Discreteness of equilibria in incomplete markets with a continuum of states

Journal of Mathematical Economics 33 Ž2000. 229–237 www.elsevier.comrlocaterjmateco

Discreteness of equilibria in incomplete markets with a continuum of states Paulo Klinger Monteiro b

a,)

, Mario ´ R. Pascoa ´

b

a COREr IMPA, Voie du Roman Pays 34, 1348 LouÕain-la-NeuÕe, Belgium Faculdade de Economia, UniÕersidade NoÕa de Lisboa, TraÕessa EsteÕao ˜ Pinto, P-1070 Lisbon, Portugal

Received 14 May 1997; received in revised form 15 January 1999; accepted 8 April 1999

Abstract We discuss the issue, raised by Mas-Colell wMas-Colell, 1991. Indeterminacy in incomplete markets economies. Economic Theory 1 Ž1., 45–61x whether local uniqueness Žrelative to the L` topology. may be a generic property of equilibria in incomplete markets economies with a continuum of states. q 2000 Elsevier Science S.A. All rights reserved. JEL classification: D52 Keywords: Equilibrium; Incomplete market economy; Continuum of states; Cardinality

1. Introduction The analysis of equilibria in incomplete markets economies with a continuum of states is still an important, open and difficult subject. The known results on existence of equilibrium use either a strong assumption on non-negativity of ex-post endowments — i.e., endowments plus real returns Žsee Hellwig, 1996; Mas-Colell and Zame, 1996; Monteiro, 1996. — or a modification of the equilibrium concept to allow for default subject to penalties Žsee Araujo et al., 1996, 1997.. The cardinality of the equilibrium set was addressed by Mas-Colell Ž1991.. )

Corresponding author. fax: q55-21-5295129; E-mail: [email protected]

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 9 9 . 0 0 0 1 7 - 8

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P.K. Monteiro, M.R. Pascoar Journal of Mathematical Economics 33 (2000) 229–237 ´

The generic finiteness of the equilibria does not extend from the complete markets case to incomplete market and the difficulty has to do with the nature of the continuum states set Žin contrast with the well-known difficulties, limited to nominal assets, faced when the states set if finite.. Mas-Colell Ž1991. gave a robust counterexample of a continuum of states incomplete market economy with uncountably many equilibria. Mas-Colell’s goal was to examine only the cardinality issue, but in his own words, ‘‘It is certainly true that for comparative statics purposes, local uniqueness is the key property.’’ The choice of topology is crucial for discreteness of equilibria: in the case of the Žseparable. L1 topology a continuum of equilibria is a non-discrete set, but in the case of the non-separable L` topology discreteness is compatible with a continuum of distinct equilibria. In Mas-Colell’s challenging words, ‘‘It could be asked if local uniqueness Žrelative to the L` topology. may still in some sense be a generic property. This is a difficult question and we have no answer to offer.’’ Our purpose is precisely to obtain results on discreteness of equilibria under the L` topology. In the case where there are no trades in assets markets, the pure spot market equilibria of the incomplete market economy are shown to be discrete, generically, on endowments and utilities. Moreover, if utilities are state-independent, discreteness is generic on endowments. Our result on the no-assets case is interesting since this case has been studied previously in the literature, by Mas-Colell Ž1991. and Davila Ž1998. having been pointed out by Mas-Colell Ž1991. that ‘‘ . . . in what concerns the indeterminacy question the trivial no assets case is not trivial at all, but it represents perfectly well what happens in the general incomplete situation.’’ When assets are traded, we show first that, generically on endowment functions and assets returns matrices, ex-post spot endowments are regular, for almost every state. In particular, this shows that the regularity assumption in the work of Mas-Colell and Monteiro Ž1996. on existence of equilibrium is actually, generic. Secondly, we establish the discreteness of equilibria with asset trades for an open dense set of endowments, provided that utility is state-independent and the set of non-regular spot endowments translated by the span of the assets return matrices has an empty interior.

2. The model and definitions I Consider a two-period economy E s ŽŽ w i ,u i . is1 , A. with a finite number I of agents. There is no consumption in the first period. There is a continuum S of states of nature that may occur in the second period affecting endowments, assets returns and possibly also preferences over G physical goods. We endow S with a s-algebra B and a non-atomic probability measure m. Preferences are assumed to be additively separable over states and the instantaneous utility functions belongs

P.K. Monteiro, M.R. Pascoar Journal of Mathematical Economics 33 (2000) 229–237 ´

231

G to the space U of C 2 functions on Rqq which are differentiably strictly concave, strictly monotone and proper; we endow U with the relative topology of C 2 uniform convergence on compacta. To define a metric to this topology, define for G each compact set K ; Rqq the C 2 Ž K . norm 5 a 5 K s max k g K < aŽ k .< q EaŽ k .< q 2 < E aŽ k .<4 . Then if K n s w1rn,n xG a metric for this topology is d U , where

dU : U = U

™ R, d

5uyÕ5 Kn

` yn U Ž u,Õ . s Ý 2

ns1

1q5uyÕ5 Kn

.

™

™

The consumer i, 1 F i F I is described by an essentially bounded measurable G endowment function w i :S Rqq and instantaneous utility map u i :S U . There are J real assets traded in the first period with returns in the second period. Assets returns are described by A s Ž A1, . . . , A J .. The returns of asset j is given by a measurable essentially bounded function A j:S R G. We denote by A the set of consumers characteristics:

™

S

G G A s Ž w,u . g Ž Rqq = U . ; w g L` Ž S,Rqq . and u is measurable .

½

5

As in ŽMas-Colell, 1991., we endow A with the complete metric d A defined by: d A Ž Ž wX ,uX . , Ž w,u . . s < w y wX < ` qesssup d U Ž u s ,uXs . . s

Naturally, the set A I is endowed with the product topology and we denote by I I I . s Ý is1 d A I the metric dŽŽ a i ,b i . is1 ,Ž c i ,d i . is1 d A ŽŽ a i ,b i .,Ž c i ,d i ... Later, we will also need the set of consumer characteristics and asset matrices, AI M s A I = L`ŽS,R G J .. We denote by d AI M the metric d AI M Ž Ž w,u, A . , Ž wX ,uX , AX . . s d A I Ž Ž wX ,uX . , Ž w,u . . q < A y AX < ` . G. We denote by SŽRq the G y 1 dimensional simplex. An equilibrium for E is a i i I . such that vector Ž p,q,Ž x , u . is1 G. 1. p:S SŽRq is measurable; G I I 2. x i :S Rq is measurable and Ý is1 x si F Ý is1 wsi for m a.e. s g S; i J I i 3. u g R for every i and Ý is1 u s 0; 4. Ž x i , u i . maximizes

™™

i s

i s

HSu Ž x . d m Ž s . subject to p Ž x y w . F p s

and qu F 0.

s

s

s

A s u i for m a.e. s

Ž 1.

I . is an equilibrium, then for m a.e. s, the vector Notice that if Ž p,q,Ž x i , u i . is1 i I Ž ps ,Ž x s . is1 . is a Walrasian equilibrium for the pure exchange economy Ž w i Ž s . q I AŽ s .u i ,u is . is1 for m a.e. s g S. Finally, we define local uniqueness.

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P.K. Monteiro, M.R. Pascoar Journal of Mathematical Economics 33 (2000) 229–237 ´

Definition 1. I (1) An equilibrium of the economy E , Ž p,q,Ž x i , u i . is1 . is locally unique if 1 I there is a neighbourhood W of Ž p, x , . . . , x . such that Ž p, x 1 , . . . , x I . is the only price and equilibrium allocation of E that belongs to W . (2) The equilibrium set of E is discrete if eÕery price and equilibrium allocations of E are locally unique. Remark 1. The definition above corresponds exactly and respectively to the mathematical notion of isolated point and discrete set.

3. Discreteness in the no assets case We will consider in this section the case where A s 0. Let us define I

G N s Ž a,b . g Ž Rqq = U . ; Ž a,b . is not a regular economy .

½

5

Theorem 1. The subset of consumer characteristics Ž w,u. g A I such that the I equilibrium set of E s ŽŽ w i ,u i . is1 ,0. is discrete contains an open and dense I subset of A . Proof. Let C be the subset of A I of economies with only isolated equilibria. Let Crs be the subset of C of simple functions with regular values. We start by showing that Crs ; intŽ C .. Take Ž w,u. g Crs . Then we can write Ž w,u. as L Ž G .I Ý ls1 w l ,u l . x T l . Since Ž wl ,u l . is regular, there exist neighbourhoods O l ; ŽRqq I and Vl ; U of w l and u l , respectively, and d ) 0 such that for any spot economy Ž w,u ˜ ˜ . g Ol = Vl , if spot equilibrium prices p, pX satisfy < p y pX < - d we must G .I have p s pX . Let e ) 0 be such that if Ž a,b . g ŽRqq = U I satisfies < a y wl < - e and d U Ž b,u l . - e , for l s 1, . . . , L, then Ž a,b . g O l = Vl , for l s 1, . . . , L. Now let Ž w,u. g A I be such that d A I ŽŽ w,u., Ž w,u.. - e . Therefore, if p 1 and p 2 are two equilibrium price functions for Ž w,u. such that < p 1 y p 2 < ` - d then p 1 s p 2 . Now, if equilibrium allocations are near, equilibrium prices will be near too. We showed that Crs ; intŽ C .. We now show that Crs is dense. It suffices to show that Crs is L Ž dense in the set of simple functions. Let Ž w,u wˆ l ,uˆ l . x T l be any simple ˆ ˆ . s Ý ls1 I Ž . function in A . The results of Smale 1974 imply that, given e ) 0, there exist, for each l, a regular spot economy Ž w lo ,u ol . such that < w lo y wˆ l < - e and d u I Ž u ol ,uˆ ol . L Ž o o. - e . Then the simple function Ž w o ,u o . s Ý ls1 w l ,u l x T l belongs to Crs and o o. Ž ŽŽ .. d A w ,u , w,u ˆ ˆ - e as claimed. Remark 2. The above theorem is in apparent contradiction with the result of Davila Ž1998. that the sets of endowments whose equilibria are determinate and indeterminate are both large. However, Davila restricts his consumers endowments to a set with a constant Edgeworth box Žnamely w0,1x 2 .. This set is a first category set in the set of all endowments and is therefore small for our purposes.

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Remark 3. In the finite dimensional case one usually obtains that the complement of the open and dense set considered above is also null. In the infinite dimensional case we lack a canonical measure and it is difficult to provide a sensible definition of null set. The mathematical notion of prevalence, a measure theoretical analog of genericity is probably the best one can do in this situation. It reduces to the notion of size given by Lebesgue’s measure in finite dimensional spaces. The concept of prevalence has been used by Anderson and Zame Ž1996. and gives sensible answers in several infinite dimensional economies. The question addressed in Theorem 1 might also be discussed along the prevalence approach. Remark 4. When the commodity space is finite-dimensional, discreteness of equilibria and continuous comparative statics go together Žsee the works of Debreu Ž1970. and Smale Ž1974. for complete markets.. In our setting we can infer some comparative statics results, but it is not possible to fully extend the finite L Ž dimensional case. For any Ž w,u. s Ý ls1 w l ,u l . x T l g Crs , there are finitely many lŽ continuous functions a i i s 1, . . . ,k l . mapping an open subset Ml of O l = Vl into kl G Rq such that Ž a il Ž w l ,u l .. is1 are the spot equilibrium prices for Ž wl ,u l . g Ml Žsee Smale Ž1974, Proposition 4... Therefore, there is an e ) 0 such that on M ' Ž w,u.; d A I ŽŽ w,u.,Ž w,u.. - e 4 , for any equilibrium price function p, Tl is partitioned into finitely many sets B il Ž p . s  s; ps s a il Ž w l , p l .4 . Generic lower hemicontinuity of the price equilibrium correspondence f Žfor the d A I metric on parameters and the L` norm on prices. is immediate. In fact, given a sequence Ž wn ,u n . n converging to Ž w,u. g Crs in the d A I metric and given any equilibrium price p for Ž w,u., take a sequence Ž pn . n such that pn < B il Ž p. s a il Ž wn l ,u nl . for i s 1, . . . , k l and l s 1, . . . , L. On M there is a continuous selection for f Žby keeping the partition  Bil 4 constant.. Upper-hemicontinuity does not hold since the above partition of Tl may change along the price sequence; the most that can be said is that, when Ž wn ,u n . converges to Ž w,u. g M and p n g f Ž wn ,u n ., any sequence contained in p nŽTl . G converges to one of k l elements of pŽTl . ; Rq , for some p g f Ž w,u..

4. Regularity of ex-post endowments In this section, we allow for asset trades. We will assume that ŽS, B . is the Lebesgue measurable space Žw0,1x, BŽw0,1x.. and m is the Lebesgue measure. We will assert that the set

 Ž w,u, A . g AI M ; ;u g ŽR J . I , Ž wsi q A s u i ,u is . iIs 1 is a regular economy a.e. 4 is a residual set Žthat is, a countable intersection of open and dense sets, therefore dense also.. This fact establishes that the regularity assumption of Mas-Colell and Monteiro Ž1996. is actually generic. This assumption was used by those authors to

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P.K. Monteiro, M.R. Pascoar Journal of Mathematical Economics 33 (2000) 229–237 ´

establish the existence of a continuous selection from the correspondence associating each portfolio with the L1 subset of equilibrium price functions for spot economies Ž wsi q A s u i .. Using this selection theorem, Mas-Colell and Monteiro show the existence of equilibrium for a continuum of states incomplete market economy. Let: F s Ž w,u, A . g AI M ; 'u g Ž R J .

½

such that m

i s

i

I

i I s is1 g N

ž ½ s g S; Ž w q A u ,u . s

5 / ) 05 .

Theorem 2. F is a first category subset of A I = L`ŽS,R G J .. Proof. Let Fm be the set

½ Ž w,u,a. g F; 'u g Ž R m

i s

i

J

I . , 5 u 5 F m, i I s is1 g N

ž ½ s g S; Ž w q A u ,u . s

5 / G 1rm5

™

and notice that F s j `ms 1 Fm . We start by showing that Fm is closed. Let Ž w n,u n, A n . g Fm for every n g N be a sequence such that Ž w n,u n, A n . Ž w,u, A.. For each n there is a u n g ŽR J . I, 5 u n 5 F m such that m s g S; Ž wsi q A s Ž u n . i , I u is . is1 g N 4 G 1rm. Without loss of generality u n u , 5 u 5 F m. If we define I T s  s g S; ŽŽWsn . i q A ns Ž u n . i . is1 g N infinitely often4 we have that m ŽT . G i I 1rm. It shall be clear that ŽWs q A s u i . is1 g N for all s g T. This proves that Fm c is closed. We now show that F is dense. Let Ž w,u, A. g AI M and e ) 0. There exist simple functions w,u ˆ ˆ and Aˆ such that d A I ŽŽ w,u ˆ ˆ .,Ž w,u.. - e and 5 A y Aˆ5 ` L ˆ - e . Let us write Ž w,u, ˆ ˆ A. s Ý ls1Ž wl ,u l , A l . x T l and, by the isomorphism theorem Žsee the work of Halmos, 1974, Theorem 41.C, page 173.., each Tl is isomorphic to w0, m ŽTl .x and also to Žw0, m ŽTl .xG . I. Then, there exist f l :Tl Be Ž0. ; ŽR G . I such that the pre-image of a null set is null. We claim that ÝTls1Ž w l q f l . x T l s w˜ together with uˆ and Aˆ are as desired, i.e., Ž w,u, ˆ Aˆ. g F c . G I I To see this, let Ml s  x g ŽRqq . ; Ž x i ,uˆ l i . is1 is a non-regular economy4 and, I given u g ŽR J . I, we have m Ž s g Tl ; Ž w˜ li q Aˆl u i . is1 g Ml 4. s m Ž s g Tl ; f l Ž s . g i 4. I i ˆ Ž . Ml y w˜ l y A l u s 0 since m Ml s 0. Therefore, m Ž s g S; Ž w˜ si q Aˆs u i ,uˆ is . is1 c J .I ˆ 4. Ž Ž . g N s 0 for any u g R , that is, w,u, ˜ ˆ A gF .

™

™

5. Discreteness when assets are traded In Section 4 we showed the genericity of the set of consumers characteristics and matrices whose ex-post spot endowments, for any portfolio, are almost surely regular. However, each portfolio choice determines a different null set of states

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where ex-post spot endowments fail to be regular. This difficulty, prevents us from providing a general result if A / 0. The difficulty persists even in the extreme case of state independent utilities and asset returns. It is true that there will be, generically, a finite number of equilibria in the associated finite dimensional spot economy. However, the possibility of an infinite number of sunspot equilibria cannot be easily dismissed Ždue to the uncountably many possible partitions of the state space into sets where each of the finitely many spot equilibria prevails.. However, we give below a partial result for the case of felicity functions and asset returns given by simple functions. Three assumptions are needed. J . A-1. There exists a short sales lower bound Õ g yRqq

A-2. The instantaneous utility and assets returns function Ž u1 , . . . ,u I, A.:S ŽR G . J is a simple function.

™U = I

Suppose T1 , . . . ,TL are the elements of the partition of S defining the simple function Ž u1, . . . ,u I, A.. Define then I

I

G Dl s w g Ž Rqq . ; Ž w i ,u li . is1 is a non-regular spot economy

½

5

and Kls

½

Ž Al u 1 , . . . , Al u I . ; u g Ž R J .

I

I

such that u i G Õ,

Ý u is0 is1

5

.

L Ž A-3. j ls1 Dl y K l . has an empty interior.

This last assumption states that the set of non-regular ex-post endowments of I the spot economy Ž w i ,u li . is1 when translated by vectors in the span of the assets returns matrices must have an empty interior. This assumption guarantees that, in GI the spot economy, the set E ; Rqq , of ex-ante endowments which may generate non-singular ex-post endowments is small, more precisely, is contained in L Ž j ls1 Dl y K l ., which is assumed to have an empty interior. If short sales were not bounded E would actually coincide with the latter, since K l will become a vector space. We do not expect A-3 to be valid in great generality. But it is valid if Dl has sufficiently low dimension. Theorem 3. Under assumptions A-1, A-2 and A-3, equilibria are locally unique for a set of ex-ante endowments which contains an open and dense set in G .I. L`ŽS,ŽRqq . G .I. Ž Proof. Let G s  w g L`ŽS,ŽRqq ; ws q K l . l Dl s B, for a.e. s g Tl ,1 F l F L4 and Grs s  w g G; w is simple4 . Finally, define

V s  w ; Ž w,u . g A I has a discrete equilibrium set 4 .

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L Ml To show that Grs ; intŽ V ., suppose w o g Grs , w o s Ý ls1 Ý ms1 wmo l x S m l where o S m l ; Tl , 1 F m F m l , 1 F l F L. By definition we have Ž wm l q K l . l Dl s B for every m,l. Since K l is compact, Dl y K l is closed. By assumption it has an empty interior. Therefore, from wmo l f Dl y K l there exists a g ) 0 such that B Ž wmo l ,g . l Ž Dl y K l . s B for 1 F m F m l , 1 F l F L. Hence, A l Dl s B where A s B Ž wmo l ,g . q K l . Then, for any x g A, there is a neighbourhood Ox and a d x ) 0 such that for every y g Ox and every spot equilibrium prices p, pX , < p y pX < - d x implies p s pX . The cover  Ox ; x g A4 has a finite subcover G .I  O 1 , . . . ,O Y 4 . Define d s min d 1 , . . . , d Y 4 . For any w g ŽRqq such that < w y wmo < 1 I - d , we conclude from w q Ž A l u , . . . , A l u . g w q K l ; A ; j Yjs1Oj that there is an e ) 0 independent of w such that for any equilibrium spot prices p and pX , for which < p y pX < - e , we must have p s pX . Therefore, for any w g L` G .I. ŽS,ŽRqq such that < w y w o < ` - d , if p and pX are equilibrium price functions G in L`ŽS,Rqq . for which < p y pX < ` - e we have p s pX . This proves that Grs ; intŽ V .. G .I. To show that Grs is dense, let w g L`ŽS,ŽRqq . Given any e ) 0 there is a simple function w˜ such that < w y w˜ < ` - er2. By the assumption the complement L Ž of j ls1 Dl y K l . is open and dense. Therefore, there is a function w g Grs such that < w y w˜ < ` - er2. This ends the proof.

Acknowledgements Monteiro gratefully acknowledges the financial support of JNICT and of the Guggenheim Foundation. Pascoa acknowledges support from FEDER and PRAXIS ´ XXI Žproject 2r2.1 Eco 28r94..

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