the structure of sunspot equilibria: the role of multiplicity

Review of Economic Studies (1999) 66, 713–732 0034-6527�99�00300713$02.00 1999 The Review of Economic Studies Limited

The Structure of SunspotEquilibria:

The Role of MultiplicityPIERO GOTTARDIUniversita di Venezia

and

ATSUSHI KAJIIUniversity of Tsukuba

First version received April 1997; final version accepted July 1998 (Eds.)

This paper examines the structure of sunspot equilibria in a standard two period exchangeeconomy with real assets. We show that for a generic choice of utility functions and endowments,there exists an open set of real asset structures whose payoffs are independent of sunspots suchthat the economy with this asset structure has a regular sunspot equilibrium. An important impli-cation of our result is that the multiplicity of non-sunspot equilibria is not necessary for theexistence of sunspot equilibria. Our technique is general and can be applied to show the existenceof sunspot equilibria in other frameworks.

1. INTRODUCTION

This paper analyses the structure of sunspot equilibria in a standard two period competi-tive exchange economy, with more than one consumption good in each period and a realasset which pays a bundle of commodities. There is no uncertainty except for sunspots.We show that for an open and dense subset of the set of utility functions and endowments,there exists an open set of real asset structures, with payoffs independent of sunspots,such that the resulting economy with an asset structure in this set has a regular sunspotequilibrium. We also show that this result is tight: in the space of utility functions, endow-ments and asset structures there is in fact an open set of economies for which sunspotequilibria do not exist. Moreover, our argument provides a systematic method whichallows us to find, generically, all the sunspot equilibria of a given economy. Thus it canbe used as a tool for studying the global structure of the set of sunspot equilibria, not justthe existence of a sunspot equilibrium.

The dependence of the existence of sunspot equilibria on the particular specificationof the payoff of the asset provides an interesting contrast with the fact that the set ofequilibrium allocations of the underlying certainty economy does not depend on the assetstructure. Recall that when markets are complete, preferences and endowments completelydetermine the set of competitive equilibria. In our setting, markets are complete in theunderlying certainty economy. So if payoffs change, the price of the asset will changeaccordingly but there is no effect on the equilibrium consumption of the certainty econ-omy. Thus the fundamentals of the certainty economy are the distribution of endowmentsand preferences among consumers, and our result implies that the existence of sunspot

713

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714 REVIEW OF ECONOMIC STUDIES

equilibria is generic in the space of fundamentals of the underlying certainty economy. Toput it differently, there is no robust condition on the fundamentals which ensures that theeconomy has no sunspot equilibria. Since certainty economies with a unique equilibriumconstitute an open set, it follows that generically, certainty economies with a unique equi-librium must also have a robust sunspot equilibrium for an appropriate choice of the realasset.

Another important implication of our analysis is that for any certainty equilibriumthere is, generically, a sunspot equilibrium arbitrarily close to it for some asset structure.The existence of these equilibria, which we shall call ‘‘local’’ sunspot equilibria by analogyto a similar notion extensively studied in the overlapping generations models, is particu-larly of interest since it questions the local uniqueness of certainty equilibria.

Thus, our result clarifies the long-standing issue of the relationship between the multi-plicity of competitive equilibria and the existence of sunspot equilibria: the multiplicity ofcertainty equilibria is not necessary for the existence of sunspot equilibria, even when assetpayoffs are independent of sunspots. There is however a more subtle relation betweenmultiplicity and sunspot equilibria, but first let us briefly discuss the model and techniquewe use.

In period 0, non-storable commodities and a single asset are exchanged. In period 1,the asset pays off a bundle of goods and commodities are exchanged again. A sunspotsignal is publicly observed at the beginning of period 1. There is no intrinsic uncertainty;that is, the parameters of the economy (preferences, endowments and asset payoffs) areindependent of sunspots. In this framework, there always exists a non-sunspot equilib-rium, where the allocation does not depend on sunspots, which can be naturally identifiedwith a certainty equilibrium. So the standard fixed point argument1 might just show theexistence of a nonsunspot equilibrium. Moreover, this equilibrium is determinate, unlikein the case of nominal assets; so there is no useful parameterization of the set of equilibriawhich might help us generate a sunspot equilibrium.

The idea of our proof is as follows. To show that a sunspot equilibrium exists, it isenough to consider two (sunspot) states. To avoid the technical difficulties describedabove, rather than directly working with the economy we described, we consider an auxili-ary economy which is identical to the original economy except for the asset payoff, whichnow depends on sunspots: say the asset pays rs units of the numeraire commodity in states. As long as the agents trade non-zero amounts of this asset, their income will dependon sunspots, so will the equilibrium allocation.2 Therefore, if we exclude exceptional casessuch as identical homothetic preferences, the spot commodity price vector ps will also bedifferent across sunspot states, since there are at least two goods in each spot. So we canfind a commodity bundle a whose value evaluated at the spot prices ps is the same as theyield of the auxiliary asset, by simply solving the linear system of equations: ps · aGrs forsG1, 2. It is straightforward to verify that the equilibrium allocation of the auxiliaryeconomy also constitutes an equilibrium allocation of the economy with a real asset thatpays the commodity bundle a obtained as above. This equilibrium is not necessarilyregular even if the corresponding auxiliary equilibrium is regular, so the last step of theargument is to show that a slight perturbation of the economy restores regularity.

Returning to the issue of multiplicity, note that sunspots can only affect allocationsthrough equilibrium prices since fundamentals (including asset payoffs) are independentof sunspots in our model. It is therefore not difficult to see that the second period spot

1. Geanakoplos–Polemarchakis (1986), for instance.2. This construction was also inspired by the argument used to show that models with nominal assets

generically have a continuum of sunspot equilibria. See discussion and references in Section 4.5.

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GOTTARDI & KAJII SUNSPOT EQUILIBRIA 715

markets where households trade their induced endowments, which are the initial endow-ment of commodities added to the returns from the asset holding, must have multipleequilibria; that is, there have to be multiple future spot equilibria after some reallocationof resources and the sunspot equilibrium is ‘‘randomizing’’ over them. Thus what accountsfor the existence of sunspot equilibria is ‘‘potential multiplicity’’ in future spot markets.3

Given an economy with multiple spot equilibria, Mas-Colell (1992) has shown, in atwo-agent model with no first period consumption, that one can find some asset structureand some endowment distribution such that for the corresponding economy, a sunspotequilibrium exists which presents a randomization over the given spot equilibria. His resulthowever does not show which economies have sunspot equilibria or how these spot equili-bria are related to the certainty equilibria of the constructed economy.4 Despite the differ-ent setting, out result addresses both: first, the set of economies is generic; secondly, thereis a robust set of economies in which sunspot equilibria are not part of certainty equilibria,and this is why an economy with a unique certainty equilibrium may have sunspot equilib-ria as well. Hence, we can show that it is ‘‘potential multiplicity’’ rather than ‘‘multi-plicity’’ that matters.

Our analysis will make it clear that at a sunspot equilibrium, the induced endowmentsin the second period tend to be far from the set of the Pareto efficient allocations of theeconomy so that the level of trades in commodities will be large; hence there is a significantincome effect in the second period, which tends to generate multiple equilibria in thesecond period economy. This finding sheds some light on the properties of the asset struc-tures for which an economy has sunspot equilibria: we should expect that the asset besuch as to induce a high volume of trade in it (and�or a high level of payoffs on theagents’ asset holdings).5 On this basis we are also led to speculate the following interestingpoint. Suppose there is a monopolist intermediary who can choose the specification of theasset a and whose objective is, as in Duffie–Jackson (1989), to maximize the tradingvolume in the asset at equilibrium. Then our result suggests that generically in utilityfunction and endowments, such a security designer may well be interested in choosing anasset a with which there is a sunspot equilibrium.

Our analysis resolves the existence problem of sunspot equilibria in economies withreal assets which has been an open question. It is also qualitatively different from theprevious results in an important respect. As far as we are aware, in the models that sharethe technical difficulties we pointed out above, i.e. with the main important exception ofnominal asset models which we discuss in Section 4.5, the available results are eitherrobust examples taking advantage of some specific feature of the model, or some argumentfor constructing a sunspot equilibrium which is only valid on a specific, non-generic setof economies (in particular on a non-generic subset of endowments and preferences whichdefine the properties of the underlying certainty economy). On the other hand, our resultrelies on a property, ‘‘potential multiplicity’’, which we show is generic in this set anddoes not use any additional structure on utility function and�or endowments other thanthe standard monotonicity and differentiability assumptions.

Furthermore, our technique appears to be applicable beyond the setting we con-sidered. We simply need to find a class of economies such that (1) every equilibrium for

3. On the other hand, when asset payoffs are sunspot dependent, not only multiplicity of equilibria of theunderlying certainty economy is not needed for the existence of sunspot equilibria as already shown by Cassand Shell (1983), but also ‘‘potential multiplicity’’ is not necessary, as established by Hens (1991).

4. Similar considerations apply to the results obtained by Balasko, Cass and Shell (1995), in a restrictedparticipation framework.

5. This will necessarily be the case if the economy is almost Pareto efficient.

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an economy in this class is a sunspot equilibrium; (2) at least one of such equilibria is alsoan equilibrium of the original economy. So we could have alternatively used the endow-ments, or preferences (e.g. beliefs) as the sunspot dependent parameters, as briefly dis-cussed in Section 4.3. The argument could then also be used to show the existence ofsunspot equilibria for other models considered in the literature.6

In the next section we present the model and state the main result. Its proof is inSection 3. In Section 4 some extensions and conjectures are discussed, and so is therelationship with the literature. The more technical proofs are collected in the Appendix.

2. THE MODEL

We consider a competitive two-period exchange economy. We assume that there are 2sunspot states in the second period, and no intrinsic uncertainty.7 To exhibit the existenceof sunspot equilibria, it is enough to consider the case of two sunspots, since a sunspotequilibrium in this setting can be naturally thought as a sunspot equilibrium of an econ-omy with an arbitrary number of sunspots.8

At the beginning of the second period, (sunspot) state sG1, 2 occurs with a publiclyknown probability πsH0. Spot commodity markets open in the first and the second per-iod. We label each spot by sG0, 1, 2, where spot zero corresponds to the first period.there are CC1 commodities in each spot, labelled by cG0, 1, 2, . . . ,C. Cn1 is assumed,i.e. at least two goods exist in each spot. Commodity 0 is a designated numeraire of theeconomy.

There is one asset, in zero net supply, which is traded in the first period. One unit ofthe asset yields a commodity bundle a ∈ �CC1 in the second period. Note that the elementsof a may be negative; thus the asset we consider is a contract which specifies some transfersof goods.

There are H households, HH1, labelled by hG1, 2, . . . ,H. Household h receivesendowments e0

h in the first period and e1h in the second period.

If the distinction among the sunspot states is ignored, we have a deterministic econ-omy. We refer to this as the certainty economy.

The following summarizes the notation (all vectors will be considered as columnvectors):

• xc,sh is the consumption of commodity c by household h in spot s; xs

h≡(xc,sh )CcG0 is

the household’s consumption plan in spot s, xh≡(xsh )

2sG0 , x≡(xh )

HhG1 .

• For a vector z ∈ �CC1, we denote by z \ the vector obtained from z by dropping thefirst element, i.e. the vector of non-numeraire goods.

• ps ∈ �CC1++ , is the (normalized) price vector of commodities in spot s, ps≡

(1, p1,s, . . . , pC,s ); p≡(ps)2sG0 defines a price vector and Π is the domain of nor-malized prices, a 3C-dimensional subset of �3(CC1)

++ .• The price of the asset is denoted by q ∈ �.• bh ∈ � is the demand for the asset by household h, b≡(bh )

HhG1 .

• λ sh will denote the Lagrange multiplier for household h which arises in the house-

hold’s utility maximization problem; λG(λ 0h , λ 1

h , λ 2h )

HhG1 .

6. Gottardi–Kajii (1996) applies this idea to the stationary overlapping generations model.7. The extension to the case in which there is also intrinsic uncertainty is briefly discussed in Section 4.3.8. See Section 4.4 for some discussion of this.

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Let Ξ≡�3(CC1)H++ B�3H

++B�HBΠB� be the space of endogenous variables with typicalelement ξG(x, λ , b, p, q); set n ≡ 3(CC1)HC3HCHC3CC1, which is the dimension ofΞ.

Household h’s preferences over consumption plans are represented by the utility func-tion Uh (xh )G∑2

sG1 πsuh (x0h , x

2h ), where uh : �CC1

++ B�CC1++ →�.

We assume:

Assumption 1. uh is C2, differentiably strictly increasing (i.e. for any xh ∈ �2(CC1)++ ,

Duh(xh )Z0), differentiably strictly concave (i.e. for any xh ∈ �2(CC1)++ , the Hessian D2uh (xh )

is negative definite), and the closure in �2(CC1) of each indifference surface is containedin �2(CC1)

++ .

We shall parameterize the economies described above by endowment vectors andutility functions. Let E≡{e≡(e0

h , e1h )HhG1 ∈ �2(CC1)H

++ } and U be the set of all functionssatisfying Assumption 1. U is endowed with the topology of C2 convergence on compactsets; let U ≡UH with the product topology. The economy is completely characterized by(e, u) ∈ E , where E ≡EBU is endowed with the natural product topology, and by the assetpayoff a ∈ �CC1. So we shall call ((e, u), a) an economy.

We consider competitive equilibria with self-fulfilling expectations: (x, b, p, q) is anequilibrium of economy ((e, u), a) if, for each h, (xh , bh ) solves the following problem givenp and q

maxxh ,bh Uh (xh ),

subject to

p0 · (x0hAe0

h )CqbhG0,

ps · (xshAe1

h )A(ps · a)bhG0, for sG1, 2,

and markets clear, i.e.

∑H

hG1 (x0hAe0

h )G0,

∑H

hG1 (xshAe1

h )G0, for sG1, 2,

∑H

hG1 bhG0.

Definition 2.1. An equilibrium is a sunspot equilibrium if x1h≠x2

h for some h.

Under Assumption 1, the utility maximization condition can be replaced with thecorresponding first order condition. For each hG1, . . . ,H, let

φ1h (ξ; ((e, u), a))G�∑2

sG1 πs (∂�∂x0h )uh (x

0h , x

sh )Aλ 0

hp0

π1 (∂�∂x1h ) uh (x

0h , x

1h )Aλ 1

hp1

π2 (∂�∂x2h ) uh (x

0h , x

2h )Aλ 2

hp2

� ∈ (�CC1)3,

φ2h (ξ; ((e, u), a))G�−p0 · (x0

hAe0h )Aqbh

−p1 · (x1hAe1

h )C(p1 · a)bh−p2 · (x2

hAe2h )C(p2 · a)bh

� ∈ �3,

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and

φ3(ξ; ((e, u), a))G�∑H

hG1 (x0\h Ae0\

h )

∑H

hG1 (x1\h Ae1\

h )

∑H

hG1 (x2\h Ae2\

h )

∑H

hG1 bh� ∈ �3CC1.

Define then Φ(ξ; ((e, u), a)) by the rule

Φ(ξ; ((e, u), a))G�···

φ1h (ξ; ((e, u), a))······

φ2h (ξ; ((e, u), a))···

φ3(ξ; ((e, u), a))

� . (2.1)

It can be readily shown that Φ((x, λ , b, p, q); ((e, u), a))G0 if and only if (x, b, p, q) is anequilibrium of the economy ((e, u), a): note that the remaining market clearing conditions(for the numeraire good in states 0, 1 and 2) are automatically satisfied. We can thenequivalently say that ξ is an equilibrium if Φ(ξ; ((e, u), a))G0.

Finally, we say that ξ is a regular equilibrium9 of economy ((e, u), a) if Φ(ξ; ((e,u), a))G0 and the nBn matrix Dξ Φ(ξ; ((e, u), a)) is invertible.

It is straightforward to show that there always exists a non-sunspot equilibrium, inwhich prices and allocations are independent of the sunspot signals. Moreover, every non-sunspot equilibrium is Pareto efficient and any Pareto efficient equilibrium is a non-sun-spot equilibrium.

Evidently, at an equilibrium, p1, p2 must be equilibrium prices for the second periodspot economy, i.e. for the exchange economy in spot sG1, 2 where each household h isendowed with e1

hCabh . Since e1hCabh is independent of sunspots, if ξ is a sunspot equilib-

rium, p1 and p2 must then be different equilibrium prices for the same economy (see Figure1).

Therefore, sunspot equilibria will not exist if: (1) the economy has a unique equilib-rium (system (2.1) has a unique solution) since a non-sunspot equilibrium always exists;(2) all the Pareto efficient allocations of the second period spot economy are supportedby the same price vector.

Case (1) arises, in particular, if e is a Pareto efficient allocation of (e, u). It is notdifficult to show that the no trade equilibrium xGe is then the unique equilibrium of((e, u), a), for all a. Moreover, it can be readily verified that the no-trade equilibrium isregular and hence we get:

Lemma 2.2. If e is a Pareto efficient allocation of ((e, u), a), there is an openneighbourhood B of ((e, u), a ) such that for any ((e′, u′ ), a′ ) ∈ B , there exists a uniqueequilibrium, which is a non-sunspot equilibrium.10

9. This definition is equivalent to the full rank condition of the Jacobian of the excess demand function.10. A claim along these lines is already in Balasko (1990).

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FIGURE 1

A proof can be found in the Appendix. Thus existence of sunspot equilibria cannotbe generic in the space EBU B�CC1, where ((e, u), a) lies.

On the other hand, an instance of case (2) is provided by the following:

Example 2.3. Suppose HG2, CG1 (i.e. there are two consumption goods); forsimplicity, assume also there is only one good in period 0. Sunspots are equally likely andhouseholds have the following utility functions: u1(x

01 , (x

111 , x12

1 ))G12 ln x0

1C12 ln x11

1 C12 ln x12

1 , u2(x02 , (x

112 , x12

2 ))Gln x02Cα ln x11

2 C(1Aα ) ln x122 .

Let αG12 . Then the agents have identical, Cobb–Douglas preferences over consump-

tion in the second period and, for any set of endowment vectors, the equilibrium spotprices for the second period spot economy are always the same. Therefore, for any e anda, economy ((e, u), a) has no sunspot equilibrium.

We will show that the situation described by cases (1), (2) is not robust to per-turbations of the agents’ utility functions, and that sunspot equilibria typically exist.Formally, define E * ⊂ E

E *G�(e, u) ∈ E :there is a ∈ �CC1 such that ((e, u), a) has

a regular sunspot equilibrium � .

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Our main result is:

Proposition 2.4. E * is an open and dense subset of E .

The result shows that for a generic choice of (e, u) we can find asset structures suchthat (e, u) has a regular sunspot equilibrium. The regularity of sunspot equilibria impliesthat the specification of the asset structure is robust to perturbations: from the ImplicitFunction Theorem (IFT),11 it follows in fact that for any (e, u) ∈ E *, there exists a non-empty open set A ⊂ �CC1 such that for any a ∈ A , ((e, u), a) has a sunspot equilibrium.

Notice that the result is consistent with Lemma 2.2 since there the neighbourhoodB depends on a, whereas in Proposition 2.4 the specification of a depends on (e, u) (seeSection 4.1). Lemma 2.2 also shows that generic existence cannot be established in thespace of ((e, u), a), nor of (e, u) for any given a, and so the above result is tight.

3. PROOF OF THE MAIN RESULT

The openness of the set E * follows from a similar application of the IFT as in the proofof Lemma 2.2, so we shall focus on the proof of the denseness of E * from now on.

3.1. Auxiliary economy

Consider an economy identical to ((e, u), a) except for the asset structure. There is nowone asset whose payoff depends on sunspots as follows: the asset pays 1 unit of thenumeraire good 0 in state 1 and r units of good 0 in state 2, where r ∈ �++ is exogenouslygiven. So an auxiliary economy is characterized by ((e, u), r).

As before, we can characterize the equilibria of the auxiliary economy by the solu-tions of a system of equations. Let ΦA (ξ; ((e, u), r)) be identical to Φ( · ) defined in (2.1)except for the terms describing the payoff of the asset: (p1 · a) is replaced with 1 and(p2 · a) with r. We say that ξ is an equilibrium of economy ((e, u), r) if ΦA (ξ; ((e, u), r))G0, and an equilibrium ξ is regular if Dξ ΦA (ξ; ((e, u), r)) is invertible.

The following result shows that a sunspot equilibrium can be readily obtained in thisauxiliary economy.

Lemma 3.1. Let ξ be an equilibrium of auxiliary economy ((e, u), r) where r≠1. Thenx1h≠x2

h for every h with bh≠0.

Proof. If bh≠0 and r≠1, the income of household h differs in state 1 and 2; henceby differentiability of uh , x

1h≠x2

h must hold. ��

Thus in the auxiliary economy, the existence of an equilibrium implies in general alsothe existence of a sunspot equilibrium. On the other hand, the equilibria of the auxiliaryeconomy are related to the equilibria of the original economy, where the payoff of theasset is sunspot independent, as follows.

11. Strictly speaking, IFT is not directly applicable since the set of economies E may not have a convenientdifferentiable structure. However, as is seen in the proof of Lemma 2.2, we can always find a subset with requireddifferentiable structure, and then the argument works as if IFT is directly applied at regular equilibria. So weshall loosely use the term IFT to mean that.

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Lemma 3.2. Let ξ be an equilibrium of the auxiliary economy ((e, u), r). If the vectora ∈ �CC1 satisfies

p1 · aG1,

p2 · aGr,(3.1)

then ξ is an equilibrium of economy ((e, u), a).Conversely, if ξ is an equilibrium of economy ((e, u), a), then there exists r such that ξ

is an equilibrium of economy ((e, u), r).

Proof. Condition (3.1) means that the real asset a yields exactly the same as thenumeraire asset of the auxiliary economy ((e, u), r) at the equilibrium prices p. By compar-ing Φ and ΦA , it is readily verified that ξ is also an equilibrium of ((e, u), a). Conversely,if ξ is an equilibrium of economy ((e, u), a), set r by the rule (3.1), and then ξ is anequilibrium of economy ((e, u), r). ��

So any sunspot equilibrium of economy ((e, u), a) can be captured by some auxiliaryeconomy.12

Notice that there does not always exist a vector a that satisfies (3.1): it may in facthappen that, at the sunspot dependent equilibrium of the auxiliary economy, the spotprices p1, p2 are collinear (recall Example 2.3). But it is intuitive that this must be a non-generic situation. Indeed, a slight perturbation of utility will break the collinearity, asshown by the following example.

Example 3.3. Consider the same setup as Example 2.3, but αH12 . Let

e11Ge1

2G(12 ,12)T and e0

1G23 , e

02G

13 .

It is not difficult to show that for any rH0, an equilibrium ξ of ((e, u), r) exists andbh≠0 for all h. Straightforward computations reveal that now the budget lines supportingthe period two Pareto efficient allocations are never parellel; Lemma 3.2 then applies andsunspot equilibria exist. For this economy all budget lines intersect at a single point vG(2(αA1)�(2αA1), 2α �(2αA1))T and it is possible to see that, for every r, a sunspot equi-librium of (e, u, r) is also a sunspot equilibrium of ((e, u), a), for aGvAe1

1 ; this is then theonly solution for a we obtain from system (3.1). Our analysis shows that this situation isalso not robust to perturbations.

In the next subsection, we shall prove that these types of perturbations are alwayspossible.

3.2. Generating a sunspot equilibrium

To show that sunspot equilibria typically exist, as claimed in the proposition, it sufficesthen to show that households trade the asset at an equilibrium of the auxiliary economy,for some r≠1, and spot prices are not collinear, generically. This is established by thefollowing two lemmas.

Lemma 3.4. Suppose ξr is a regular equilibrium of ((e, u), r*), for r*≠1, and brh≠0 forall h. Then for any neighbourhood V of (e, u), there exists (e, u) ∈ V and an equilibrium ξof ((e, u), r*) with p1,1≠p1,2.

12. See Section 4.3 for more on this.

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The result will be established by using a transversality argument (the proof is in theAppendix). Notice that in sunspot models perturbations of the economy are severelyrestricted and standard techniques (as in, e.g. Geanakopolos–Polemarchakis (1986)) can-not be applied. We now show that the condition required by Lemma 3.4 is genericallysatisfied.

Lemma 3.5. There is an open dense subset E r ⊂ E such that for any (e, u) ∈ E r thereexists a regular sunspot equilibrium ξ of ((e, u), r*) for some r*≠1 and brh≠0 for every h.

Proof. The result will be established here for r* close to 1.13 Notice that the equilib-rium system of the auxiliary economy is quite similar to the system studied in Cass (1992).It can be easily verified that the argument of his Lemma 2 and Theorem 1 applies to ourframework, and establishes the generic regularity of non-sunspot equilibria for rG1. Moreprecisely, we can show that there is an open dense subset E r ⊂ E such that for any (e, u) ∈E r , whenever ξr is a non-sunspot equilibrium of ((e, u), 1), Dξ ΦA (ξr, (e, u), 1) is non-singularand brh≠0 for every h. The result then follows by the IFT. ��

Therefore, combining Lemmas 3.1, 3.2, 3.4 and 3.5, we can conclude that the set ofeconomies (e, u) such that for some a, ((e, u), a) has a sunspot equilibrium is dense in E .

Unfortunately, the sunspot equilibrium obtained by the above argument may not bea regular equilibrium of ((e, u), a). To see why, consider Example 3.3 again: recall that forevery r, the only solution with respect to a of system (3.1) is given by aGvAe1

1 . Thusthere is only one asset structure a such that ((e, u), a) has a sunspot equilibrium; also((e, u), a) has a continuum of sunspot equilibria (obtained by letting r vary). Evidentlythese sunspot equilibria cannot be expressed as a (local) one-to-one function of a and socannot be regular equilibria of the original economy (since otherwise the IFT would beviolated). Note that this is true even though these sunspot equilibria are all regular equilib-ria of the auxiliary economy.14 The next result shows that this case is exceptional: if theconstructed equilibrium fails to be regular, we can always perturb the utility functions toobtain a regular sunspot equilibrium. Example 3.3 has in fact a very unusual property ofall supporting budget lines intersecting at a single point.

Lemma 3.6. Suppose there is a sunspot equilibrium of ((e, u), a). Then for any neigh-bourhood V of (e, u), there exists (e, u) ∈ V such that ((e, u), a) has a regular sunspotequilibrium.

A proof is in the Appendix. This completes the proof of Proposition 2.4.

The generic case of the proposition is illustrated by the following example:

Example 3.7. Consider the following modification of the utility functions ofExample 3.3:

u1(x01 , (x

111 , x12

1 ))Gln x01C√x11

1 C12√x12

1 , u2(x02 , (x11

2 , x122 ))Gln x0

2C12√x11

2 C12√x12

2 .

The region W of period two endowment distributions for which there are multiplespot equilibria has now positive measure (see Figure 2). Let A be the set of asset

13. Other possibilities are discussed in the next section.14. This type of singularity is related to the regularity properties of the induced second period equilibrium

(and these properties are affected when we vary the asset structure supporting the same equilibrium). See Kajii(1998), Gottardi–Kajii (1995b) for further discussion on this.

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structures for which the economy under investigation has a sunspot equilibrium; this setcan be naturally seen as a subset of W A{e1

1}. It can be shown that in this case A is atwo-dimensional, open subset of W A{e1

1 }, and that the sunspot equilibria obtained bythe previous lemmas will be generically regular.15

the structure of sunspot equilibria: the role of multiplicity

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