi .; i. Hence, m
m
h Ž l , ´ . G Ý l i u i Ž wXi . G Ý l i u i Ž wi . s 0, i
i
by assumption H2. Since the function hŽ l , P. is concave, finite valued and bounded below, it is continuous in W . B We shall simply denote by D s dom hŽP, ´ . and by U` the asymptotic cone of UŽ ´ .. For ´ g W , observe that E´ h Ž l , ´ . s z g F X N h Ž l , ´ . y h Ž l , ´ y u . G zu, ; u, such that ´ y u g W 4 . Let z verify h Ž l , ´ . y h Ž l , ´ y u . G zu, ; u, such that ´ y u g W . Let y g F. Then there exists 0 - m - 1 such that ´ y m y g W , hence, we have, h Ž l , ´ . y h Ž l , ´ y m y . G m Ž zy . . As hŽ l , P. is concave, hŽ l , ´ y m y . G m hŽ l , ´ y y . q Ž 1 y m . hŽ l , ´ . hence,
m Ž h Ž l , ´ . y h Ž l , ´ y y . . G h Ž l , ´ . y h Ž l , ´ y m y . . G m Ž zy . which, after division by m , is to be proven. We shall now state the properties of E´ hŽ l , 0.. Proposition 3.2. Assume H1-H2-H3. (a) ;l g D, ;´ g W , E´ h( l , ´ ) is non-empty, conÕex, and weakly compact, (b) ;l g intD, there exists a weakly compact subset K of F X such that E´ h( lX , 0) : K, ;lX g V( l ) a neighborhood of l , and (c) the correspondence l g intD E´ h( l , 0) is upper hemi- continuous in the weak topology of F.
™
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Proof. Since by Proposition 3.1, hŽ l , P. is continuous in W , E´ hŽ l , ´ . is non-empty, convex, and weakly compact Žsee Aubin, 1982, p. 108.. To prove Žb., we have h Ž lX , 0 . y h Ž lX , ´ . G E´ h Ž lX , 0 . Ž y´ . , h Ž lX , 0 . y h Ž lX , y´ . G E´ h Ž lX , 0 . Ž ´ . . Let V Ž l . be a compact neighborhood of l such that V Ž l . : int D. Since hŽP, 0. is continuous in V Ž l ., < hŽ lX , 0.< F M. We also have h Ž lX , ´ . G 0, ;lX g V Ž l . , ;´ g W . Hence < E´ h Ž lX , 0 . ´ < F h Ž lX , 0 . F M , ;lX g V Ž l . , ;´ g W and E´ hŽ lX , 0. is compact by Alaoglu theorem. Lastly to prove Žc., since hŽP, 0. and hŽP, ´ . are continuous on int D, the correspondence l g int D E´ hŽ l , 0. has a closed graph and from Žb. is upper hemi-continuous for the weak topology of F. B
™
In the next proposition, we prove the existence of Pareto-optima. We first need a lemma. Lemma 3.3. Let C( l ) s {u g U(0) N l u G 0}. Assume H1-H2-H4. Then the correspondence C: intD R mq is conÕex, compact Õalued and u.h.c. Assume furthermore H3, then it is continuous. The Proof may be found in the Appendix.
™
Proposition 3.4. (a) Assume H1-H2. Let l g D. If h( l , 0) s l Õ, then there exists x g A(0) such that h( l , 0) s Ý i l i u i (x i ). If l i ) 0, Õi is unique and there exists a unique x i such that Õi s u i (x i ). (b) Assume H1-H2-H3-H4. Then intD s { l 4 0 N ' Õ g U(0), h( l , 0) s l Õ} s { l 4 0 N ' x Pareto-optimal such that h( l , 0) s Ý i l i u i (x i )}. (c) Assume H1-H2-H3-H4. Then the map intD Argmax{ l u N u g U(0)} is continuous. (d) Assume H1-H2.
™
(i) If x is Pareto-optimal, then there exists l g D such that h( l , 0) s Ý i l i u i (x i ). (ii) Assume furthermore H4. Then H5 intD /œ 0 existence of a Pareto-optimum. (iii) Assume furthermore H3-H4-H6 and X i s F, ; i, then x is Pareto-optimal, iff there exists l g intD such that h( l , 0) s Ý i l i u i (x i ).
m
m
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405
Proof. If hŽ l , 0. s l Õ, then there exists x g AŽ0. such that Õi F u i Ž x i .. Hence hŽ l , 0. s Ý i l i u i Ž x i .. If l i ) 0, Õi s u i Ž x i . and x i is unique since each u i is strictly concave. To prove Žb., let l g int D. Then by Lemma 3.3, C Ž l . is compact and hŽ l , 0. s sup l u N u g C Ž l .4 s l Õ for some Õ g UŽ0.. Since l g int D, l 4 0. From Ža., Õi s u i Ž x i ., ; i and obviously x is Pareto-optimal. Conversely, let us assume that hŽ l , 0. s l Õ for some l 4 0 and Õ g UŽ0.. If l f int D, then there exists p g U` _ 04 such that l p s 0. However, hŽ l , 0. s l Õ s l Ž Õ q p . which contradicts Ža.. Hence, l g int D. To prove Žc., we use the fact that hŽ l , 0. s sup l u N u g C Ž l .4 s l Õ, the lemma, and the maximum theorem. Lastly, Žd. Ži. is well-known. The Proof of Žii. may be found in the Appendix. To prove Žiii., let us prove that if x is Pareto-optimal, then l 4 0. If not, assume that l 1 s 0, l 2 s 0, . . . , l r s 0. Let k rq1 be as in H6. Let xX g A be defined as follows: xX1 s x 1 y k rq1 , xXrq1 s x rq1 q k rq1 , xXj s x j , ; j / 1, r q 1, then
Ý l i u i Ž xXi . ) Ý l i u i Ž x i . i
i
a contradiction.
B
For l g int D, let Õ Ž l . s Arg max l u N u g U Ž 0 . 4 and let x i Ž l . be such that Õi Ž l . s u i Ž x i Ž l ... Let us recall the definition of the excess utility correspondence. For l g int D, ; i s 1, . . . , m, let Ei Ž l . s
½
z P Ž x i Ž l . y wi .
li
5
, z g E´ h Ž l , 0 . .
Proposition 3.5. Assume H1-H5. For l g intD, E is a conÕex, compact, non-empty Õalued, upper hemi-continuous correspondence, which satisfies l E( l ) s 0 (Walras Law). Proof. Clearly, E is convex and non-empty valued. Let O Ž l . be a compact neighborhood of l g int D. Then ; i, ; z g E´ hŽ lX , 0., lX g O Ž l ., X
X
Õi Ž l . s u i Ž x i Ž l . . y u i Ž w i . G
z P Ž x i Ž lX . y wi .
li.
Hence t i Fmax< Õi Ž lX . < F A i , OŽ l.
; t i g Ei Ž lX . , ;lX g O Ž l .
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for some A i , by Proposition 3.4Žc.. Since Ý i lXi Ei Ž lX . s 0, 'Bi g R such that t i G Bi ; t i g Ei Ž lX . , lX g O Ž l . . Hence EŽ lX . has values in a fixed compact set of R m for lX g O Ž l .. Let us finally prove that E has closed graph. Let l n l g int D and Ž z n P Ž x i Ž l n . y wi ..rl ni j i , ; i, z n g E´ hŽ l n , 0..
™
™
; i , Õi Ž l n . y Õi Ž l . G
zn P Ž x i Ž l n . y x i Ž l . .
l ni
Since by Proposition 3.2, E´ hŽ l , 0. is u.h.c., some subsequence of z n converges weakly to z g E´ hŽ l , 0.. Hence, since Õi Ž l n . Õi Ž l ., by Proposition 3.4Žc., we get
™
zw s Ý zx i Ž l . G Ýlimsup z n x i Ž l n . G limsupÝ z n x i Ž l n . s zw, i
i
i
hence zx i Ž l . s limsup z n x i Ž l n .; i and j s Ž z P Ž x i Ž l . y wirl i . and E is upper hemi-continuous. B Proof of Theorem 3.1. Let D n s l g int D, N 5 l 5 s 1, F Ž l . F Žy1rn.4 and K n be the cone generated by D n . It follows from a generalized version of Gale–Nikaido–Debreu’s lemma proven in Florenzano and Le Van Ž1986. that ; n, 'l n g D n , 'e n g EŽ l n . such that yl e n F 0, ;l g K n . Let l be a limit point of l n . There are three cases. Case Ž1. l g int D. Then by Proposition 3.5, e n e g EŽ l .. Thus yl e F 0, ;l g D and hence, ye g U` . Since l g int D, yl e - 0 if e / 0 which contradicts Walras Law. Hence, e s 0. There exists z g E´ hŽ l,0. such that 0 s ŽŽ z P Ž x i Ž l . y wi ..rl i .; i.
™
; i , ; x i g X i , l i Ž u i Ž x i Ž l . . y u i Ž x i . . G z Ž x i Ž l . y x i . s z Ž wi y x i . . m Thus zx i F zwi implies u i Ž x i . F u i Ž x i Ž l ... Hence, wŽ x i Ž l .. is1 , zx is an equilibrium. Case Ž2. l g ED and 5 e n 5 `. Since e i n F Õi Ž l n ., ; i, e n g UŽ0., ; n. Hence, Ž e i nr5 e n 5. t g U` _ 04 . Since yl e n F 0, ;l g K n , we get at the limit yl t F 0, ;l g D, which is impossible. Case Ž3. l g ED and some subsequence of e n e. As in Case Ž1., we have l e s 0. Since e i n F Õi Ž l n ., ; i, Õ Ž l n . is bounded below. By Proposition 2.1, it is bounded above. Without lost of generality, let us assume that Õ Ž l n . converges to Õ. Since
™
™
™
Ý l ni Õi Ž l n . G Ý l ni u i , i
i
we have
l Õ G l u ; u g U Ž 0. ,
; u g U Ž 0. ,
R.-A. Dana, C. Le Van r Journal of Mathematical Economics 34 (2000) 397–413
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hence ` ) hŽ l , 0. s l z. Since l e s 0, hŽ l , 0. s l Ž z q e .. By Proposition 3.4Ža., l i ) 0 implies e i s 0. Hence, l i e i s 0, ; i. By Proposition 3.2 and its Proof, we still have E´ h Ž l n , 0 . ´ F < h Ž l n , 0 . < F M , ; n, ;´ g W ,
™
since hŽ l n , 0. hŽ l n , 0.. We may therefore assume that if e in s Ž z n P Ž x i Ž l n .. y wi .rl ni , z n g E´ hŽ l n , 0., then z n z weakly. Let x Ž l . be such that hŽ l , 0. s Ý i l i u i Ž x i Ž l ... Then as in the Proof of Proposition 3.5, z n x i Ž l n . zx i Ž l ., ; i. Hence
™
™
; i , z Ž x i Ž l . y wi . slime inl ni s l i e i s 0. n
m Let us prove that wŽ x i Ž l .. is1 , z x is a quasi-equilibrium. Since
l ni u i Ž x i Ž l n . . y u i Ž x i . G z n Ž x i Ž l n . y x i . , ; i , if l i ) 0,
l i u i Ž x i Ž l . . y u i Ž x i . G z Ž x i Ž l . y x i . s z Ž wi y x i . . Let x i be such that u i Ž x i . ) u i Ž x i Ž l .. Žby assumption H6 such an x i exists.. m Then zx i ) zwi . Hence, z / 0. If l i s 0, then zwi F zx i . Hence, wŽ x i Ž l .. is1 , z x is a quasi-equilibrium. Since ` ) hŽ l , 0., l g D and from the last inequality z g E´ hŽ l , 0.. Assume furthermore ;l g D, ; z g E´ hŽ l , 0. _ 04 , ; i, zwi ) inf zX i . Let us m , z x is an equilibrium, equivalently that u i Ž x i . ) u i Ž x i Ž l .. prove that wŽ x i Ž l .. is1 implies z Ž wi y x i . - 0. If l i ) 0, it follows from the proof above. Assume l i s 0 and zwi s zx i . Let yi g X i be such that zyi - zwi . Then for t gx0, 1w, u i Ž tx i q Ž1 y t . yi . ) tu i Ž x i . q Ž1 y t . u i Ž yi . and ztx i q Ž1 y t . yi - zwi , hence, for t sufficiently close to one, u i Ž tx i q Ž1 y t . yi . ) u i Ž x i Ž l .. and ztx i q Ž1 y t . yi - zwi a m contradiction. Hence, wŽ x i Ž l .. is1 , z x is an equilibrium. Conversely, if there exists an equilibrium, then there exists a Pareto-optimum. It follows from Proof of Proposition 3.4Žd-ii. that int D / f . B
4. Comments on our assumptions and links with the literature Ž1. In order to obtain support prices, Brown and Werner Ž1995., Dana et al. Ž1997. assume that ; x g AŽ0., Pi Ž x i . / 0, ; i and there exists j with int Pj Ž x j . / 0. In this paper, we only assume ; i, Pi Ž x i . / 0, ; x g AŽ0. ŽH6. but in order to obtain support prices, we have to assume H3. Assumption H3 is fulfilled if int X i / 0, ; i, and u i is continuous for every i. Ž2. The standard assumption on utilities is to assume that they are strictly quasi-concave. We assume that utilities are concave to get a convex utility set and
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strictly concave in order to have that to a set of utility weight vectors, there corresponds a unique Pareto-optimum. Ž3. Assumption H4 may seem very strong since in the topological Negishi’s approach, one only uses the set V Ž0. of individually rational utilities. Brown and Werner Ž1995., Dana et al. Ž1997. only assume V Ž0. compact. If V Ž0. is assumed to be compact, one can substitute to the original economy another economy with the same quasi-equilibria for which H4 is fulfilled. Indeed, consider the economy EX where the consumption sets X i are replaced by: X iX s x g X i N u i Ž x i . G u i Ž wi . 4 . If V Ž0. is compact, the set U X Ž 0 . s Õ g R m N Õi F u i Ž x i . , ; i , for some x g AX Ž 0 . 4 where AX Ž0. s Ž x 1 , . . . , x m . g P i X iX <Ý mis1 x i s w4 is obviously closed since U X Ž0. m s V Ž0. q Ry . Hence, H4 is fulfilled for the economy EX which has a quasi-equilibrium. That quasi-equilibrium is a quasi-equilibrium for the initial economy. Ž4. In order to link our results to those of Chichilnisky and Heal Ž1993., let m
½
AŽ t , 0. s x s Ž x 1 , . . . , x m . gŁXi N Ý x i i
is1
5
s v , u i Ž x i . G u i Ž ti . , ; i , t gŁXi . i
The following conditions are similar to those introduced by Chichilnisky and Heal Ž1993.. G1. F is a reflexive space and X i s F, ; i. G2. For any t g F m , AŽ t, 0. is norm bounded. G3. ; i, u i is norm continuous and strictly concave. Proposition 4.1. Assume G1-G2-G3, then H4 and H5 are fulfilled. Proof. Since u i is norm continuous for every i, for any t g F m , AŽ t, 0. is norm closed and norm bounded, hence weakly compact, since F is reflexive. Since u i is weakly u.s.c., ; i, V Ž0. is compact and H5 is fulfilled. Lastly, for any t g F m , let V Ž t , 0 . s z g R m N u i Ž t i . F z i F u i Ž x i . , ; i , for some x g A Ž 0 . 4 . Similarly, for any t g F m , V Ž t, 0. is compact. Since u i is defined on F, inf x g F u i Ž x . s y`, ; i. Hence, UŽ0. s jt g F m V Ž t, 0. and closed, thus, H4 is fulfilled. B
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It turns out that G2 is not generally fulfilled. In fact, Cheng Ž1991. shows that it is not fulfilled in the case of L p and Von Neumann–Morgenstern utilities. Ž5. In order to link our results to those of Brown and Werner Ž1995., let us recall the following definition: let p g F X _ 04 . A sequential arbitrage opportunity for i with respect to p is a sequence xˆ n such that wi q xˆ n g X i , lim u i Ž wi q xˆ n . s sup x i g X i u i Ž x i . and lim pxˆ n F 0. A price system is arbitrage-free for the economy if for every agent, there is no sequential arbitrage. Assuming H1-H2, Brown and Werner Ž1995. prove that the existence of an arbitrage-free price for the economy implies H5. We slightly strengthen their result. Proposition 4.2. Assume H1-H2-H4 and sup x i g X i u i (x i ) s `, ; i. Then, the existence of a fair utility weight Õector (i.e. intD / f ) is equiÕalent to H5 which in turn is equiÕalent to the existence of an arbitrage-free price for the economy. Proof. We prove here that the existence of a fair utility weight vector is equivalent to the existence of an arbitrage-free price for the economy. Let l be fair utility weight vector and x Ž l . be the associated Pareto-optimum by Proposition 3.4Žb.. Let p g E´ hŽ l, 0. and let Ž x in . be a sequence such that u i Ž wi q x in . `. Then
™
l i ui Ž x i Ž l . . y ui Ž px in
™
wi q x in
. GpŽ xi y Ž
wi q x in
..
hence, `, which implies that p is arbitrage-free price for the economy. The converse is proven in Brown and Werner Ž1995.. The reader can check that it does not require the continuity of the utility functions. B
5. Arbitrage and duality in finite dimensional economies In this section, we shall show that in finite dimensional economies, the demand approach which is based on the assumption of existence of a no-arbitrage price and the duality approach, which is based on the assumption of existence of a fair utility weight vector, are equivalent. We shall maintain the following assumptions: F1. X i : R l is closed and convex, ; i.
™
F2. ; i, u i : X i R is strictly concave and continuous and does not have a satiation point and u i Ž wi . s 0. F3. w is not weakly Pareto-optimal. Remark. Clearly, F1 is equiÕalent to H1, F2 implies H2 and H6, and F2-F3 imply H3. A vector t g R l is useful for i if u i Ž x q t . G u i Ž x ., ; x g X i . Let Wi denote the set of useful vectors for i. It is a standard result that Wi is a closed and convex
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cone and that it is the asymptotic cone of the set x g X i , u i Ž x i . G a i , ; i4 for any a g R n. It follows from F2 that Wi / 04 . A vector p g R l is a no-arbitrage price if there exists no x / 0, x g Ý iWi such that px F 0 Žequivalently, x g Ý iWi _ 04 implies px ) 0.. Hence, p g R l is a no-arbitrage price if p g intŽyÝ i ŽWi .8.. Symmetrically, we may introduce the following definition. Assume UŽ0. is closed. A vector l g R m is a fair utility weight Õector if there exists no t / 0, t g U` such that l t G 0 Žequivalently t g U` _ 04 implies l t - 0.. Hence, l g R m is a fair utility weight vector if UŽ0. is closed and if l g intU`8Ž0.. For further use, let Õ g R m and A Ž Õ, 0 . s Ž x 1 , . . . , x m . g A Ž 0 . < u i Ž x i . G Õi , ; i 4 be the set of attainable allocations with utilities bounded below by Õ. Theorem 5.1. Assume F1-F2, then the following statements are equiÕalent. 1. 2. 3. 4. 5.
there exists a no-arbitrage price, A(Õ, 0) is compact for any Õ, H4 and H5 are fulfilled, there exists a fair utility weight Õector and H4 is fulfilled, and there exists a Pareto-optimal allocation.
Any preÕious condition implies that there exists a quasi-equilibrium. Assume furthermore ;l g D, ; z g E´ h( l , 0) _ {0}, ; i, pwi ) infpX i , then the assumption of existence of a no-arbitrage price is equiÕalent to the assumption of existence of a fair utility weight Õector and H4 is fulfilled and is equiÕalent to the existence of an equilibrium. Proof. Let us sketch these equivalences. Statement Ž1. above implies Ž2.. Since u i is concave and continuous, AŽ Õ, 0. is closed and convex for any Õ and its asymptotic cone is x g P iWi N Ý i x i s 04 s 04 . Assume there exists x I g Wi , ; i, x j / 0, for some j, Ý i x i s 0. Let p be no-arbitrage price. Then pÝ i x i s 0 while px j ) 0, px i G 0, ; i / j, a contradiction. Ž2. implies Ž3.. Since V Ž0. s AŽ0, 0., H5 is fulfilled. To prove H4, let Õ n g UŽ0. Õ. Then ; n, ; i, there exists x in such that Õin F u i Ž x in ., ;i, ;´ , for n large enough, x n g AŽ Õ y ´ , 0.. Hence, a subsequence of x n converges to x and Õi F u i Ž x i ., ; i. Ž3. implies Ž4. and Ž4. implies Ž5. following from Proposition 3.4 Žii.. Ž5. implies Ž1.. See for example, Dana et al. Ž1999.. B
™
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Acknowledgements We are very grateful to the referee for his comments and suggestions.
Appendix A
Lemma 3.3. Let C( l ) s {u g U(0)< l u G 0}. Assume H1-H2-H4. Then, the correspondence C: intD R mq is conÕex, compact Õalued, and u.h.c. Assume furthermore H3, then it is continuous.
™
Proof. Let
F Ž l . s sup l p N p g U` , 5 p 5 s 1 4 . m< Ž . m< Ž . Let us first prove that D s l g Rq F l F 04 and int D s l g Rq F l - 04 . m Indeed, since D s U`8 g Rq , the proof of the first statement is obvious. Since F is continuous,
l g Rqm N F Ž l . - 0 4 : int D Conversely, let l g int D and assume that F Ž l . s 0. Let p be such that F Ž l . s p l and ´ g R m be such that p´ ) 0. For t ) 0 sufficiently small l q t ´ g D but F Ž l q t ´ . G Ž l q t ´ . p ) 0, a contradiction. Let us next prove the statement in Lemma 3.3. Since C Ž l . is clearly closed and convex, let us show that if l g int D, C Ž l .` s 04 . Let t g C Ž l .` . Then t g U` and l t G 0, hence, l t s 0. However, F Ž l . s 0 which contradicts the fact that l g int D. Hence, C is compact-valued. Obviously, C has closed graph. Let us next show that C is u.h.c. Let V Ž l . be a compact neighborhood of l g int D. Let us prove that C has values in a fixed compact set. Suppose not, then there exists l n l , u n g C Ž l n . such that 5 u n 5 q`. Then u nr5 u n 5 u g U` _ 04 and l u G 0. Since l g int D, l u - 0, a contradiction. Hence, C has values in a fixed compact set in a neighborhood of l and is u.h.c. Let us lastly show that C is l.h.c. Let u g C Ž l . and l n l . If l u ) 0, then l n u ) 0, for n large enough and C is l.h.c. at l . If l u s 0, let Õ g intV Ž0.. Then Õ 4 0 and l Õ ) 0, hence, l n Õ ) 0 for n large enough. Assume l n u - 0, ; n. Let un gx0, 1w be defined by l nw un u q Ž1 y un . Õ x s 0. The sequence un being bounded, un u and lw u u q Ž1 y u . Õ x s Ž1 y u . l Õ s 0. Hence, u s 1 and un u q Ž1 y un . Õ u with un u q Ž1 y un . Õ g C Ž l n . for n large enough. B
™
™
™
™
™ ™
Proof of Proposition 3.4(d-ii). Assume first V Ž0. bounded, hence, compact and int D s f . Since D8 s U` , U` contains a line tÕ, t g R4 . Let I1 s i < Õi ) 04 , I2 s i < Õi - 04 and I3 s i < Õi s 04 . Without lost of generality, we may assume that
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I1 / f . Let i 0 g I1. Let z g V Ž0. be such that z i 0 s max z i0 , z g V Ž0.4 . There exists x g AŽ0., xX g AŽ0. such that ; i , z i q Õi F u i Ž x i . z i y Õi F u i Ž xXi . . Let i g I1. If x i / xXi , x i q xXi zi - ui . 2
ž
/
If x i s xXi , then z i - z i q Õi F u i Ž x i . s u i
ž
x i q xXi 2
/
,
a contradiction if i s i 0 . Therefore, U` contains no line and int D / f . Hence, H5 implies int D / f . It follows from Proposition 3.4Žb. that if int D / f , then there exists a Pareto-optimum. Hence, int D / f implies existence of a Pareto-optimum. m Assume V Ž0. unbounded. Then, V Ž0.` s UŽ0.` l Rq / 04 . Let Õ g V Ž0.` , Õ / 0 and x g AŽ0. be a Pareto-optimum. Let z be such that z i s u i Ž x i ., ; i. Then z q Õ g V Ž0., hence, there exists xX g AŽ0. such that z i q Õi F u i Ž xXi ., ; i. Hence, u i Ž x i . F u i Ž xXi ., ; i with a strict inequality for at least some i since Õ / 0, a contradiction. Hence, existence of a Pareto-optimum implies H5. B
References Aliprantis, C.D., Brown, D.J., Polyrakis, I.A., Werne, J., 1998. Portfolio dominance and optimality in infinite asset markets. Journal of Mathematical Economics 30, 347–366. Aubin, J.P., 1982. Mathematical Methods of Game and Economic Theory. North-Holland. Brown, D.J., Werner, J., 1995. Arbitrage and existence of equilibrium in infinite asset markets. Review of Economics Studies 62, 101–114. Cheng, H., 1991. Asset market equilibrium in infinite dimensional complete markets. Journal of Mathematical Economics 20, 137–152. Chichilnisky, G., Heal, G.M., 1993. Competitive equilibrium in Sobolev spaces without bounds on short sales. Journal of Economic Theory 59, 364–384. Dana, R.A., Le Van, C., Magnien, F., 1997. Asset equilibrium in assets markets with and without short-selling. Journal of Mathematical Analysis and Applications 206, 567–588. Dana, R.A., Le Van, C., Magnien, F., 1999. On the different notions of arbitrage and existence of equilibrium, Journal of Economic Theory, in press. Florenzano, M., Le Van, C., 1986. A note on Gale–Nikaido–Debreu lemma and the existence of general equilibrium. Economics Letters 22, 107–110. Grandmont, J.M., 1977. Temporary general equilibrium theory. Econometrica 45, 535–572. Grandmont, J.M., 1982. Temporary general equilibrium theory, Handbook of Mathematical Economics vol. 2 North-Holland. Green, J.R., 1973. Temporary general equilibrium in a sequential trading model with spot and future transactions. Econometrica 41, 1103–1123.
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Hart, O., 1974. On the existence of an equilibrium in a securities model. Journal of Economic Theory 9, 293–311. Page, F.H. Jr., 1996. Arbitrage and asset prices. Mathematical Social Sciences 31, 183–208. Page, F.H. Jr., Wooders, M.H., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162. Rockafellar, R.T., 1970. Convex Analysis. Princeton Univ. Press, Princeton, NJ. Werner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55, 1403–1418.
Arbitrage, duality and asset equilibria Rose-Anne Dana a , Cuong Le Van b,) a
Ceremade, Place du marechal De Lattre de Tassigny, 75775 Paris, France ´ b CNRS, Cermsem, 106-112 bouleÕard de l’Hopital, 75013 Paris, France
Received 27 April 1998; received in revised form 2 August 1999; accepted 17 April 2000
Abstract In finite dimensional economies, it was proven by Werner wWerner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55, 1403–1418.x, that if there exists a no-arbitrage price Žequivalently, under standard assumptions on agents’ utilities, if aggregate demand exists for some price., then there exists an equilibrium. This result does not generalize to the infinite dimension. The purpose of this paper is to propose a ‘‘utility weight’’ interpretation of the notion of ‘‘of no-arbitrage price’’. We define ‘‘fair utility weight vectors’’ as utility weight vectors for which the representative agent problem has a unique solution. They correspond to no-arbitrage prices. The assumption that there exists a Pareto-optimum, can be viewed as the equivalent of the assumption of existence of aggregate demand. We may then define in the space of utility weight vector, the excess utility correspondence, which has the properties of an excess demand correspondence. We use a generalized version of Gale–Nikaido–Debreu’s lemma to prove the existence of an equilibrium. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Arbitrage; Duality; Asset
1. Introduction The problem of existence of an Arrow–Debreu equilibrium in economies with consumption sets unbounded below, has appeared in different economic settings, in particular in the theory of temporary equilibrium Žsee Green, 1973 and )
Corresponding author. Tel.: q33-1-40-77-8400; fax: q33-1-44-24-3857. E-mail address: [email protected] ŽC. Le Van..
0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 0 0 . 0 0 0 4 9 - 5
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Grandmont 1977, 1982. and in the asset market equilibrium theory Žsee Hart, 1974.. It was later considered as an abstract equilibrium theory problem Žfor a survey of the literature, see Page, 1996.. In finite dimensional economies, various conditions have been given for the existence of an equilibrium, in particular Ø the assumption of existence of a no-arbitrage price Žwith different concepts of absence of arbitrage., Ø the assumption of absence of unbounded and utility increasing trades, Žsee e.g. Page and Wooders, 1996. and Ø the assumption that the individually rational utility set is compact. Under standard assumptions on agents’ utilities, Žin particular that indifference curves contain no-half lines and that preferred sets have same asymptotic cone and that there is no satiation., these assumptions are equivalent to each other and also to the assumption that the individually rational attainable set is compact or to the existence of a Pareto-optimum Žfor a comparison of hypotheses and methods, see Dana et al., 1999.. In particular, Grandmont Ž1977, 1982. and later Werner Ž1987. gave an existence proof based on the demand approach. Werner Ž1987. assumes that there exists a ‘‘no-arbitrage’’ price or equivalently Žunder standard assumptions on agents’ utilities. that aggregate demand exists at some price. The problem of existence of an Arrow–Debreu equilibrium in infinite dimensional economies with consumption sets unbounded below has been discussed by Cheng Ž1991., Brown and Werner Ž1995., Chichilnisky and Heal Ž1993., Dana et al. Ž1997. and Aliprantis et al. Ž1998.. As it is well known, the assumptions stated above are not equivalent. The assumption of absence of free lunch is too weak, the assumption that individually rational attainable set is compact is generally not fulfilled. The standard assumption has been to assume that individually rational utility set is compact. A first attempt to understand the link between arbitrage theory and the existence of equilibrium has been made by Brown and Werner Ž1995. who introduced ‘‘arbitrage free prices’’ and proved that the assumption of existence of an arbitrage free price implies that the individually rational utility set is bounded. They also proved that if the utility set is closed and under standard assumptions on the economy, the existence of an equilibrium is equivalent to the existence of arbitrage free prices. In proving the existence of equilibrium in infinite dimensional economies with consumption sets unbounded below, all authors cited above have used Negishi’s approach in its topological version making no reference to arbitrage concepts in their proof. The purpose of this paper is to propose a ‘‘dual’’ Žin the sense of utility weight. interpretation of the notion of ‘‘no-arbitrage’’. In finite dimensional economies, a no-arbitrage price for the economy is a price such that pwi ) 0, ; wi g Wi _ 04 where Wi denotes the asymptotic cone of agents’ preferred sets. The existence of a no-arbitrage price is known to be equivalent to intÝ iWi8 / f.
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Similarly Žif the utility set is closed., a ‘‘fair utility weight vector’’ is a vector l such that l z - 0, ;z g U` _ 04 where U` denotes the asymptotic cone of agents’ utility set. The existence of a fair utility weight vector is equivalent to intU`8 / 0. In finite dimensional economies, this similarity is even more striking, a no-arbitrage price is a price for which aggregate demand exists. When there exists no-arbitrage price, one can define the excess demand correspondence and use a generalized version of Gale–Nikaido–Debreu’s lemma to prove the existence of an equilibrium. Fair utility weight vectors are utility weights for which the representative agent problem has a unique solution Žwhich is Pareto-optimal.. We may then similarly define in the space of utility weight vectors, the excess utility correspondence, which has the properties of an excess demand correspondence and use a generalized version of Gale–Nikaido–Debreu’s lemma to prove the existence of an equilibrium. The paper is organized as follows. In Section 2, we set the model and introduce some notation. In Section 3, we prove our main result which is the equivalence between existence of equilibrium and existence of a fair utility weight vector. This is done in several steps. We first prove the existence of Pareto-optima, we then define and characterize the excess utility correspondence. Lastly, we apply a generalized version of Gale–Nikaido–Debreu’s lemma to prove the existence of an equilibrium. Comments on our assumptions and links with the literature are given in Section 4. Section 5 is devoted to finite dimensional economies. We show the equivalence between the existence of a no-arbitrage price, existence of a fair utility weight vector, and compactness of the set of individually rational attainable allocations.
2. The model and notation We shall use the following notation. Given a subset C of R n , intC, EC, and C denote its interior, its boundary, and its closure. For a convex subset C : R n, int r C denotes its relative interior, when C is regarded as a subset of its affine hull. We consider a pure exchange economy with a commodity space F assumed to be a locally convex, topological space with dual F X . There are m agents. Agent i is described by a consumption set X i : F and an initial endowment wi g X i . The preferences of agent i are represented by a utility function u i : X i R. Let w s Ý mis1wi denote aggregate endowment. For the sake of completeness, we recall two definitions.
™
A pair Ž x, p . g P i X i = F X _ 04 is an quasi-equilibrium if Ži.; i, u i Ž x i . ) u i Ž x i ., implies px i G pwi , and Žii. Ý i x i s w.
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A pair Ž x, p . g P i X i = F X _ 04 is an equilibrium if Ži. ; i, u i Žx i . G u i Ž x i ., for every x i such that px i F pwi , and Žii. Ý i x i s w. We shall make the following assumptions about the agent’s characteristics. H1. X i is closed and convex, ; i. H2. ; i, u i : X i
™ R is strictly concave and u Ž w . s 0. i
i
H3. Ž w1 , . . . , wm . is not weakly Pareto-optimal and furthermore, there exists a symmetric neighborhood W : F of zero such that ;´ g W , there exists wX s Ž wX1 , . . . , wmX . g P i X i with Ý mis1wXi s w q ´ such that u i Ž wXi . G u i Ž wi ., ; i. For ´ g W , let m
½
AŽ ´ . s Ž x 1 , . . . , x m . gŁXi < Ý x i s w q ´ i
is1
5
be the set of attainable allocations when aggregate endowment is w q ´ ; U Ž ´ . s Õ g R m < Õi F u i Ž x i . , ; i ,
for some x g A Ž ´ . 4
be the utility set, V Ž 0 . s z g R m < u i Ž wi . F z i F u i Ž x i . , ; i ,
for some x g A Ž 0 . 4
be the set of individually rational utilities. H4. UŽ0. is closed. H5. V Ž0. is bounded. H6. ; x g AŽ0., 'Ž k 1 , . . . , k n . g F m , such that u i Ž x i q k i . ) u i Ž x i ., ; i. We postpone the comments on our assumptions to Section 4. Let us already remark that H1 is standard. As far as H3 is concerned, it is usual to assume that Ž w1 , . . . , wm . is not weakly Pareto-optimal, in other words that there exists wX g AŽ0. such that u i Ž wiX . ) u i Ž wi ., ; i. If not, it is well known that there exists a price p such that ŽŽ w 1 , . . . , wm ., p . is a quasi-equilibrium. Assumption H3 is stronger. H6 means that if x s Ž x 1 , x 2 , . . . . g AŽ0., then agent i is not satiated at x i. Let V be a closed convex set of R m , then V` s x g R m < x s lim n l n x n , l n G 0, l n 0, x n g V 4 is its asymptotic cone and V8 s y g R m < Õy F 0, ; Õ g V 4 its polar cone.
™
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Since we do not assume UŽ ´ . closed, we shall need to consider U`Ž ´ . the asymptotic cone of UŽ ´ . and U`8Ž ´ . the polar of the asymptotic cone U`Ž ´ .. For further use, let us remark the following: for a g R m , let V Ž a, 0 . s Õ g R m < a i F Õi F u i Ž x i . , ; i , x g A Ž 0 . 4 . The purpose of next proposition is to show that any subset of the utility set which is bounded below is bounded above. Proposition 2.1. Assume H1-H2-H4-H5, then V(a, 0) is compact, ;a. Proof. Since UŽ0. is closed, V Ž a, 0. is closed, ;a. Hence, it is bounded iff its m asymptotic cone equals 04 . V Ž a, 0.` s UŽ0.` l Rq s V Ž0.` s 04 by H5. B
3. Fair utility weight vectors and existence of equilibria In finite dimensional economies, one of the methods to prove the existence of an equilibrium when consumption sets are unbounded below consists first in defining prices for which aggregate demand exists Žno-arbitrage prices.. One applies subsequently a generalized version of Gale–Nikaido–Debreu’s lemma to prove the existence of an equilibrium. As it is well known, this method does not generalize to infinite dimensional economies even if consumption sets are bounded below. To prove the existence of an equilibrium in infinite dimensional economies where consumption sets are bounded below, Negishi’s method has often been used either in its topological version with a separation argument to find supporting prices or in its differential version. Supporting prices are then elements of the subdifferential of the support function of the utility set also called ‘‘the aggregate utility’’. In this paper, we follow the second approach. Our aim is to define the excess utility correspondence. We shall show that this correspondence is defined on the set of strictly positive weights for which there exists Pareto-optima. This is why we call these weights ‘‘fair utility weight vectors’’. We shall prove that it has all the properties of an excess demand correspondence in finite dimensional economies. While the excess demand correspondence is defined on the set of ‘‘no-arbitrage prices’’, the excess utility correspondence is defined on the set of ‘‘fair utility weight vectors’’. As for the excess demand correspondence, we apply a generalized version of Gale–Nikaido–Debreu’s lemma to prove the existence of an equilibrium. In order to define the excess utility correspondence, let us introduce some notation.For every l g R m , ´ g F, let h Ž l , ´ . s sup
Ýl i z i
zgU Ž ´ . i
be the support function of UŽ ´ .. Trivially, hŽ l , P. is concave.
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m Let DŽ ´ . s dom hŽ l , ´ . s l g R m N hŽ l , ´ . - `4 . Since U`Ž ´ . > Ry , it has non-empty interior. It follows from Aubin Ž1982. p. 34, Proposition 9 that m. m DŽ ´ . s U`8Ž ´ . ; ŽRy 8 s Rq . We shall prove that DŽ ´ . is independent of ´ and hence, denote it by D. For l g int D, we shall prove that there exists a unique x Ž l . Pareto-optimal such that hŽ l , 0. s Ý i l i u i Ž x i Ž l ... More precisely, For l g int D, let Õ Ž l . s Argmax l u N u g UŽ0.4 and x i Ž l . be such that Õi Ž l . s u i Ž x i Ž l ... We call int D the set of fair utility weight vectors.
For ´ g W ,
let E´ h Ž l , ´ .
s z g F X N h Ž l , ´ . y h Ž l , ´ y u . G zu, ; u g F 4 . We shall prove that E´ hŽ l , 0. is not empty, that it is convex, weakly compact, and u.h.c. We may now define the excess utility correspondence. For l g int D, ; i s 1, . . . , m, let Ei Ž l . s
½
z P Ž x i Ž l . . y wi
li
5
, z g E´ h Ž l , 0 . .
Our main result is the following. Theorem 3.1. Assume H1-H2-H3-H4-H6. (a) If there exists a fair utility weight Õector, then there exists a quasi-equilibrium (x, z) with z g E´ h(l , 0) and l g D. Assume furthermore ;l g D, ; p g E´ h( l ,0) _ {0}, ; i, pwi ) infpX i , then, (b) the existence of a fair utility weight Õector is equiÕalent to the existence of an equilibrium.
The Proof of Theorem 3.1 is done in four steps, ŽProposition 3.1, 3.2 and 3.4.. In particular in Proposition 3.4, we give a sufficient condition for the existence of fair utility weight vectors. Proposition 3.1. (a) Assume H1-H2. Then ;´ g W , h(P, ´ ) is conÕex and lower semi-continuous. (b) Assume H1-H2. Then ;´ g W , ´ X g W , D(´ ) s D(´ X ). (c) Assume H1-H2-H3. Then ;l g D(´ ), h( l , P) is continuous in W .
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Proof. To prove Ža. the function hŽP, ´ . is convex, and lower semi-continuous as a support function Žsee Rockafellar, 1970, Theorem 13.2.. To prove Žb. assume that hŽ l , ´ . - ` and there exists ´ X g W such that hŽ l , ´ X . s `. Let ´ Y g W be such that ´ s t ´ X q Ž1 y t .´Y with t gx0, 1w. We then have: q` ) h Ž l , ´ . G th Ž l , ´ X . q Ž 1 y t . h Ž l , ´ Y . s q`, since hŽ l , ´ Y . ) y`, a contradiction. Hence, hŽ l , ´ Y . - q` and DŽ ´ X . s DŽ ´ .. Let us lastly prove that hŽ l , P. is continuous in W . Let ´ g W . It follows from H3 that there exists wX g AŽ ´ . such that u i Ž wiX . G u i Ž wi .; i. Hence, m
m
h Ž l , ´ . G Ý l i u i Ž wXi . G Ý l i u i Ž wi . s 0, i
i
by assumption H2. Since the function hŽ l , P. is concave, finite valued and bounded below, it is continuous in W . B We shall simply denote by D s dom hŽP, ´ . and by U` the asymptotic cone of UŽ ´ .. For ´ g W , observe that E´ h Ž l , ´ . s z g F X N h Ž l , ´ . y h Ž l , ´ y u . G zu, ; u, such that ´ y u g W 4 . Let z verify h Ž l , ´ . y h Ž l , ´ y u . G zu, ; u, such that ´ y u g W . Let y g F. Then there exists 0 - m - 1 such that ´ y m y g W , hence, we have, h Ž l , ´ . y h Ž l , ´ y m y . G m Ž zy . . As hŽ l , P. is concave, hŽ l , ´ y m y . G m hŽ l , ´ y y . q Ž 1 y m . hŽ l , ´ . hence,
m Ž h Ž l , ´ . y h Ž l , ´ y y . . G h Ž l , ´ . y h Ž l , ´ y m y . . G m Ž zy . which, after division by m , is to be proven. We shall now state the properties of E´ hŽ l , 0.. Proposition 3.2. Assume H1-H2-H3. (a) ;l g D, ;´ g W , E´ h( l , ´ ) is non-empty, conÕex, and weakly compact, (b) ;l g intD, there exists a weakly compact subset K of F X such that E´ h( lX , 0) : K, ;lX g V( l ) a neighborhood of l , and (c) the correspondence l g intD E´ h( l , 0) is upper hemi- continuous in the weak topology of F.
™
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Proof. Since by Proposition 3.1, hŽ l , P. is continuous in W , E´ hŽ l , ´ . is non-empty, convex, and weakly compact Žsee Aubin, 1982, p. 108.. To prove Žb., we have h Ž lX , 0 . y h Ž lX , ´ . G E´ h Ž lX , 0 . Ž y´ . , h Ž lX , 0 . y h Ž lX , y´ . G E´ h Ž lX , 0 . Ž ´ . . Let V Ž l . be a compact neighborhood of l such that V Ž l . : int D. Since hŽP, 0. is continuous in V Ž l ., < hŽ lX , 0.< F M. We also have h Ž lX , ´ . G 0, ;lX g V Ž l . , ;´ g W . Hence < E´ h Ž lX , 0 . ´ < F h Ž lX , 0 . F M , ;lX g V Ž l . , ;´ g W and E´ hŽ lX , 0. is compact by Alaoglu theorem. Lastly to prove Žc., since hŽP, 0. and hŽP, ´ . are continuous on int D, the correspondence l g int D E´ hŽ l , 0. has a closed graph and from Žb. is upper hemi-continuous for the weak topology of F. B
™
In the next proposition, we prove the existence of Pareto-optima. We first need a lemma. Lemma 3.3. Let C( l ) s {u g U(0) N l u G 0}. Assume H1-H2-H4. Then the correspondence C: intD R mq is conÕex, compact Õalued and u.h.c. Assume furthermore H3, then it is continuous. The Proof may be found in the Appendix.
™
Proposition 3.4. (a) Assume H1-H2. Let l g D. If h( l , 0) s l Õ, then there exists x g A(0) such that h( l , 0) s Ý i l i u i (x i ). If l i ) 0, Õi is unique and there exists a unique x i such that Õi s u i (x i ). (b) Assume H1-H2-H3-H4. Then intD s { l 4 0 N ' Õ g U(0), h( l , 0) s l Õ} s { l 4 0 N ' x Pareto-optimal such that h( l , 0) s Ý i l i u i (x i )}. (c) Assume H1-H2-H3-H4. Then the map intD Argmax{ l u N u g U(0)} is continuous. (d) Assume H1-H2.
™
(i) If x is Pareto-optimal, then there exists l g D such that h( l , 0) s Ý i l i u i (x i ). (ii) Assume furthermore H4. Then H5 intD /œ 0 existence of a Pareto-optimum. (iii) Assume furthermore H3-H4-H6 and X i s F, ; i, then x is Pareto-optimal, iff there exists l g intD such that h( l , 0) s Ý i l i u i (x i ).
m
m
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Proof. If hŽ l , 0. s l Õ, then there exists x g AŽ0. such that Õi F u i Ž x i .. Hence hŽ l , 0. s Ý i l i u i Ž x i .. If l i ) 0, Õi s u i Ž x i . and x i is unique since each u i is strictly concave. To prove Žb., let l g int D. Then by Lemma 3.3, C Ž l . is compact and hŽ l , 0. s sup l u N u g C Ž l .4 s l Õ for some Õ g UŽ0.. Since l g int D, l 4 0. From Ža., Õi s u i Ž x i ., ; i and obviously x is Pareto-optimal. Conversely, let us assume that hŽ l , 0. s l Õ for some l 4 0 and Õ g UŽ0.. If l f int D, then there exists p g U` _ 04 such that l p s 0. However, hŽ l , 0. s l Õ s l Ž Õ q p . which contradicts Ža.. Hence, l g int D. To prove Žc., we use the fact that hŽ l , 0. s sup l u N u g C Ž l .4 s l Õ, the lemma, and the maximum theorem. Lastly, Žd. Ži. is well-known. The Proof of Žii. may be found in the Appendix. To prove Žiii., let us prove that if x is Pareto-optimal, then l 4 0. If not, assume that l 1 s 0, l 2 s 0, . . . , l r s 0. Let k rq1 be as in H6. Let xX g A be defined as follows: xX1 s x 1 y k rq1 , xXrq1 s x rq1 q k rq1 , xXj s x j , ; j / 1, r q 1, then
Ý l i u i Ž xXi . ) Ý l i u i Ž x i . i
i
a contradiction.
B
For l g int D, let Õ Ž l . s Arg max l u N u g U Ž 0 . 4 and let x i Ž l . be such that Õi Ž l . s u i Ž x i Ž l ... Let us recall the definition of the excess utility correspondence. For l g int D, ; i s 1, . . . , m, let Ei Ž l . s
½
z P Ž x i Ž l . y wi .
li
5
, z g E´ h Ž l , 0 . .
Proposition 3.5. Assume H1-H5. For l g intD, E is a conÕex, compact, non-empty Õalued, upper hemi-continuous correspondence, which satisfies l E( l ) s 0 (Walras Law). Proof. Clearly, E is convex and non-empty valued. Let O Ž l . be a compact neighborhood of l g int D. Then ; i, ; z g E´ hŽ lX , 0., lX g O Ž l ., X
X
Õi Ž l . s u i Ž x i Ž l . . y u i Ž w i . G
z P Ž x i Ž lX . y wi .
li.
Hence t i Fmax< Õi Ž lX . < F A i , OŽ l.
; t i g Ei Ž lX . , ;lX g O Ž l .
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for some A i , by Proposition 3.4Žc.. Since Ý i lXi Ei Ž lX . s 0, 'Bi g R such that t i G Bi ; t i g Ei Ž lX . , lX g O Ž l . . Hence EŽ lX . has values in a fixed compact set of R m for lX g O Ž l .. Let us finally prove that E has closed graph. Let l n l g int D and Ž z n P Ž x i Ž l n . y wi ..rl ni j i , ; i, z n g E´ hŽ l n , 0..
™
™
; i , Õi Ž l n . y Õi Ž l . G
zn P Ž x i Ž l n . y x i Ž l . .
l ni
Since by Proposition 3.2, E´ hŽ l , 0. is u.h.c., some subsequence of z n converges weakly to z g E´ hŽ l , 0.. Hence, since Õi Ž l n . Õi Ž l ., by Proposition 3.4Žc., we get
™
zw s Ý zx i Ž l . G Ýlimsup z n x i Ž l n . G limsupÝ z n x i Ž l n . s zw, i
i
i
hence zx i Ž l . s limsup z n x i Ž l n .; i and j s Ž z P Ž x i Ž l . y wirl i . and E is upper hemi-continuous. B Proof of Theorem 3.1. Let D n s l g int D, N 5 l 5 s 1, F Ž l . F Žy1rn.4 and K n be the cone generated by D n . It follows from a generalized version of Gale–Nikaido–Debreu’s lemma proven in Florenzano and Le Van Ž1986. that ; n, 'l n g D n , 'e n g EŽ l n . such that yl e n F 0, ;l g K n . Let l be a limit point of l n . There are three cases. Case Ž1. l g int D. Then by Proposition 3.5, e n e g EŽ l .. Thus yl e F 0, ;l g D and hence, ye g U` . Since l g int D, yl e - 0 if e / 0 which contradicts Walras Law. Hence, e s 0. There exists z g E´ hŽ l,0. such that 0 s ŽŽ z P Ž x i Ž l . y wi ..rl i .; i.
™
; i , ; x i g X i , l i Ž u i Ž x i Ž l . . y u i Ž x i . . G z Ž x i Ž l . y x i . s z Ž wi y x i . . m Thus zx i F zwi implies u i Ž x i . F u i Ž x i Ž l ... Hence, wŽ x i Ž l .. is1 , zx is an equilibrium. Case Ž2. l g ED and 5 e n 5 `. Since e i n F Õi Ž l n ., ; i, e n g UŽ0., ; n. Hence, Ž e i nr5 e n 5. t g U` _ 04 . Since yl e n F 0, ;l g K n , we get at the limit yl t F 0, ;l g D, which is impossible. Case Ž3. l g ED and some subsequence of e n e. As in Case Ž1., we have l e s 0. Since e i n F Õi Ž l n ., ; i, Õ Ž l n . is bounded below. By Proposition 2.1, it is bounded above. Without lost of generality, let us assume that Õ Ž l n . converges to Õ. Since
™
™
™
Ý l ni Õi Ž l n . G Ý l ni u i , i
i
we have
l Õ G l u ; u g U Ž 0. ,
; u g U Ž 0. ,
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hence ` ) hŽ l , 0. s l z. Since l e s 0, hŽ l , 0. s l Ž z q e .. By Proposition 3.4Ža., l i ) 0 implies e i s 0. Hence, l i e i s 0, ; i. By Proposition 3.2 and its Proof, we still have E´ h Ž l n , 0 . ´ F < h Ž l n , 0 . < F M , ; n, ;´ g W ,
™
since hŽ l n , 0. hŽ l n , 0.. We may therefore assume that if e in s Ž z n P Ž x i Ž l n .. y wi .rl ni , z n g E´ hŽ l n , 0., then z n z weakly. Let x Ž l . be such that hŽ l , 0. s Ý i l i u i Ž x i Ž l ... Then as in the Proof of Proposition 3.5, z n x i Ž l n . zx i Ž l ., ; i. Hence
™
™
; i , z Ž x i Ž l . y wi . slime inl ni s l i e i s 0. n
m Let us prove that wŽ x i Ž l .. is1 , z x is a quasi-equilibrium. Since
l ni u i Ž x i Ž l n . . y u i Ž x i . G z n Ž x i Ž l n . y x i . , ; i , if l i ) 0,
l i u i Ž x i Ž l . . y u i Ž x i . G z Ž x i Ž l . y x i . s z Ž wi y x i . . Let x i be such that u i Ž x i . ) u i Ž x i Ž l .. Žby assumption H6 such an x i exists.. m Then zx i ) zwi . Hence, z / 0. If l i s 0, then zwi F zx i . Hence, wŽ x i Ž l .. is1 , z x is a quasi-equilibrium. Since ` ) hŽ l , 0., l g D and from the last inequality z g E´ hŽ l , 0.. Assume furthermore ;l g D, ; z g E´ hŽ l , 0. _ 04 , ; i, zwi ) inf zX i . Let us m , z x is an equilibrium, equivalently that u i Ž x i . ) u i Ž x i Ž l .. prove that wŽ x i Ž l .. is1 implies z Ž wi y x i . - 0. If l i ) 0, it follows from the proof above. Assume l i s 0 and zwi s zx i . Let yi g X i be such that zyi - zwi . Then for t gx0, 1w, u i Ž tx i q Ž1 y t . yi . ) tu i Ž x i . q Ž1 y t . u i Ž yi . and ztx i q Ž1 y t . yi - zwi , hence, for t sufficiently close to one, u i Ž tx i q Ž1 y t . yi . ) u i Ž x i Ž l .. and ztx i q Ž1 y t . yi - zwi a m contradiction. Hence, wŽ x i Ž l .. is1 , z x is an equilibrium. Conversely, if there exists an equilibrium, then there exists a Pareto-optimum. It follows from Proof of Proposition 3.4Žd-ii. that int D / f . B
4. Comments on our assumptions and links with the literature Ž1. In order to obtain support prices, Brown and Werner Ž1995., Dana et al. Ž1997. assume that ; x g AŽ0., Pi Ž x i . / 0, ; i and there exists j with int Pj Ž x j . / 0. In this paper, we only assume ; i, Pi Ž x i . / 0, ; x g AŽ0. ŽH6. but in order to obtain support prices, we have to assume H3. Assumption H3 is fulfilled if int X i / 0, ; i, and u i is continuous for every i. Ž2. The standard assumption on utilities is to assume that they are strictly quasi-concave. We assume that utilities are concave to get a convex utility set and
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strictly concave in order to have that to a set of utility weight vectors, there corresponds a unique Pareto-optimum. Ž3. Assumption H4 may seem very strong since in the topological Negishi’s approach, one only uses the set V Ž0. of individually rational utilities. Brown and Werner Ž1995., Dana et al. Ž1997. only assume V Ž0. compact. If V Ž0. is assumed to be compact, one can substitute to the original economy another economy with the same quasi-equilibria for which H4 is fulfilled. Indeed, consider the economy EX where the consumption sets X i are replaced by: X iX s x g X i N u i Ž x i . G u i Ž wi . 4 . If V Ž0. is compact, the set U X Ž 0 . s Õ g R m N Õi F u i Ž x i . , ; i , for some x g AX Ž 0 . 4 where AX Ž0. s Ž x 1 , . . . , x m . g P i X iX <Ý mis1 x i s w4 is obviously closed since U X Ž0. m s V Ž0. q Ry . Hence, H4 is fulfilled for the economy EX which has a quasi-equilibrium. That quasi-equilibrium is a quasi-equilibrium for the initial economy. Ž4. In order to link our results to those of Chichilnisky and Heal Ž1993., let m
½
AŽ t , 0. s x s Ž x 1 , . . . , x m . gŁXi N Ý x i i
is1
5
s v , u i Ž x i . G u i Ž ti . , ; i , t gŁXi . i
The following conditions are similar to those introduced by Chichilnisky and Heal Ž1993.. G1. F is a reflexive space and X i s F, ; i. G2. For any t g F m , AŽ t, 0. is norm bounded. G3. ; i, u i is norm continuous and strictly concave. Proposition 4.1. Assume G1-G2-G3, then H4 and H5 are fulfilled. Proof. Since u i is norm continuous for every i, for any t g F m , AŽ t, 0. is norm closed and norm bounded, hence weakly compact, since F is reflexive. Since u i is weakly u.s.c., ; i, V Ž0. is compact and H5 is fulfilled. Lastly, for any t g F m , let V Ž t , 0 . s z g R m N u i Ž t i . F z i F u i Ž x i . , ; i , for some x g A Ž 0 . 4 . Similarly, for any t g F m , V Ž t, 0. is compact. Since u i is defined on F, inf x g F u i Ž x . s y`, ; i. Hence, UŽ0. s jt g F m V Ž t, 0. and closed, thus, H4 is fulfilled. B
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It turns out that G2 is not generally fulfilled. In fact, Cheng Ž1991. shows that it is not fulfilled in the case of L p and Von Neumann–Morgenstern utilities. Ž5. In order to link our results to those of Brown and Werner Ž1995., let us recall the following definition: let p g F X _ 04 . A sequential arbitrage opportunity for i with respect to p is a sequence xˆ n such that wi q xˆ n g X i , lim u i Ž wi q xˆ n . s sup x i g X i u i Ž x i . and lim pxˆ n F 0. A price system is arbitrage-free for the economy if for every agent, there is no sequential arbitrage. Assuming H1-H2, Brown and Werner Ž1995. prove that the existence of an arbitrage-free price for the economy implies H5. We slightly strengthen their result. Proposition 4.2. Assume H1-H2-H4 and sup x i g X i u i (x i ) s `, ; i. Then, the existence of a fair utility weight Õector (i.e. intD / f ) is equiÕalent to H5 which in turn is equiÕalent to the existence of an arbitrage-free price for the economy. Proof. We prove here that the existence of a fair utility weight vector is equivalent to the existence of an arbitrage-free price for the economy. Let l be fair utility weight vector and x Ž l . be the associated Pareto-optimum by Proposition 3.4Žb.. Let p g E´ hŽ l, 0. and let Ž x in . be a sequence such that u i Ž wi q x in . `. Then
™
l i ui Ž x i Ž l . . y ui Ž px in
™
wi q x in
. GpŽ xi y Ž
wi q x in
..
hence, `, which implies that p is arbitrage-free price for the economy. The converse is proven in Brown and Werner Ž1995.. The reader can check that it does not require the continuity of the utility functions. B
5. Arbitrage and duality in finite dimensional economies In this section, we shall show that in finite dimensional economies, the demand approach which is based on the assumption of existence of a no-arbitrage price and the duality approach, which is based on the assumption of existence of a fair utility weight vector, are equivalent. We shall maintain the following assumptions: F1. X i : R l is closed and convex, ; i.
™
F2. ; i, u i : X i R is strictly concave and continuous and does not have a satiation point and u i Ž wi . s 0. F3. w is not weakly Pareto-optimal. Remark. Clearly, F1 is equiÕalent to H1, F2 implies H2 and H6, and F2-F3 imply H3. A vector t g R l is useful for i if u i Ž x q t . G u i Ž x ., ; x g X i . Let Wi denote the set of useful vectors for i. It is a standard result that Wi is a closed and convex
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cone and that it is the asymptotic cone of the set x g X i , u i Ž x i . G a i , ; i4 for any a g R n. It follows from F2 that Wi / 04 . A vector p g R l is a no-arbitrage price if there exists no x / 0, x g Ý iWi such that px F 0 Žequivalently, x g Ý iWi _ 04 implies px ) 0.. Hence, p g R l is a no-arbitrage price if p g intŽyÝ i ŽWi .8.. Symmetrically, we may introduce the following definition. Assume UŽ0. is closed. A vector l g R m is a fair utility weight Õector if there exists no t / 0, t g U` such that l t G 0 Žequivalently t g U` _ 04 implies l t - 0.. Hence, l g R m is a fair utility weight vector if UŽ0. is closed and if l g intU`8Ž0.. For further use, let Õ g R m and A Ž Õ, 0 . s Ž x 1 , . . . , x m . g A Ž 0 . < u i Ž x i . G Õi , ; i 4 be the set of attainable allocations with utilities bounded below by Õ. Theorem 5.1. Assume F1-F2, then the following statements are equiÕalent. 1. 2. 3. 4. 5.
there exists a no-arbitrage price, A(Õ, 0) is compact for any Õ, H4 and H5 are fulfilled, there exists a fair utility weight Õector and H4 is fulfilled, and there exists a Pareto-optimal allocation.
Any preÕious condition implies that there exists a quasi-equilibrium. Assume furthermore ;l g D, ; z g E´ h( l , 0) _ {0}, ; i, pwi ) infpX i , then the assumption of existence of a no-arbitrage price is equiÕalent to the assumption of existence of a fair utility weight Õector and H4 is fulfilled and is equiÕalent to the existence of an equilibrium. Proof. Let us sketch these equivalences. Statement Ž1. above implies Ž2.. Since u i is concave and continuous, AŽ Õ, 0. is closed and convex for any Õ and its asymptotic cone is x g P iWi N Ý i x i s 04 s 04 . Assume there exists x I g Wi , ; i, x j / 0, for some j, Ý i x i s 0. Let p be no-arbitrage price. Then pÝ i x i s 0 while px j ) 0, px i G 0, ; i / j, a contradiction. Ž2. implies Ž3.. Since V Ž0. s AŽ0, 0., H5 is fulfilled. To prove H4, let Õ n g UŽ0. Õ. Then ; n, ; i, there exists x in such that Õin F u i Ž x in ., ;i, ;´ , for n large enough, x n g AŽ Õ y ´ , 0.. Hence, a subsequence of x n converges to x and Õi F u i Ž x i ., ; i. Ž3. implies Ž4. and Ž4. implies Ž5. following from Proposition 3.4 Žii.. Ž5. implies Ž1.. See for example, Dana et al. Ž1999.. B
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Acknowledgements We are very grateful to the referee for his comments and suggestions.
Appendix A
Lemma 3.3. Let C( l ) s {u g U(0)< l u G 0}. Assume H1-H2-H4. Then, the correspondence C: intD R mq is conÕex, compact Õalued, and u.h.c. Assume furthermore H3, then it is continuous.
™
Proof. Let
F Ž l . s sup l p N p g U` , 5 p 5 s 1 4 . m< Ž . m< Ž . Let us first prove that D s l g Rq F l F 04 and int D s l g Rq F l - 04 . m Indeed, since D s U`8 g Rq , the proof of the first statement is obvious. Since F is continuous,
l g Rqm N F Ž l . - 0 4 : int D Conversely, let l g int D and assume that F Ž l . s 0. Let p be such that F Ž l . s p l and ´ g R m be such that p´ ) 0. For t ) 0 sufficiently small l q t ´ g D but F Ž l q t ´ . G Ž l q t ´ . p ) 0, a contradiction. Let us next prove the statement in Lemma 3.3. Since C Ž l . is clearly closed and convex, let us show that if l g int D, C Ž l .` s 04 . Let t g C Ž l .` . Then t g U` and l t G 0, hence, l t s 0. However, F Ž l . s 0 which contradicts the fact that l g int D. Hence, C is compact-valued. Obviously, C has closed graph. Let us next show that C is u.h.c. Let V Ž l . be a compact neighborhood of l g int D. Let us prove that C has values in a fixed compact set. Suppose not, then there exists l n l , u n g C Ž l n . such that 5 u n 5 q`. Then u nr5 u n 5 u g U` _ 04 and l u G 0. Since l g int D, l u - 0, a contradiction. Hence, C has values in a fixed compact set in a neighborhood of l and is u.h.c. Let us lastly show that C is l.h.c. Let u g C Ž l . and l n l . If l u ) 0, then l n u ) 0, for n large enough and C is l.h.c. at l . If l u s 0, let Õ g intV Ž0.. Then Õ 4 0 and l Õ ) 0, hence, l n Õ ) 0 for n large enough. Assume l n u - 0, ; n. Let un gx0, 1w be defined by l nw un u q Ž1 y un . Õ x s 0. The sequence un being bounded, un u and lw u u q Ž1 y u . Õ x s Ž1 y u . l Õ s 0. Hence, u s 1 and un u q Ž1 y un . Õ u with un u q Ž1 y un . Õ g C Ž l n . for n large enough. B
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Proof of Proposition 3.4(d-ii). Assume first V Ž0. bounded, hence, compact and int D s f . Since D8 s U` , U` contains a line tÕ, t g R4 . Let I1 s i < Õi ) 04 , I2 s i < Õi - 04 and I3 s i < Õi s 04 . Without lost of generality, we may assume that
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I1 / f . Let i 0 g I1. Let z g V Ž0. be such that z i 0 s max z i0 , z g V Ž0.4 . There exists x g AŽ0., xX g AŽ0. such that ; i , z i q Õi F u i Ž x i . z i y Õi F u i Ž xXi . . Let i g I1. If x i / xXi , x i q xXi zi - ui . 2
ž
/
If x i s xXi , then z i - z i q Õi F u i Ž x i . s u i
ž
x i q xXi 2
/
,
a contradiction if i s i 0 . Therefore, U` contains no line and int D / f . Hence, H5 implies int D / f . It follows from Proposition 3.4Žb. that if int D / f , then there exists a Pareto-optimum. Hence, int D / f implies existence of a Pareto-optimum. m Assume V Ž0. unbounded. Then, V Ž0.` s UŽ0.` l Rq / 04 . Let Õ g V Ž0.` , Õ / 0 and x g AŽ0. be a Pareto-optimum. Let z be such that z i s u i Ž x i ., ; i. Then z q Õ g V Ž0., hence, there exists xX g AŽ0. such that z i q Õi F u i Ž xXi ., ; i. Hence, u i Ž x i . F u i Ž xXi ., ; i with a strict inequality for at least some i since Õ / 0, a contradiction. Hence, existence of a Pareto-optimum implies H5. B
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