Equilibrium in Economies with Financial Markets: Uniqueness of Expectations and Indeterminacy

Journal of Economic Theory  ET2188 journal of economic theory 71, 183208 (1996) article no. 0114

Equilibrium in Economies with Financial Markets: Uniqueness of Expectations and Indeterminacy* Tito Pietra Department of Economics, Rutgers University, New Brunswick, New Jersey 08903; and Dipartimento di Economia Politica, Universita di Modena, Modena, Italy

and Paolo Siconolfi Graduate School of Business, Columbia University, New York, New York 10027 Received May 28, 1994; revised November 28, 1995

We consider two-period, pure exchange economies with uncertainty and complete or incomplete asset markets. Assets are nominal. If the number of agents and of period-zero commodities is large enough, there is a dense, residual set of economies (parametrized by utility functions) such that, for each pair of distinct financial equilibria, spot zero equilibrium prices are different. Agents, observing first-period equilibrium prices, can formulate exact forecasts on future equilibrium prices, notwithstanding the real indeterminacy of the set of financial equilibria. If the asset market is complete, the result is true for an open and dense set of economies. Journal of Economic Literature classification numbers: D52, D80.  1996 Academic Press, Inc.

1. INTRODUCTION Most economic models display multiplicity, or even indeterminacy, of the set of equilibrium outcomes. In intertemporal, sequential economies, multiplicity of equilibria may create a logical difficulty for the rational expectations hypothesis, because it could make it problematic for agents to formulate exact expectations on the equilibrium value of the future variables (see, [7]). * The first author acknowledges the support of an ``Innocenzo Gasparini'' scholarship, of the Ministero dell'Universita e della Ricerca Scientifica e Tecnologica (MURST 600) and of the Consiglio Nazionale delle Ricerche (CNR). The second author acknowledges the support of the Graduate School of Business, Columbia University. The paper was completed while both authors were visiting CORE, Universite Catholique de Louvain. We wish to thank David Cass and an associate editor of this journal for helpful suggestions.

183 0022-053196 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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Multiplicity of equilibria is common in ArrowDebreu economies. In that framework, though, from the point of view of individual behavior, multiplicity of equilibria is irrelevant, because agents do not need to formulate expectations. Agents are price takers. They observe equilibrium prices and act optimally. The issue is more complicated in sequential economies. Consider an intertemporal economy with two periods and a sequence of spot commodity markets. In general, ``today'' consumption and portfolio choices are a function of ``today'' commodity ( p 0 ) and asset (q) prices, and of ``tomorrow'' commodity prices ( p 1 ). Individuals observe ( p 0, q), while they form expectations on p 1. The hypothesis of perfect foresight (or rational expectations) requires that individual expectations coincide with future equilibrium prices. Yet, if today observed (equilibrium) prices are consistent with several of tomorrow (equilibrium) prices, it is not clear how agents can coordinate their expectations by selecting one particular future price. We consider the issue of how perfect forecasts can be formulated by taking into account the sequential nature of trades. When trade takes place on a sequence of markets, ``today'' prices play a double role: They affect the allocation of consumption and portfolios and they may also reveal information on future equilibrium prices. Since financial economies are subject to several degrees of nominal indeterminacy, several future equilibrium prices p 1 will, in general, correspond to an observed equilibrium price ( p 0, q). However, individuals do not need to take into account pure nominal redundancies which do not affect either consumption or portfolio optimal choices. Therefore, in this introduction, whenever we talk about future prices p 1, we implicitly assume that pure nominal redundancies have been eliminated. A necessary and sufficient condition for the observable relative prices ( p^ 0, q^ ) to ``reveal'' future prices p^ 1 (hence making perfect forecasts plausible) is that ( p^ 0, p^ 1, q^ ) is the only intertemporal equilibrium of the economy with ( p 0, q)=( p^ 0, q^ ). Hereafter, equilibria with this property will be called forecastable rational expectations equilibria (for short, forecastable equilibria). To the contrary, if financial equilibria were not forecastable, the rational expectations hypothesis would appear questionable, because of the absence of any coordination mechanism allowing the agents to focus on the same expected future (equilibrium) prices. The paper analyzes an intertemporal economy, with two periods and uncertainty. There is an asset market in the first period that can be either complete or incomplete. Assets are nominal; i.e., their yields are fixed in terms of abstract units of account. Hence, the economy is potentially subject to indeterminacy of equilibrium allocations (see [3] and [5]). In the first part of the paper, we find sufficient conditions which guarantee that, typically (i.e., for a dense, residual set of economies, if the

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economy displays real indeterminacy, for an open and dense set otherwise), all financial equilibria are forecastable. These conditions establish a lower bound, related to the number of degrees of real indeterminacy, on the number of (types of) individuals (H) and of period-zero commodities (L). The number of assets (I ) does not play any role in the argument. In the second part of the paper, we establish a lower bound for the number of observable first-period prices, i.e., both commodity and asset prices. If the number of observable first-period prices is above this lower bound, typically, all equilibria are forecastable. For economies with an incomplete asset market, this lower bound is less severe that the one found using only commodity prices. However, if the asset market is complete, any noarbitrage asset price vector can support a given equilibrium allocation. Hence, in this case, the observation of asset prices does not convey any information on the future second-period equilibrium prices. All the results of the paper extend immediately to economies with real numeraire assets. The changes in the arguments are trivial and therefore not reported. We indicate in the text how the lower bounds on the number of observable prices have to be adjusted to guarantee that the theorems apply also to economies with real assets. It is evident that there are economies without forecastable equilibria. Consider an economy where preferences are time-separable and, in the first period, identical across consumers and homothetic. Suppose this economy has an incomplete asset market and nominal assets. Typically, for any given asset price vector, this economy will display several dimensions of equilibrium allocations. However, all the equilibria will be supported by the same first-period equilibrium prices. This example makes clear that the elimination of these economies requires preference perturbations. Therefore, the space of economies is defined in terms of utility functions, endowed with the C 2 compact-open topology. We also give an example of an open set of economies for which the dimensionality conditions are violated and with equilibria which are not forecastable. Since time-invariant, Von NeumannMorgenstern utility functions are widely used in applications, in the last section, we extend the analysis to economies with this type of preferences. For this class of economies, there are severe restrictions on the set of acceptable utility perturbations. Therefore, we now augment the parameterization of the economy by introducing initial endowments. We establish that, if the asset market is complete, typically (for an open and dense set of preferences and endowments) all equilibria are forecastable. It is an open issue if we can generalize our results to this class of economies when the asset market is incomplete. We do not consider the issue of how the particular, realized financial equilibrium of an economy is selected out of the set of equilibria. Though,

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any theory of price formation in intertemporal economies must deal with individual expectations and their coordination. Therefore, we believe that our analysis is useful in that direction as well. The issue we consider is somewhat similar to the problem of the role of prices in revealing information in general equilibrium economies with asymmetric information (see, [11] and [1]). The results presented here, however, are not implied and do not imply these well-known results. In particular, to construct equilibria with fully revealing prices, it suffices to perturb the parameters of the economy affected by the information realization. On the contrary, in our model, there is no random variable whose realizations, prior to the opening of the markets, affect the fundamentals of the economy.

2. THE MODEL Consider a pure exchange economy which lasts two periods. The superscript 0 denotes ``today,'' while (1, s), s=1, ..., S, denotes one of the S different states of nature which might realize ``tomorrow.'' There are H households, indexed by a subscript h, h=1, ..., H. At each period and state of the world there is a complete set of spot markets for L commodities, indexed by the subscript c, with c=1, ..., L. Let l#(S+1) L. A commodity bundle is an l-dimensional column vector x# (x 0, ..., x 1, s, . . .), where the L-dimensional column vector x 0 =(. . ., x 0c , . . .) is a bundle of first-period commodities, while the L-dimensional column vecs tor x 1, s =(. . ., x 1, c , . . .), s=1, ..., S, is a bundle of commodities in state s in the second period. The SL-dimensional column vector x 1 =(. . ., x 1, s, . . .) is a bundle of commodities in the second period. Commodity prices are p#( p 0, p 1 ), an l-dimensional row vector. p 0 denotes first-period prices and p 1 =(. . ., p 1, s, . . .) denotes second-period prices for all possible realizations of the state of the world. At spot s=0, there are markets for I assets, indexed by a superscript i, i=1, ..., I. Assets are nominal; i.e., asset yields are fixed in the same abstract units of account as commodity prices. y i #(. .., y i, s, . . .) denotes the vector of payoffs of asset i, a column vector of dimension S, and Y denotes the (S_I )-dimensional matrix of asset payoffs. Asset prices are q#(. . ., q i, . . .), a row vector of dimension I, and b#(. . ., b i, . . .) is a portfolio, a column vector of dimension I. Also, with span[Y] we denote the linear subspace generated by the columns of Y. To simplify the proofs of the various theorems, we will maintain the following assumption:

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Assumption 1. The asset yield matrix Y is in general position and span[Y] & R s++ {<. General position means that each selection of I rows of the matrix Y has maximal rank I. This is a standard assumption which is typically satisfied in the space of (S_I )-dimensional matrices. It could be dispensed with at the cost of a substantial technical and notational burden. Moreover, span[Y] & R s++ {< implies the existence of a portfolio with strictly positive returns in each state ``tomorrow.'' For each household h, the consumption space is R l++ and the endowment vector is e h # R l++ . Preferences are described by a utility function u h # C , strictly monotone, differentiably strictly quasi-concave, and satisfying the boundary condition: the set [x # R l++ | u h(x h )u h(x*h )] is closed in R l++ , for each x* h r0. Let U* be the space of utility functions satisfying the stated assumptions. Endow U* with the C 2 compact-open (weak) topology. The space of economies, endowed with the product topology, is U#(U*) H. An economy is a collection [u h ] H h=1 #u # U. Endowments are fixed. 2.1. The Definition of Forecastable Equilibria The set of normalized commodity prices is P#[ p # R l++ | p 01 #1 and p #1, for s=1, ..., S]. We parameterize equilibria in terms of future commodity price levels, g # R S++ . Let D(g) be a diagonal matrix of dimension S_S with diagonal elements d s( g)#1g s, s=1, ..., S. At prices ( p, q), p # P, and parameters g, the individual maximization problem is 1, s 1

max u h(z h +e h )

subject to

p 0z 0h +qb h =0,

p 1 g z 1h =D( g) Yb h ,

(1)

where z h #(x h &e h ) and p 1 g z 1h #(. . ., p 1, sz 1,h s , . . .). Let f h( p, q, g, u h ) be consumer h's demand function. Definition 1. A financial equilibrium of an economy u # U is a price system ( p*, q*, g*) and an associated consumption and portfolio allocation (x*, b*)#(. .., (x* h , b* h ), . . .) such that: (i) (x*&e h , b *) solves the individual maximization problem (1), at h ( p*, q*, g*, u h ); (ii)

 h (x h*&e h )=0 and  h b h*=0.

Let (x u( p, q, g), b u( p, q, g))#(. . ., (x hu( p, q, g), b hu( p, q, g)), . . .) be the equilibrium allocation and portfolio associated with the equilibrium price system ( p, q, g) of an economy u # U.

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An equilibrium ( p*, q*, g*) of an economy u # U may fail to be forecastable for two different reasons: (i) There might be a different equilibrium ( p, q, g) with x u( p, q, g){ x u( p*, q*, g*) and ( p 0*, q*)=( p 0, q). (ii) There might be second-period price levels g, g{g*, such that x u( p*, q*, g*)=x u( p*, q*, g), while b u( p*, q*, g*){b u( p*, q*, g). The first problem is related to the real indeterminacy (or, at least, multiplicity) of equilibrium allocations. The second problem is generated by the presence of purely nominal indeterminacy. In both cases, the observation of the first-period equilibrium prices (and the knowledge of the set of equilibrium prices) does not suffice to coordinate individuals on choices compatible with a general equilibrium of the economy. Definition 2. A financial equilibrium ( p*, q*, g*) of an economy u # U is forecastable if there is no other financial equilibrium ( p, q, g) with (x u( p, q, g), b u( p, q, g)){(x u( p*, q*, g*), b u( p*, q*, g*)) such that ( p 0*, q*) =( p 0, q). Economies with nominal (as well as real) assets always display (S+1) degrees of indeterminacy, parameterized by the first-period price levels and by the vector g. For our analysis, it is convenient to separate purely nominal indeterminacy from the indeterminacy that might affect equilibrium allocations. Independently of the number (and nature) of assets, two degrees of indeterminacy are always purely nominal. More precisely, if ( p*, q*, g*) is an equilibrium price vector supporting the allocation x u( p*, q*, g*), u # U, then, for all positive constants c 1 , the price vector (c 1 p 0*, p 1*, c 1 q*, g*) supports the same allocation. Evidently, this degree of indeterminacy does not affect individual optimal choices of commodity bundles and portfolios. Therefore, we did not loose any generality by imposing the normalization p 01 =1. Furthermore, since the matrix Y is in general position, span[D(g) Y]= span[D( g$) Y] if and only if g and g$ are collinear. Hence, for all positive constants c 2 , the price vectors ( p*, q*c 2 , c 2 g*) and ( p*, q*, g*) support the same allocation. When 0
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of real indeterminacy that an economy can display. Given Assumption 1, K(I ) is equal to (S&1) if the asset market is incomplete. It is equal to zero otherwise. To avoid confusing nominal and real indeterminacy, we denote by + the typical element of M. Let D#(S&I). Define the map Z h( p, q, g, u h ), Z h : P_R I_R S++_ U*  R l&D&1 given by agent h's excess demand for all commodities, but commodity 1 in the first period and in the second period in states s= 1, ..., D. Let Z: P_R I_R S++ _U  R l&D&1 be the aggregate excess demand function for the same commodities. Given that each agent faces (S+1) budget constraints, by the modified form of Walras' Law, ( p, q, g) is an equilibrium of the economy u # U if and only if Z( p, q, g, u)=0. Let Q be the set of noarbitrage asset prices, an open subset of R I. The set of ``normalized'' prices is P#[( p, q, +) # P_R I_M: q # Q], an open subset of P_R I_M. Hence, P is a smooth manifold without boundary, diffeomorphic to an open subset of R l&D&1+K(I ). To simplify notation we will denote by ? the elements of the set P. The definition of the set P eliminates only nominal redundancies. Hence, all the equilibrium allocations of any economy u # U can be supported by some ? # P. Furthermore, for all non-empty, compact subsets M*/M and I for each u # U, Z &1 u (0) & ((P_R _M*) & P){< and, by the boundary &1 I conditions, Z u (0) & ((P_R _M*) & P) is compact. 2.2. The Parameterization of the Space of the Economies Given that most of our results hold for some subset of the space of economies, it is necessary to be more specific about the relevant properties of the space U and to introduce an explicit parameterization of this space. First, bear in mind that the topological space U is metrizable, separable, and complete. Therefore, the intersection of any countable family of open, dense setsi.e., each residual setis dense (see, for instance, [9, Proposition 2.4.5]). Since economies are identified with utility functions, the notion of Lebesgue measure (as well as any natural concept of measure) cannot be applied. As usual in these circumstances, we use topological notions to define negligible sets. Sets are negligible if their complements are either open and dense or residual. Recall that, since U, endowed with the C 2 compact-open (weak) topology, is a complete metric space, the complement of a residual set, i.e., the countable union of closed and nowhere dense subsets of U, is nowhere dense in U (see [9], for a discussion of these concepts). We use (locally) finite dimensional, linear perturbations of the utility functions and therefore, when we need to perturb (or differentiate) utility functions, we take U to be a finite dimensional manifold.

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For each h, let V 1h and V 2h be open (possibly empty) sets such that cl V hj /R l++ , for j=1, 2. Also, assume that, given V hj , for j=1, 2, there is an open set V =jh which satisfies cl V hj /V =jh /cl V =jh /R l++ and =2 cl V =1 h & cl V h =<. Given u # U, sets V hj and V =jh , for j=1, 2, and vectors $ 1h # R l and $ 2h # R l, define u $ the economy where, for each h, u $h is obtained by replacing u h with the function 1 2 =2 2 u $h(x h )#u h(x h )+% 1(x h , V =1 h )($ h x h )+% (x h , V h )($ h x h ),

where, for each j, % j (x h , V =jh ) is a smooth ``bump'' function, % j: R l++  [0, 1], which takes the value 1 if x h # V hj and the value 0 if x h  cl V =jh (see, [8, p. 41]). Evidently, if u h # U*, u $h # U* for $ 1h and $ 2h small enough. The parameterization adopted allows us to use independent perturbations of the utility function on the disjoint sets V hj , for j=1, 2.

3. FORECASTABLE EQUILIBRIA AND COMMODITY PRICES In this section, we find sufficient conditions for the existence of a residual set of economies (which is open and sense if I=S) such that, for each economy in this set, all equilibria are forecastable. To simplify the exposition, in this section we prove the result just specifying a lower bound, L*, on the number of first period commodities. If L>L*, for a residual set of economies, all equilibria are forecastable. First we state and prove the result for economies with an incomplete asset market (Theorem 1) and then, at the end of the section, we study economies with I=S (Corollary 1). In the next section, we prove that, if 0(D+S) and L>(2S&1). Under the maintained assumptions, there is a residual set of economies, U /U, such that, for each pair of equilibrium prices ( p*, q*, g*) and ( p, q, g), ( p*, q*) and ( p, q) # P_Q, of the economy u # U , (i)

p 0*{p 0, if x u( p, q, g){x u( p*, q*, g*),

(ii) b u( p, q, g) = b u( p*, q*, g*), if x u( p, q, g) = x u( p*, q*, g*) and ( p 0*, q*)=( p 0, q). Hence, for each economy u # U, all financial equilibria are forecastable.

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The complement of U contains the set of economies with some equilibria which fail to be forecastable. Moreover, it contains economies whose equilibria are all forecastable when one takes into consideration asset prices, but with some equilibria whose period-zero commodity prices are identical. We deal with the role of asset prices in the next section. As already mentioned, the complement of U is nowhere dense in U. Therefore, it is topologically negligible. Theorem 1 establishes that different equilibrium allocations are associated with different first-period commodity prices as well as that, given first-period prices, the same equilibrium allocation cannot be supported by two distinct portfolios. Let ?=( p, q, +) # Z &1 u (0) & P be an equilibrium of the economy u # U and x u(?) be the corresponding equilibrium allocation. Define G(x u(?))#[(q$, g, g$): g{g$, x u(?)=x u( p, q$, g)=x u( p, q$, g$), and

b u( p, q$, g){b u( p, q$, g$)].

If G(x u(?)){<, there exist at least two distinct price vectors, ( p, q$, g) and ( p, q$, g$), supporting the same equilibrium allocation x u(?). Moreover, ( p, q$, g) and ( p, q$, g$) have identical first-period prices, but they require different supporting portfolios. Hence, if G(x u(?)){<, the economy u has equilibria which are not forecastable. To prove Theorem 1, it suffices to show three generic properties of the set of equilibria: (a)

Different vectors + generate distinct equilibrium allocations.

(b) The set G(x u(?)) is empty. (c) Different equilibrium allocations are associated with different first-period equilibrium commodity prices. The argument evolves in two steps. After a preparatory result on demand functions (Lemma 1), we study a and b when + is restricted to lying in a compact subset of M. Then, adopting a technique of proof borrowed from [10], we extend these results to M and we prove c. The crucial step in the proof of a and b is to show that, generically, at each equilibrium ?, the associated (H_(S+1))-dimensional matrix of Lagrange multipliers, 4 u(?), has (maximal) rank(D+1). Once this is established, a straightforward spanning argument suffices to prove the two results. The proof of the genericity of this rank condition is based on the transversality theorem, applied to a suitable function. The domain of this function involves the set M. Unfortunately, M is not compact (and, since +r0, there is no way to represent it as a compact set) and, therefore, the set Z &1 u (0) & P is not compact. Moreover, on the boundary of M, when

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some entries of the vector + are equal to zero, the matrix D(+) Y might loose rank, inducing discontinuities of the excess demand function. However, for each nonempty and compact subset of M, say M*, the set I Z &1 u (0) & (P_R _M*) is a compact subset of P (hence, the excess demand function is continuous also on the boundary of this set). In Lemma 2, we prove that, for each arbitrary M*, there exists an open and dense set of economies for which a and b hold for + # M*. Hence, we prove that the equilibrium allocation map is typically globally injective on M*. The proofs of real indeterminacy of the set of equilibria in economies with nominal assets are based on the local version of a (see, [3] or [5]). As a preliminary step, we need to characterize the properties of the excess demand function Z. 1 1, S Let * h(?, u h )#(* 0h , * 1, h , ..., * h )(?, u h ) be the (S+1)-dimensional row vector of Lagrange multipliers associated with the constraints of the individual maximization problem (1). Lemma 1. (i)

Let u # U and ? # P. Then,

rank(D $h Z h(?, u h ))=(l&D&1), for each h,

1 1, D (ii) rank(D $h * h(?, u h ))#rank(D $h(* 0h , * 1, )(?, u h ))=(D+1), h , ..., * h for each h.

Proof of Lemma 1. The argument is a straightforward modification of the proofs in [4] and [6] and is, therefore, omitted. K Lemma 1 and the parameterization of the utility functions proposed above have a simple and useful implication, used in the proofs of Claim 1 and Theorem 1: Let P$ be a nonempty subset of P and suppose that there exists an open set V 1h such that f h(P$, u h )/V 1h and cl V 1h /R l++ . Then, for all ? # P$, rank(D $h Z h(?, u h ))=(l&D&1). Let M*/M be any compact manifold. Define P(M*)#[? # P : + # M*], a smooth manifold, possibly, with boundary. Lemma 2. Let 0(D+S). For each compact manifold M*/M, there is an open, dense set of economies, U M* /U, such that, for each u # U M*, (i)

if ?, ?$ # P(M*) & Z &1 u (0) and ?{?$, then x u(?){x u(?$),

(ii)

G(x u(?))=<, for ? # P(M*) & Z &1 u (0).

The assumption on the number of individuals, H>(D+S), needs some elaboration. Balasko and Cass [3] parameterize economies in endowment space, 0, and, assuming H>(D+1), they show that, for each endowment in an open and dense subset of 0, there exists an open subset of M such

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that i in Lemma 2 holds. However, the definition of forecastable equilibria requires i to be true for all + # M* (and for each M*). It is trivial to check, from our argument, that H>(D+1) suffices to prove the result for an open and dense subset of U_M*. The proof of Lemma 2 is based on the following observation. For u # U and ? # P & Z &1 u (0), let 4 u(?) be the matrix of dimension H_(S+1) with rows equal to * h(?, u h ), h=1, ..., H. At each ? # P(M*) & Z &1 u (0), the firstorder conditions of the individual maximization problem imply that &q 4 u(?)( D(+) Y )=0. If rank 4 u(?)=(D+1), the rows of the matrix 4 u(?) &q span the orthogonal complement to the linear subspace span [( D(+) Y )]. Hence, if 4 u(?) has maximal rank, changes in + affecting span[D(+)Y] must affect equilibrium allocations. Therefore, the crucial step of the proof of Lemma 2 is the following Claim, whose proof is deferred to Appendix A. Claim 1. Let 0(D+S). For each compact manifold M*/M, there is an open, dense set of economies, U M* /U, such that, for each u # U M*, if ? # P(M*) & Z &1 u (0), then, rank 4 u(?)=(D+1). Proof of Lemma 2. Pick an economy u in the open and dense set U M*, for which Claim 1 holds. Let ? # P(M*) & Z &1 u (0), by the first-order conditions of the individual maximization problem 4 u(?)

&q

\D(+) Y+ =0.

(2)

Since rank 4 u(?)=(D+1), (2) implies that the rows of the matrix 4 u(?) &q span the orthogonal complement to the linear subspace span [( D(+) Y )]. &1 M* Consider two distinct equilibria, ?, ?$ # P(M*) & Z u (0), u # U . If, by contradiction, x u(?)=x u(?$), since commodity prices are normalized in P, the first-order conditions of optimization problem (1) imply that p=p$ and * h(?, u h )=* h(?$, u h ), for all h. But then ?{?$ and p=p$ imply that either &q &q$ +{+$ or q{q$ or both. In each case, span[( D(+) Y )]{span[( D(+$) Y )], which, together with rank 4 u(?)=(D+1) and * h(?, u h )=* h(?$, u h ), for all h, contradicts (2). Hence, ?, ?$ # P(M*) & Z &1 u (0) and ?{?$ imply x u (?) {x u(?$). To show ii, suppose, by contradiction, that G(x u(?)){<. Then, there exist g, g$, g{g$, and q$ such that x u(?)=x u( p, q$, g)=x u( p, q$, g$) and b u( p, q$, g){b u( p, q$, g$). Since u # U M*, Claim 1, Eq. (2), and x u(?)= x u( p, q$, g)=x u( p, q$, g$) (hence, 4 u(?)=4 u( p, q$, g)=4 u( p, q$, g$)) imply that span

&q

&q$

&q$

_\D(+) Y+& =span _\D(g) Y+& =span _\D( g$) Y+& .

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(3)

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Since Y is in general position, (3) implies that (q, +)=c(q$, g)=c$(q$, g$) for some strictly positive constants c and c$. But then, since (by Assumption 1) 0  Q, b u( p, q$, g)=b u( p, q$, g$). A contradiction. K To prove Theorem 1, we need to extend Lemma 2 to M and then to show that, typically, if ? and ?$ are distinct equilibrium prices of an economy u # U, p 0 {p 0$ (as well as G(x u(?))=<). However, the set M is not compact. Moreover, distinct vectors + and +$, and, therefore, distinct equilibrium prices ? and ?$, can be arbitrarily close, creating problems for the perturbations of the utility functions. We overcome these difficulties by adopting a strategy of proof which is borrowed from [10]. This consists in picking a countable cover of compact subsets of J#[(?, ?$) # P_P : ?{?$], i.e., of the set of distinct price vectors. Restrict attention to prices in an element of the cover. Let [#, #$] be any pair of two disjoint, closed (relative to P) balls contained in P. Evidently, #_#$ is contained in J. Consider equilibrium price vectors (?, ?$) # #_#$. Then, either ( p, q){ ( p$, q$) or +{+$ or both. In both cases, by Lemma 2 (and the compactness of #_#$), x u(?){x u(?$), for u # U 1, an open and dense set of economies. Moreover, for u # U 1, both G(x u(?))=< and G(x u(?$))=<. Then, applying the transversality theorem to a suitable function, we show that, for an open and dense subset U 1, say U$ 1, first-period commodity prices are different, when equilibrium prices are restricted to 1. Since the countable collection of sets 1 covers J, the intersection of the countable collection of open and dense sets of economies U$ 1 generates a residual set of economies whose equilibria are all forecastable. However, there is an additional problem that we have to deal with. When 0
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Let U 1 be the open and dense subset of U which satisfies Lemma 2 for M*=# M _ #$M . Bear in mind that G(x u(?))=<, for all ? # Z &1 u (0) & (# _ #$) and u # U 1. Define the map 9: #_#$_U 1  R 2(l&D&1)+(L&1), given by 9(?, ?$, u)#(Z(?, u), Z(?$, u), p 01 &p 0$ 1 ), where p 01 #( p 02 , ..., p 0L ). Let 9 u be the map 9 for given u. Suppose that, for an open and dense subset of U 1, U$ 1, 9 u & 0. Then, since 9 u maps a smooth manifold of dimension 2(l&D+S&2) into R 2(l&D&1)+(L&1), L>(2S&1) and 9 u & 0 1 imply that 9 &1 u (0) & (#_#$)=<, for u # U$ . The map 9 is smooth and the set #_#$ is compact. Hence, the set U$ 1 is open. We now need to show that U$ 1 is dense. If, for all open sets B/U 1,  0 and, hence, there exists u* # B such that 9 &1 u* (0) & (#_#$)=<, 9 u* & U$ 1 is dense. Therefore, suppose, by contradiction, that there exists an open subset B$ such that 9 &1 u (0) & (#_#$){<, for all u # B$. We want to show that this implies that 9 u & 0, for u # B*, a nonempty subset of B$. As mentioned above, this would contradict 9 &1 u (0) & (#_#$){< and, therefore, it would establish the density of U$ 1. The proof of the existence of the set B* is based on the transversality theorem and, therefore, requires the perturbation of the utility functions of some individuals. As already explained, we have to make sure that different perturbations do not have intersecting domains of definition. This is guaranteed by the following claim, whose proof is deferred to Appendix A. Claim 2. There exists a finite open covering of #_#$, [N(t), t=1, ..., N], an open subset of B$, B, and a one-to-one map h: [1, ..., N]  [1, ..., H] such that &f h(t)(?, u h(t) )&f h(t)(?$, u h(t) )&>v*>0, for all t and (?, ?$, u) # 9 &1(0) & (N(t) & (#_#$))_B. j and Claim 2 implies that, for all t=1, ..., N, there are open sets V h(t) V , j=1, 2, with the properties defined in Section 2.1 and satisfying fh(t)(?, u h(t) ) # V 1h(t) and f h(t)(?$, u h(t) ) # V 2h(t) , for all (?, ?$, u) # 9 &1(0) & (N(t)&(#_#$))_B. Hence, at each (?, ?$, u) # 9 &1(0)&(N(t)&(#_#$))_B, =2 we can perturb u h(t) independently on the two neighborhoods V =1 h(t) and V h(t) . Consider the restriction of the map 9 on (N(t) & (#_#$))_B, 9 N(t) . Let D L&1(1) denote the identity matrix of dimension (L&1). Since (l&D+S&2)>L, even if (?, ?$) are on the boundary of N(t), there exist directions (?)  and (?$)  such that D (?)  ( p 0 &p 0$)=D L&1(1) and D (?$)  ( p 0 &p 0$)= &D L&1(1). Hence, for all (?, ?$, u) # 9 &1(0) & (N(t) & (#_#$))_B, the matrix D ((?)  , (?$)  , $) 9(?, ?$, u), t=1, ..., N, contains the sub-matrix =j h(t)

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\

D (?) Z(?, u)

0

0 D (?$) Z(?$, u) (D L&1(1), 0) &(D L&1(1), 0)

D $1h(t) Z(?, u) 0 0

0 D

2 $h(t)

+

Z(?$, u) . 0

j Z(?, u) is always invertible, j=1, 2. Hence, 9 By Lemma 1, D $h(t)  0. N(t) & By the transversality theorem, 9 N(t), u & 0, for u # B*(t), a dense subset of B. Moreover, since (N(t) & (#_#$)) are bounded subsets of J, and 9 N(t) is continuous also on the boundary of (N(t) & (#_#$)), the set B*(t) is open. Let B*= t B*(t). Since [N(t), t=1, ..., N] is a finite open cover of (#_#$), B* is an open and dense subset of B. Hence, for all u # B*, 9 u & 0. As already explained, this means that U$ 1 is dense set. Let U # 1 # 1 * U$ 1. By construction, U is a residual subset of U. By definition of the map 9, ? and ?$ # Z &1 u (0) & P and x u(?){x u(?$) imply that p 0 {p 0$, u # U . Moreover, by construction of the set U, G(x u(?))=<  . Therefore, all equilibria of the economies for all ? # Z &1 u (0) & P, u # U u # U are forecastable. K

Now, as a corollary, we prove that, when the asset market is complete, two first-period commodities suffice to guarantee the existence of an open and dense set of economies whose equilibria are all forecastable. We are able to get an open and dense set, as contrasted with a residual one, because, when markets are complete, equilibria are typically locally isolated. Hence, loosely speaking, if an economy has both forecastable and isolated equilibria this property survives to perturbations, i.e., this property is open. Corollary 1. Let L>1 and S=I. Under the maintained assumptions, the set U , defined in Theorem 1, is an open and dense subset of U. Proof of Corollary 1. Let U R be the set of regular economies, an open and dense subset of U. For each u # U R, the set of equilibrium consumption allocations has finite cardinality. It is then straightforward to show that the results of Lemma 2 hold independently of the number of agents, for all u # U R. Since I=S, P#[( p, q): q # Q] is an open subset of P_R I. Define #, 1#[#, #$] and 1* as in the proof of Theorem 1. Observe that we do not need Claim 2 to establish that 9 & 0, because, for each u # U R, equilibria are locally isolated. Denote by U R the residual subset of U R whose existence and properties follow from the transversality argument of Theorem 1. To prove that U R is open, pick u # U R. Since u is a regular economy, the equilibria, (? u ) t, t=1, ..., T, are finite, and since u # U R, p t, 0 {p t$, 0, for t{t$. Moreover, there exists a neighborhood of u, B(u)/U R, such that the equilibria are a continuous function of u$ # B(u)

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and their number is locally constant. Hence, for some open set 0  is open. K B$(u)/B(u), p t,u$ 0 {p t$, u$ , for each u # B$(u). Hence, U From the proof of the corollary, it should be clear that the same result holds true for economies with an incomplete asset market and real numeraire assets. For these economies, in fact, equilibria are typically isolated and economies are typically regular. These are the only properties needed to establish the openness of U .

4. FORECASTABLE EQUILIBRIA AND ASSET PRICES Theorem 1 only exploits first-period commodity prices to find sufficient conditions for the existence of economies with only forecastable equilibria. The purpose of this section is to study how the observation of asset prices weakens the conditions of Theorem 1. This section will deal only with economies with an incomplete asset market, because, when the asset market is complete, the observation of asset prices does not convey any information on future equilibrium prices. In fact, given any q # Q, we can choose normalized commodity prices p # P, and second-period price levels, g # R S++ , so that ( p, q, g) is any one of the equilibria of the economy u # U. Therefore, complete asset market economies with forecastable equilibria are characterized by Corollary 1. Let ? # P be an equilibrium and x u(?) be the associated allocation of an economy u # U. Definition 3. Two equilibria ?, ?* # P of an economy u # U are confounding if x u(?){x u(?*), p 0 =p 0* and if there exists (q^, g, g*) such that x u( p, q^, g)=x u(?), x u( p*, q^, g*)=x u(?*). Obviously, if two equilibria are confounding, they are also nonforecastable. However, while non-forecastability is defined for a particular equilibrium (and implies the existence of some other equilibrium with the same observable prices), the notion of confounding equilibrium refers to a pair of equilibria. In order to rule out the existence of confounding equilibria, we need to characterize the supporting price systems ( p, q$, g) to a given equilibrium allocation x u(?). A sufficient condition for ( p$, q$, g), p$ # P, to support x u(?) as an equilibrium allocation is p=p$ and span

&q

&q$

_\D(+) Y( p )+& =span _\D( g) Y( p )+& . 1

1

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(4)

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Hence, what matters for the forecastability of equilibria is the number of asset prices vectors q$ which satisfy (4), for some g # R S++ . Equation (4) becomes a necessary condition if an equilibrium ? of u # U satisfies rank(4 u(?))=(D+1) (see Lemma 2). Lemma 3. Let 00. Similarly, ( p*, q", g*) supports x u(?*) if and only if (q", g*)= '*(q*, +*), for some '*>0. Therefore, ?, ?* are confounding if and only if there exist scalars ' and '* # R ++ such that q$='q='*q*=q". Hence, the scalar '$#''* satisfies q'$=q*. K Theorem 2. Let 0(D+S) and (L+I )>2S. There exists a residual set of economies, U N /U, such that, for each economy u # U N, all financial equilibria are forecastable. Proof of Theorem 2. Define #, #$, 1, U 1, and 1* as in the proof of Theorem 1. Consider the map 9 N: #_#$_U 1_R ++  R 2(l&D&1)+(I+L&1), defined as 9 N (?, ?$, u, ')#(Z(?, u), Z(?$, u), ( p 01 , 'q)&( p 0$ 1 , q$)).  0, for u # U$ 1, an open and dense subset of We need to show that 9 N u & &1 (0)=<. U . Then, since (L+I )>2S, for all u # U$ 1, (9 N u) 1 &1 &1 Since for all u # U , (Z u (0), Z u (0)) & (#_#$) is compact, the set Q 1u # &1 [(q, q$): ((?, u), (?$, u)) # (Z &1 u (0), Z u (0)) & (#_#$), for some ( p, +, p$, +$)] is compact. Consider the map v(q, q$, '): Q 1u _R ++  R I, defined by v(q, q$, ')#q'&q$. Obviously, v &1(0) & Q 1u _R ++ is a compact set. (0) & (#_#$_R ++ ) is compact and, therefore, Hence, for u # U 1, 9 N&1 u since 9 N is smooth, U$ 1 is open. To show density, we apply the same argument used in the proof of Theorem 1. Hence, we skip details and we go directly to the computation of the matrix D ((?)  , (?$)  , $) 9 N (?, ?$, u, '), at a point (?, ?$, ') # (0) & (#_#$_R ++ ). 9 N&1 u Let D I (') denote the (I_I)-dimensional diagonal matrix with elements d i (')=', for all i. Since (l&D+(S&1))>(L+I ), even if ? or ?$ are on the boundary of # or #$, there exist directions (?)  and (?$)  such that 1

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D (?)  ( p 0 &p 0$)=D L&1(1), D (?$)  ( p 0 &p 0$)= &D L&1(1), D (?)  ('q&q$)= (0) & D I ('), and D (?$)  ('q&q$)= &D I (1). Hence, at (?, ?$, ') # 9 N&1 u (#_#$_R ++ ), the matrix D ((?)  , (?$)  , $) 9 N (?, ?$, u, ') contains the submatrix

\

D (?) Z(?, u) 0 (D L&1(1), 0, 0) (0, 0, D I ('))

0 D (?$) Z(?$, u) &(D L&1(1), 0, 0) (0, 0, &D I (1))

D $1h Z(?, u) 0 0 D $2h Z(?$, u) . 0 0 0 0

+

Hence, D ((?)  , (?$)  , $) 9 N (?, ?$, u, ') is a matrix of full column rank. Therefore, the transversality theorem implies the density of U$ 1. The remaining part of the argument is identical to the one used in the proof of Theorem 1 and is therefore omitted. K A minor modification of the proof of Theorem 2 implies that, with real numeraire assets, the condition (L+I )>2 suffices to guarantee the existence of an open and dense subset of economies whose equilibria are all forecastable.

5. A ROBUST EXAMPLE OF ECONOMIES WITH EQUILIBRIA WHICH ARE NOT FORECASTABLE In this section we construct a robust example of an economy with a one-dimensional manifold of different equilibrium prices (and allocations) for which period-zero prices are invariant, i.e., with a one-dimensional manifold of equilibria which are not forecastable. We consider an economy with three consumers and three states of nature in the second period. There is one financial asset, inside money, with payoffs (1, 1, 1). In period zero there are two commodities. Hence, L=2 and K=2. To simplify computations, there is a unique commodity at each spot in period one. For each consumer, preferences are described by a Von Neumann Morgenstern utility function u h(x h ) # ; 0h1 ln x 0h1 + (1 & ; 0h1 ) ln x 0h2 + s s ; 1h  s>0 : s ln x 1, h , where : #13 is the (objective) probability of each state of nature in period 1. Without loss of generality, we normalize p 01 #q#1. Exploiting the sequence of budget constraints, the indirect utility function of the portfolio is given by V h(b h , p)#k h +ln (e 0h1 +p 02e 0h2 &b h )+ s 1, s )), where k h is a function which does not (13) ; 1h  s>0 ln (e 1, h +(b h p depend upon b h .

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Using the function V h(b h , p), the (necessary and sufficient) first-order conditions for an optimal choice of b n are given by D b V h(b h , p)# s 1, s ))=0. &1(e 0h1 +p 02e 0h2 &b h )+(13) ; 1h  s>0 (1p 1, s )(e 1, h +(b h p Let b h( p) be the solution to the first-order conditions. Assume that e 1 #(1, 1, 2, 1, 1), ; 011 #13, and ; 11 #16, e 2 #(2, 1, 1, 1, 1), ; 021 #23, and ; 12 #12, e 3 #(1, 2, 1, 1, 1), ; 031 #12, and ; 13 #12. It is easy to check that, given these values of the parameters, there is a financial equilibrium with prices ( p^, q^ )=(1, 1, 1, 1, 1, 1) and portfolio allocation b 1 = &23, b 2 = b 3 =13. It is also easy to check that financial equilibria are the zeros of the map E( p)#(7 h b h( p), p 02 &1112&(2b 2( p)+b 3( p))12). Given that ( p^, q^ )=(1, 1, 1, 1, 1, 1) is a financial equilibrium, by the implicit function theorem, if det (D ( p1, 1, p1, 2 ) E( p)){0, at ( p^, q^ ), then, for each value of p 1, 3 in some open neighborhood B(1), there is an equilibrium p~ with p~ 02 =1. Hence, the set of equilibria (which exhibits two degrees of real indeterminacy) contains a smooth one-dimensional manifold of distinct financial equilibria with the same period-zero prices, which are, therefore, not forecastable. By direct computation, at ( p^, q^ ), det(D ( p1, 2, p1, 2 ) E( p))= &844050< &0.2. This establishes our claim. The inequality above is preserved by small perturbations of the ``fundamentals.'' Hence, there is an open neighborhood of economies for which a similar result holds.

6. ECONOMIES WITH A COMPLETE ASSET MARKET AND TIME-INVARIANT, VON NEUMANNMORGENSTERN PREFERENCES In this section we specialize our analysis to the class of complete market economies where each agent has time-invariant, Von Neumann Morgenstern preferences. Since this is the most restrictive setting for preference perturbations, the analysis of this section generalizes immediately to general Von NeumannMorgenstern utility functions. We show that there exists an open and dense set of economies whose equilibria are all forecastable. Since this class of economies is characterized by severe restrictions on the set of possible perturbations of the utility functions, we augment the parameterization of the economy, introducing initial endowments as an additional parameter. Since I=S, the economy with an asset market is isomorphic to the corresponding ArrowDebreu economy. Hence, to simplify notation and without any loss of generality, we drop the asset market and we consider the standard maximization problem with a unique budget constraint, normalizing commodity prices in P$# [ p # R l++ : p 01 #1]. The conditions for forecastability are unaffected by our dropping the asset markets, because (as pointed out above), with complete markets, asset prices do not play any essential role in our analysis.

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Let e^ #(.. ., e h , . . .) # R lH ++ denote the initial distribution of endowments. Also, let : # R S++ be the vector of (objective) probabilities of the various states of nature in period 1. Utility functions u h are Von NeumannMorgenstern and invariant over time. Therefore, the (cardinal) index v h : R L++  R is both time and state invariant. v h is differentiably increasing, differentiably strictly concave and v h # C . In Addition, for each h, the utility function u h(x h )#v h(x 0h )+ s  s : sv h(x 1, h ) satisfies the assumptions of Section 2. Denote by V* the space of indexes v h satisfying the stated assumptions. Endow V* with the C 2 compact-open (weak) topology. A utility profile is H [v h ] H h=1 #v # V, where V#(V*) . An economy is (e^, v) # E. Perturbations of v h are restricted to be (locally) linear and finite dimensional. We use the same apparatus introduced in Section 2.2, with two additional, important qualifications: (i) to display full rank perturbations of the utility function of individual h around a given consumption bundle x h , we need that s 1, s 1, s$ , for s{s$. x 0h {x 1, h , for all s, and that x h {x h (ii) to display independent, full rank perturbations of the utility function of individual h around two given and distinct consumption bundles x^ h s$ and x~ h , we need that both x^ h and x~ h satisfy i and that x^ 0h {x~ 0h , x^ 0h {x~ 1, , h 1, s 0 1, s 1, s$ x^ h {x~ h , and x^ h {x~ h , for all s and s$. Corollary 1 follows from the possibility of perturbing independently distinct equilibrium allocations, whenever they have identical first-period prices. This requires independent, full rank perturbations of the utility function of at least one individual around the distinct equilibrium allocations. Hence, any two equilibrium allocations must satisfy requisites i and ii. Lemma 4 shows that requisite i is generically satisfied at an equilibrium allocation. Lemma 5 shows that property ii is typically satisfied at an equilibrium. Its proof is based on the generic property of sequential regularity of ArrowDebreu equilibria, established in [2]. The proofs of the two lemmata are in Appendix B. Lemma 4. Let L>1. Then, there exists an open and dense set of economies, E$, such that, for each (e^, v) # E$, each equilibrium allocation x e^, v( p)#(. .., x h( p, e h , v h ), . . .) satisfies s 1, s 1, s$ (i) x 0h( p, e h , v h ){x 1, h ( p, e h , v h ), x h ( p, e h , v h ){x h ( p, e h , v h ), for all h, s and s$, s{s$.

(ii)

s s 1, s$ s$ and p 1, sp 1, p 1, , for each s and s$, s{s$. p 0 {p 1, sp 1, 1 1 {p 1

Lemma 5. There exists an open and dense set of economies, E", E"/E$, such that, for each (e^, v) # E", for each distinct pair of equilibrium allocations

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x e^, v( p^ ) and x e^, v( p~ ), either p^ 0 {p~ 0 or, for some h, x 0h( p^, e h , v h ){ s 0 1, s x 0h( p~, e h , v h ), x 0h( p^, e h , v h ){x 1, h ( p~, e h , v h ), x h( p~, e h , v h ){x h ( p^, e h , v h ) and 1, s 1, s$ x h ( p^, e h , v h ){x h ( p~, e h , v h ), for all s and s$. For this class of economies, the existence of an open and dense set of economies whose equilibria are forecastable follows immediately from Lemma 5. Restrict attention to economies (e^, v) # E". If two distinct equilibria are not forecastable, the equilibrium allocation of at least one individual is, by Lemma 5, different at each spot. This allows us to exploit (even with time-invariant Von NeumannMorgenstern utility functions) the same full rank utility perturbation adopted in the proof of Corollary 1. Hence, there exists an economy (e^, v) arbitrarily close to (e^$, v$) such that all its equilibria are forecastable. While Lemma 4 is of immediate generalization to economies with I
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Let T u be the map T for a given value of u. Since M* and W are compact sets, the boundary conditions, satisfied by individual utility functions, imply that T u&1(0) & (P(M*)_W) is, for all u # U, a compact set. Then, by the transversality theorem, if T & 0, there is an open and dense subset of economies, U M* /U, such that T u & 0, for u # U M*. Evidently, rank D (?, w)T u(?, w)  (l & D & 1) + (S & 1) + D < (l + S & 1). Hence, if T u & 0, it must be (T u ) &1(0)=<. Then, for each u # U M*, at each equilibrium, the matrix 4 D (?, u) has full row rank (D+1) and, hence, there exist (D+1) individuals with linearly independent Lagrange multipliers * h(?, u), which is the thesis. Let us show that T & 0. Since M*/M is a compact manifold, for all u # U, the set of equilibria Z &1 u (0) & P(M*) is a compact set. Then, for all u # U, there exists an open neighborhood of u, B(u)/U, such that, for all h and u$ # B(u), x hu$(Z &1 u$ (0) & P(M*))/V h , for some open set V h , with cl V h /R l++ . Hence, by choosing appropriately the set V =h /R l++ , we can construct, for each economy u$ # B(u), a well-defined ``bump'' function %(x h , V =h ), which is constant (and equal to 1) in a neighborhood of each equilibrium allocation (associated with ? # Z &1 u$ (0) & P(M*)). By choosing $ h small enough, we then obtain that the ``perturbed'' economy u $ is in the set B(u) and, therefore in U. Define $ # ($ 1 , ..., $ D+2 ). Bear in mind that 4 D (?, u) w # s s (. . .,  D s=0 w * h (?, u h ), . ..). Then, the matrix of derivatives D $ T(?, w, u) is given by

\

D $D+2 Z(?, u D+2 ) D $1 Z(?, u h ) } } } D $h Z(?, u h ) } } } 0 } } } wD $h * h(?, u h ) } } } 0 . 0 }}} 0 } } } wD $D+2 * D+2(?, u D+2 )

+

By Lemma 1, both D $1 Z(?, u 1 ) and D $h * h(?, u h ) are full row rank matrices. Moreover, given that w # W, w{0. Therefore, for each h, wD $h * h(?, u h ){0. Hence, T & 0. K Proof of Claim 2. Pick u # B$ and let C h(?, ?$, u h )#max(& f h(?, u h )& fh(?$, u h )&, &9(?, ?$, u))&). Consider the following programming problem: min (?, ?$) # #_#$

max C h(?, ?$, u h ).

(A1)

h

For each h, the map C h(?, ?$, u h ) is continuous. Therefore, given that the number of agents is finite, the map max h C h(?, ?$, u h ) is certainly lower semi-continuous. Hence, given that #_#$ is compact, there always exists a solution to (A1). Let v be the value of the map C h(?, ?$, u h ) computed at one of the optimal solutions (?*, ?$*, u h* ). Since # & #$=<, by Lemma 2,

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if Z(?*, u)=0 and Z(?$*, u)=0, then, for some agent h*, x h*u(?*){ xh*u(?$*), while if either Z(?*, u){0 or Z(?$*, u){0, &9(?, ?$, u))&{0. Hence, v>0. For h=1, ..., H, consider the sets N$(h)#[(?, ?$) # #_#$: C h(?, ?$, u h )> v2]. By definition of v, [N$(h), h=1, ..., H] is a finite cover of #_#$. Then, by continuity of the individual demand functions, there are open neighborhoods of u h , B(u h ), and open subsets of J, N(h), N$(h)/N(h), such that C h(?, ?$, u$h )>v4, for all (?, ?$, u$h ) # N(h)_B(u h ). [N(h): h=1, ..., H] is a finite open cover of #_#$ and B=6 h B(u h ) & B$ is an open subset of B. Let [h 1 , ..., h N ] be the subset of [1, ..., H] such that N(h) & #_#${< and let h be the map that associates with t, t=1, ..., N, h(t)=h t . Denote by N(t) the set N(h t ). By construction, h: [1, ..., N]  [1, ..., N] is one-to-one, [N(t): t= 1, ..., N] is an open cover of #_#$, and C h(?, ?$, u$h )>v*, v*=v4, for all (?, ?$, u) # N(t)_B(u h ). Hence, if (?, ?$, u$h ) # 9 &1(0) & (N t & (#_#$))_B, then &f h(t)(?, u h(t) )&f h(t)(?$, u h(t) )&>v*, for all t. K

APPENDIX B Throughout this appendix, to simplify notation, we drop the time index and we identify with s=0, period 0, and with s>0, period 1 in state s. Proof of Lemma 4. For each possible choice of s and s$, s{s$, define the map Q s, s$: P$_E  R l&1+L&1, by s$ Q s, s$( p, e^, v)#(Z( p, e^, v), p 1, sp 11, s &p 1, s$p 1, 1 ). s 1, s 1, s Let p 1, 1 #( p 2 , ..., p L ). Evidently,

D (e^, p1,1s ) Q s, s$( p, e^, v)#

\

D e^ Z( p, e^, v) D p1,1s Z( p, e^, v) . s 0 D L&1(1p 1, 1 )

+

Given that D e^ Z( p, e^, v) has full row rank, Q s, s$ & 0. Let Q s,e^, s$v be the map for given (e^, v) # E. Q s,e^, s$v maps a manifold of dimension (l&1) into Q l+L&1 . Therefore, since (Q s,e^, s$v ) &1(0) & P$ is, for (e^, v) # E, a compact set R and L>1, by the transversality theorem, Q s, s$ & 0 implies that there is an open and dense subset of economies, E s, s$ /E, such that (Q s,e^, s$v) &1(0)=<. s$ . In Evidently, this implies that, at each equilibrium, p 1, sp 1,1 s {p 1, s$p 1, 1 turn, by the first-order conditions of optimization problem (1), this implies s 1, s$ that, for each h, the associated equilibrium allocation satisfies x 1, h$ {x h$ . Let S*#[(s, s$): s and s$=0, ..., S and s{s$]. S* is finite and, hence, E# & (s, s$) # S* E s, s$ is an open and dense subset of E. K s, s$

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To prove Lemma 5 we need some more notation: Let S(1) and S(2) be any two nonempty, disjoint and exhaustive subsets of the time-state indexes 0, 1, ..., S. Let Z(1)( p, e^, v)#(Z s( p, e^, v)) s # S(1) and Z(2)( p, e^, v)# (Z s( p, e^, v)) s # S(2) . Similarly, let p(1) ( p(2)) denote commodity prices referring to spot markets s # S(1)(S(2)). For each set S(1) and S(2), with a simple permutation of rows, D p Z( p, e^, v) can be written as D p(1) Z(1)( p, e^, v) p(1) Z(2)( p, e^, v)

\D

D p(2) Z(1)( p, e^, v) . D p(2) Z(2)( p, e^, v)

+

Proposition 11, in [2], shows that, given S(1) and S(2), there exists an open and dense subset of the set of regular economies such that both D p(1)Z(1)( p, e^, v) and D p(2)Z(2)( p, e^, v) are invertible at each equilibrium price p. Since the number of the possible sets S(1) and S(2) is finite, there exists an open and dense set of economies such that D p(1)Z(1) and D p(2)Z(2) are invertible for each possible choice of the sets S(1) and S(2). An economy with such a property is sequentially regular. Let E SR be the set of sequentially regular economies. Observe that, for given p(2) and (e^, v) # E SR, Z(1)( p(1), p(2), e^, v) is a  R L*S(1)&k , where k=1 if 0 # S(1) and 0, otherwise. map from R L*S(1)&k ++ ++ The property of sequential regularity implies that, in a neighborhood of a competitive equilibrium of (e^, v) # E SR, the equilibrium prices of the spots s # S(1) adjust continuously to perturbations of the economy, even when the other spot markets are out of equilibrium. This property is the key to the proof of Lemma 5. Proof of Lemma 5. E" is obviously open. To establish density, define s s$ E*#[e^ # R lH ++ : 7 h e h {7 h e h , for s{s$], an open and dense subset of E. R, Pick an economy (e^, v) # E #E SR & E$ & E*, an open and dense subset of E. Assume that there exist at least two equilibrium prices of (e^, v), p^ and p~, with p^ 0 =p~ 0. We prove the statement for an arbitrary pair of distinct equilibria. Since the number of equilibria is generically finite, this implies the thesis of the lemma. Let (. . ., x^ h , . . .) and (. . ., x~ h , .. .) be the equilibrium allocations associated with p^ and p~, respectively. Then, either (i) x^ 0h$ {x~ 0h$ , for some individual h$, or (ii) x^ 0h =x~ 0h , for all h. The proof evolves in two steps; first we consider (i) and then (ii). Since we only consider regular economies, equilibrium prices are locally a smooth function of the parameters describing the economy and, moreover, their number is locally constant. In particular, given two equilibria p^ and p~ of an economy (e^*, v*) # E R, , let p^(e^, v) and p~(e^, v) be the two smooth maps describing (locally) the equilibria. Step 1. Let p^ and p~ be two distinct equilibrium prices of a regular economy (e^, v) and suppose that for some individual, say h=1, x^ 01 {x~ 01 .

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Then, there exists an economy (e^$, v$), arbitrarily close to (e^, v), such that p^ 0(e^$, v$){p~ 0(e^$, v$). s Proof of Step 1. By Lemma 4 and by p^ 0 =p~ 0, p~ 0 {p^ 1, sp^ 1, 1 , for all s>0. Hence, by the first-order conditions of the individual optimization problem and by the form of the utility functions, x~ 01 {x^ s$1 , for all s$, and x~ 01 {x~ s1 , for s>0. Pick p 0" arbitrarily close to p~ 0, p 0" {p~ 0, and consider the system of equations

Z s( p 0", ( p s ) s>0 , (e^, v))=0,

s>0.

(B1)

Since (e^, v) # E R, , by the implicit function theorem, there exists ( p s") s>0 arbitrarily close to ( p~ s ) s>0 that solves (B1). Set p"#( p 0", ( p s") s>0 ) and 0 0 x 0" 1 # h e h & h>1 f h( p", e h , v h ). By construction, the consumption bundle 0" s" (x 1 , ( f 1 ( p", e 1 , v 1 )) s>0 ) satisfies the budget constraint of individual 1 at prices p". By the properties of the allocations x^ 1 and x~ 1 , there are open sets 0 0= 0= L 0" 0 V 01 , V 0= 1 which satisfy cl V 1 /V 1 /cl V 1 /R ++ and such that x 1 # V 1 0 0 s 0= s 0= and x~ 1 # V 1 , while x^ 1  cl V 1 , for all s, and x~ 1  cl V 1 , for all s>0. Let *"1 be the Lagrange multiplier of agent 1 when p=p". Define $ 01 # (*"1 p 0"&D x0 v 1(x 01")). By choosing p 0" close enough to p~ 0, the function v $1 , 0 v $1(x 1 )#v 1(x 1 )+%(x 1 , V 0= 1 ) $ 1 x1 ,

can be made arbitrarily close to v 1 . By construction, x"1 (with associated Lagrange multipliers *"1 ) is agent 1's solution to optimization problem (1) at prices p" and given the utility function v $1 . Let v$#(v $1 , (v h ) h>1 ). Then, for (e^, v$) close enough to (e^, v), p"#p~(e^, v$) and p^ are equilibrium prices of (e^, v$) and p^ 0 {p 0". K If x^ 0h =x~ 0h for all h, then * h and * h , agent h's Lagrange multipliers associated with the two equilibria, satisfy * h =* h for all h. Given * h and x sh , there exists a unique p s that solves : s D xs v h(x sh )=* h p s. Hence, by the firstorder conditions of optimization problem (1) and by * h =* h , if x^ sh =x~ s$h , for some h, s, and s$, then x^ sh =x~ s$h , for all h. Therefore, since p^ {p~ and since 7 h e sh is one-to-one in s, neither x^ sh =x~ sh , for all s, nor x^ sh =x~ s$h , for s{s$. Therefore, there exists an individual h, say h=1, such that S(2)# [s: x^ s1 {x~ s$1 , s$=0, ..., S]{<. Also, let S(1)=[s: s  S(2)]. The previous observation is the key to the following step and depends crucially upon the assumption of market completeness. Otherwise, invariance of * 0h across equilibria would not imply invariance of * sh across equilibria for s>0.

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Step 2. Let p^ and p~ be two distinct equilibrium prices of a regular economy (e^, v) and suppose that for all h, x^ 0h =x~ 0h . Then, there exists an economy (e^*, v*) arbitrarily close to (e^, v) such that S(1)=<. Proof of Step 2. Define a perturbation of the endowments of agents 1 and 2, 2e 1 and 2e 2 , arbitrarily close to zero and such that (2e 1 +2e 2 )=0, p^ 2e 1 =p^ 2e 2 =0, while p~ 2e 1 {0 (and, hence, p~ 2e 2 {0). Also, let e* h #e h , #(e +2e ) and e* #(e +2e ). Since total endowments if h>2, while e* 1 1 1 2 2 2 are unchanged and p^ 2e 1 =p^ 2e 1 =0, p^ is an equilibrium price vector of the economy (e^*, v). Fix p~(2) and consider the system of equations Z(1)( p(1), p~(2), e^*, v)=0.

(B2)

Since (e^, v) # E R, , D p(1) Z(1)( p, e^, v) and D p(2) Z(2)( p, e^, v) are invertible. Hence, by the implicit function theorem, for 2e 1 small enough, there exists p"(1) arbitrarily close to p~(1) that solves (B2). Let p"=( p"(1), p~(2)). s* We need to show that x s* 1 ( p", e 1 , v 1 ){x 1 ( p~, e 1 , v 1 ), for some s* # S(1). s Suppose, by contradiction, that x 1( p", e 1 , v 1 )=x s1( p~, e 1 , v 1 ), for all s # S(1). Then, since 0 # S(1) and x^ 01 =x~ 01 , the first-order conditions of the optimization problem (1) imply that p"(1)=p~(1). Since, by construction, p~(2)=p"(2), p~ =p". Therefore, f s1( p", e 1 , v 1 )=x~ s1 , for all s. Since, p~ 2e 1 {0, this is impossible. Hence, there is s* # S(1) such that s* s* s$ x s* 1 ( p", e 1 , v 1 ){x 1 ( p~, e 1 , v 1 ). Therefore, x 1 ( p", e 1 , v 1 ){x 1 ( p^, e 1 , v 1 ) for s* s$ each s$ such that x 1 ( p~, e 1 , v 1 )=x 1 ( p^, e 1 , v 1 ). Moreover, for 2e 1 small s$ s* enough, x s* 1 ( p", e 1 , v 1 ){x 1 ( p^, e 1 , v 1 ) for all s$ such that x 1 ( p~, e 1 , v 1 ){ s$ s* x 1 ( p^, e 1 , v 1 ). The two sets of inequalities imply that x 1 ( p", e 1 , v 1 ){ x s$1( p^, e 1 , v 1 ) for each s$. s s For s # S(2), define the vector x s" 1 # h e h & h>1 f h ( p", e h , v h ). Observe s" s that, by construction, ((x 1 ) s # S(2) , ( f 1( p", e 1 , v 1 )) s # S(1) ) satisfies the budget constraint of individual 1 at prices p". Let *"1 be agent 1's Lagrange multiplier at the optimal solution to problem (1) when p=p". By the same argument used in the proof of the previous step, the following perturbation of the utility function of agent 1 is well defined: Let $ s #(*"1 p 1, s"&D xs v 1(x s1")), s # S(2). Then, choosing 2e 1 small enough, the cardinality index v $1 , s s v $1(x s1 )#v 1(x s1 )+ : % s1(x s1 , V s= 1 ) $1x1, s # S(2)

can be made arbitrarily close to v 1 . x"1 (with associated Lagrange multiplier *"1 ) is agent 1's solution to optimization problem (1) at prices p" and given the utility function v $1 .

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Let v*#(v $1 , (v h ) h>1 ). (e^*, v*) is arbitrarily close to (e^, v) and both p^ and p"#p~(e^*, v*) are equilibrium prices of (e^*, v*). Moreover, for this economy, the set S(2) includes all the states of nature in the original set S(2) and at least one additional state of nature s*. Since the number of states of nature is finite, the iteration of this procedure implies the thesis. K

REFERENCES 1. B. Allen, Generic existence of completely revealing equilibria for economies with uncertainty when prices convey information, Econometrica 49 (1981), 11731199. 2. Y. Balasko, The expectational stability of Walrasian equilibria, J. Math. Econ. 23 (1994), 179203. 3. Y. Balasko and D. Cass, The structure of financial equilibrium with exogenous yields: The case of incomplete markets, Econometrica 57 (1989), 135163. 4. Y. Balasko and D. Cass, Regular demand with several general budget constraints, in ``Equilibrium and Dynamics: Essays in Honor of David Gale'' (M. Majumdar, Ed.), pp. 2942, MacMillan, London, 1991. 5. J. D. Geanakoplos and A. Mas-Colell, Real indeterminacy with financial assets, J. Econ. Theory 47 (1989), 2238. 6. J. D. Geanakoplos and H. Polemarchakis, Existence, regularity and constrained suboptimality of competitive allocations when the asset market is incomplete, in ``Uncertainty, Information and Communication: Essays in Honor of K. J. Arrow'' (W. Heller, R. Starr, and D. Starret, Eds.), Vol. III, pp. 6596, Cambridge Univ. Press, Cambridge, 1986. 7. F. Hahn, A remark on missing markets, Cambridge University Working Papers, 1991. 8. M. V. Hirsch, ``Differential Topology,'' Springer-Verlag, BerlinHeidelbergNew York, 1976. 9. A. Mas-Colell, ``The Theory of General Economic Equilibrium. A Differentiable Approach,'' Cambridge Univ. Press, Cambridge, 1985. 10. A. Mas-Colell and J. H. Nachbar, On the finiteness of the number of critical equilibria, with an application to random selections, J. Math. Econ. 20 (1991), 397410. 11. R. Radner, Rational expectations equilibrium: Generic existence and the information revealed by prices, Econometrica 47 (1979), 655678.

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