Stationary Markov equilibria on a non-compact self-justified set

Journal of Mathematical Economics 42 (2006) 269–290

Stationary Markov equilibria on a non-compact self-justified set Norio Takeoka Department of Economics, University of Rochester, Rochester, NY 14627, USA Received 23 September 2003; received in revised form 22 February 2005; accepted 5 August 2005 Available online 3 March 2006

Abstract The existence of stationary processes of temporary equilibria is examined in an OLG model, where there are finitely many commodities and consumers in each period, and endowments profiles and expectations profiles are subject to stochastic shocks. A state space is taken as the set of all payoff-relevant variables, and dynamics of the economy is captured as a stochastic process in the state space. In our model, however, the state space does not necessarily admit a compact-truncation consistent with the intertemporal restrictions because distributions over expectations profiles may have non-compact supports. As shown in Duffie et al. [Duffie, D., Geanakoplos, J., Mas-Colell, A., McLennan, A., 1994. Stationary Markov equilibria. Econometrica 62, 745–781), such a compact-truncation, called a self-justified set, is essential for the existence of stationary Markov equilibria. We extend their existence theorem so as to be applicable to our model. © 2006 Elsevier B.V. All rights reserved. JEL classification: C62; D50 Keywords: Stationary Markov equilibria; Self-justified sets; Ergodicity; Overlapping generations; Temporary equilibria

1. Introduction We examine a monetary stationary equilibrium in a stochastic OLG model with finitely many commodities and agents in each period. Our model is based on the temporary equilibrium theory, such as Grandmont (1977), where an agent makes a forecast of the equilibrium price and of her endowment in the future, conditional on currently available information. This forecast rule, called an expectation function, is taken as a primitive. We consider the situation where, in every period, initial endowments and forecast rules of the agents are subject to stochastic shocks, and show the existence of a stationary process of temporary equilibria. E-mail address: [email protected] (N. Takeoka). 0304-4068/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2005.08.005

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For the purpose, we adapt the technique provided by Duffie et al. (1994) (hereafter DGMM). They consider a state space S and a correspondence G : S → P(S), where P(S) is the set of Borel probability measures over S. This correspondence is interpreted as the intertemporal consistency, derived from some particular model, embodying exogenous shocks, feasibility conditions, and the optimality of dynamic programming problems of the agents. A measurable subset J ⊂ S is said to be self-justified if G(s) ∩ P(J) = ∅ for all s ∈ J. Under regular assumptions on G, DGMM show that, if there exists a non-empty compact self-justified set J ⊂ S, then G admits a measurable selection Π : J → P(J) having an ergodic measure µ on J. The triplet (J, Π, µ) is called an ergodic Markov equilibrium for G. In our economic model, the state space S is taken as the set of payoff-relevant variables in one period (that is, endowments, expectation functions, action profiles of the agents, and a price system) satisfying the market clearing condition. Given the current state s, the intertemporal restriction G(s) is defined as the set of distributions over future states consistent with the exogenous shocks and the requirements for temporary equilibria. In our model, however, one of the assumptions of DGMM’s theorem fails to hold. Since the set of expectations functions is not compact, the distribution of a stochastic shock to the expectations functions may have a non-compact support. Thus there does not necessarily exist a compact self-justified set. Our main contribution is to extend DGMM’s existence theorem so as to admit non-compact selfjustified sets. In a general setting, we consider a state space S ≡ S1 × S2 and a correspondence G : S1 → P(S), where both S1 and S2 are complete separable metric spaces and S2 is assumed as the set of variables having no serial autocorrelation. For convenience, we regard G as a correspondence G : S → P(S). As a candidate for a non-compact self-justified set, we focus on a non-empty closed self-justified set J ⊂ S of which image under the projection mapping from S onto S1 is a compact set. As shown below, however, this weaker requirement for self-justified sets is not enough for the existence of ergodic Markov equilibria, and hence we need an additional assumption. Say that a self-justified set J satisfies the marginal restriction if there exists a non-empty compact subset M ⊂ P(S2 ) such that, for any s ∈ J and ν ∈ G(s) ∩ P(J), the marginal distribution of ν on S2 belongs to M. This condition does not require the support of ν ∈ G(s) ∩ P(J) to be compact. We show the existence of an ergodic Markov equilibrium for G under the above two assumptions on a non-empty closed self-justified set together with the same regular assumptions on G as in DGMM. Their theorem corresponds to the case where S2 is a singleton set. As an application of this extension, we show the existence of ergodic processes of monetary temporary equilibria in our economic model. We take as S2 the set of profiles of expectations functions because stochastic shocks to expectation functions are assumed to have no serial autocorrelation. The other payoff-relevant variables may have serial autocorrelation and hence are incorporated into S1 . Our result can be shown under regular assumptions on the stochastic shocks. Precisely, stochastic shocks are modeled by a transition probability P : E → P(E) and a conditional probability system Q : E × ∆ → P(S2 ), where E is a compact subset of endowments profiles and ∆ is the set of admissible price systems. We allow any continuous transition probability P such that P(e) is atomless, and any continuous conditional probability system Q. 1.1. Related literature The existence of stationary processes of temporary equilibria is not new. In OLG models based on the temporary equilibrium theory, Grandmont and Hildenbrand (1974), Grandmont (1977),

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Hellwig (1980) and Blume (1982) have shown the existence. Our result is an extension of these literature. In Grandmont and Hildenbrand (1974), a stochastic shock is given as a Markov process on a compact set of endowments profiles. An expectations profile does not change over time. This assumption corresponds to the case where Q is the constant function equal to the degenerate distribution at a fixed expectations profile. Hellwig (1980) considers a more general framework under the assumption that an exogenous shock is given as a time-homogeneous Markov process in a compact set. Blume (1982) allows non-compact self-justified sets. His existence theorem corresponds to the special case where a self-justified set satisfies the marginal restriction with respect to M = {λ} for some fixed λ ∈ P(S2 ). In other words, this is the case where an expectations profile follows an i.i.d. distribution. Details of the comparison with Blume (1982) are relegated to Appendix B. In OLG or more general one-step forward looking models, Knieps (1979), Spear (1985), Spear and Srivastava (1986), Spear (1988), Spear et al. (1990), Chiappori and Guesnerie (1992), Cass et al. (1992), Wang (1994), Gottardi (1996), Krebs (1997) and Magill and Quinzii (2003) study the existence of stationary rational expectations equilibria, where expectations are endogenously determined and consistent with the future temporary equilibria. DGMM also provide several examples of rational expectations models. These literature rely on a compact state space or a compact truncation of the original state space. Our extension may be applicable also in these contexts. 2. Mathematical preliminaries The following notational conventions, definitions, and facts are employed throughout the paper. The σ-algebra over a topological space X is always assumed to be the Borel σ-algebra, denoted by B(X). Given (X, B(X)), the set of Borel probability measures on X, denoted by P(X), is endowed with the weak convergence topology, that is, µn → µ if and only if:

f dµn → f dµ, for any real-valued bounded continuous function f on X (Parthasarathy (1967), p. 40). Given the weak convergence topology, X is a compact metric space if and only if P(X) is a compact metric space (Parthasarathy (1967), p. 45). Let X and Y be complete separable metric spaces. A measurable function P : X → P(Y ) is called a conditional probability system. In addition, if X = Y , P is called a transition probability on X. For any conditional probability system P : X → P(Y ) and λ ∈ P(X), there exists a unique probability measure on P(X × Y ), denoted by λ ⊗ P, satisfying:
λ ⊗ P(A × B) = P(x)(B) dλ(x), A

for all A ∈ B(X) and B ∈ B(Y ) (Dudley (1989), pp. 268–270). In addition, if P is the constant function equal to µ ∈ P(Y ), then λ ⊗ P is denoted by λ ⊗ µ and is called a product measure on P(X × Y ). For any ν ∈ P(X × Y ), the marginal distribution of ν on X is the probability measure µ ∈ P(X) defined by µ(A) = ν(A × Y ) for any A ∈ B(X). The marginal distribution of ν on Y is similarly defined. Especially, the marginal distribution of λ ⊗ P on Y is denoted by P dλ, and the value   of P dλ at B ∈ B(Y ) is denoted by P(B) dλ.

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Let F : X → Y be a correspondence. A function f : X → Y is called a selection of F if m f (x) ∈ F (x) for all x. A measurable selection of F is denoted by f ∼ F . Say that F has a closed graph if xn → x and yn → y with yn ∈ F (xn ) imply y ∈ F (x). Say that F is upper semi continuous (u.s.c.) if, for any x and any open set U with F (x) ⊂ U, there exists an open neighborhood V of x such that F (x ) ⊂ U for all x ∈ V . If Y is compact and F has a closed graph, then F is u.s.c. 3. The economic model 3.1. The environment Time is discrete and indexed by t ≥ 0. In every period, I agents are born, each of whom lives for two periods, and indexed by i. One group, born at t − 1, is called the old and the other group, born at t, is called the young. Subscripts j = 1, 2 are used for denoting the young and the old, respectively. There are L perishable consumption goods in each period. Agent i receives an endowment  initial i be the set of L . Let E ≡ eij ∈ Eji for j = 1, 2, where Eji ⊂ RL is a compact subset of R E ++ i,j j all possible endowments profiles in a period. i Agent i’s consumption set in a single period is Xi ≡ RL + . She has a lifetime utility function u : i i i X × X → R. Assume that u is bounded, continuous, strictly increasing, and strictly concave. 3.2. Market structure and actions In each period, markets for the current consumption goods are open. There is an asset, called money, which can be stored from one period to the next. We assume that the total stock of money M > 0 is constant over time. The set of admissible price systems is given by    L   L ∆ ≡ p = (q, r) ∈ R+ × R+  ql + r = 1 ,  l=1

where q is a price system of consumption goods and r is a price of money. Let ∆+ ≡ {p ∈ ∆|ql > 0, l = 1, · · · , L} and ∆++ ≡ {p ∈ ∆+ |r > 0}. A pair a = (x, m) ∈ Xi × R+ is called an action, where x is the current consumption and m is be the set of actions. the saving in terms of money. Let Ai ≡ Xi × R+ = RL+1 + 3.3. Expectations A young agent forecasts the equilibrium price system and her endowment in the next period. The expectation of agent i is given by a continuous function ψi : ∆ × ∆ × E1i → P(∆+ × E2i ). That is, the expectation depends on the current price system, the equilibrium price system which prevailed in the previous period, and the current endowment. We exclude the case where agent i expects some goods to have zero price in the future. Let Ψ i be the  set of all continuous expectation functions, endowed with the compact-open topology.1 Let Ψ ≡ i Ψ i be the set of expectations profiles. A subbase for this topology is given by sets of the form {ψi |ψi (D) ⊂ U} for any compact subset D of ∆ × ∆ × E1i and any open subset U of P(∆+ × E2i ). Under this topology, Ψ i is a complete separable metric space. (Kuratowski (1968), pp. 93–94). 1

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3.4. Stochastic shocks There exist two kinds of exogenous shocks in this economy. First, the endowments profile evolves over time according to a time-homogeneous Markov process with a transition probability P : E → P(E) such that (i) P is continuous and (ii) P(e) is atomless for all e ∈ E. Second, the expectations profile stochastically depends on the price system and the endowments profile in the previous period. Precisely, we assume a continuous conditional probability system Q : ∆ × E → P(Ψ ). A pair (P, Q) completely characterizes the stochastic nature of the model. 3.5. Demands First consider the decision problem of the old. Given the current price p, the endowment ei2 , i and the action a1i,−1 = (x1i,−1 , mi,−1 1 ) ∈ A taken in the previous period, agent i chooses an action a2i ∈ Ai maximizing her utility function ui (x1i,−1 , ·) subject to the budget constraint: qx2i + rmi2 = qei2 + rmi,−1 1 . Since ui is strictly concave, an optimal action must be unique. Let the optimal action be denoted by ξ2i (p, ei2 , a1i,−1 ) ∈ Ai . Since mi2 must be zero for the maximum, ξ2i (p, ei2 , a1i,−1 ) can be denoted by (ϕi (p, ei2 , a1i,−1 ), 0), where ϕi (p, ei2 , a1i,−1 ) ∈ Xi is an optimal consumption. Since ui is strictly increasing, no optimal action exists if p ∈ ∆ \ ∆+ . Next consider the decision problem of the young. Given the current price p, the previous equilibrium price p−1 , the endowment ei1 , and the expectation function ψi , agent i chooses an action a1i maximizing the expected lifetime utility function defined by
ui (x1i , ϕi (·, ·, a1i )) dψi (p, p−1 , ei1 ), vi (a1i , p, p−1 , ei1 , ψi ) ≡ ∆+ ×E2i

under the budget constraint qx1i + rmi1 = qei1 . The set of optimal actions is denoted by ξ1i (p, p−1 , ei1 , ψi ) ⊂ Ai . Properties of ξ1i and ξ2i have been studied in temporary equilibrium theory. The following two propositions are standard. Proofs are omitted (see Grandmont and Hildenbrand (1974) and Christiansen and Majumdar (1977)). Proposition 3.1. Under the assumptions in Section 3.1: (i) ϕi : ∆+ × E2i × Ai → Xi is a continuous function. in i i i + i + (ii) Let (pn , ein 2 , a1 ) ∈ ∆ × E2 × A be a sequence converging to (p, e2 , a1 ) with p ∈ ∆ \ ∆ . in in i n Then ϕ (p , e2 , a1 ) diverges to infinity. The intuition behind Proposition 3.1 (ii) is that, if p · (ei2 , mi1 ) > 0 and ui is strictly increasing, agent i infinitely increases consumption of commodities, as prices of those commodities go to zero. Proposition 3.2. Under the assumptions in Sections 3.1 and 3.3: (i) ξ1i : ∆+ × ∆ × E1i × Ψ i → Ai has a closed graph; (ii) ξ1i : ∆++ × ∆ × E1i × Ψ i → Ai is a continuous function; (iii) Take any (p, p−1 , ei1 , ψi ) ∈ (∆+ \ ∆++ ) × ∆ × E1i × Ψ i . Then:

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(a) ξ1i (p, p−1 , ei1 , ψi ) = ∅ if and only if ψi (p, p−1 , ei1 ) assigns probability zero to ∆++ × E2i , (b) if a1i = (x1i , mi1 ) ∈ ξ1i (p, p−1 , ei1 , ψi ), then (x1i , αmi1 ) ∈ ξ1i (p, p−1 , ei1 , ψi ) for any nonnegative real number α. The intuition behind Proposition 3.2 (iii) is as follows: suppose r = 0. If agent i anticipates that money has positive value tomorrow, she will demand an infinitely large amount of money. Hence, there exists no optimal solution to the maximization problem. On the other hand, whenever there exists an optimal action a1i = (x1i , mi1 ) ∈ ξ1i (p, p−1 , ei1 , ψi ), she should anticipate that money will be valueless tomorrow. Since r = 0, any level of money holding is indifferent, and hence (x1i , αmi1 ) ∈ ξ1i (p, p−1 , ei1 , ψi ) for any α ≥ 0. 3.6. State space and temporary equilibria Let Z≡

 i=1

Ai ×



Ai × ∆

i=1

be the space of the endogenous variables, that is, actions profiles and price systems. Let S1 be the subset of E × Z satisfying the market clearing conditions, that is:        i i i i i S1 ≡ s1 = (e, z) ∈ E × Z  (x1 + x2 ) = (e1 + e2 ), m1 = M ,  i

i

i

and S2 be the set of expectations profiles, that is, S2 ≡ Ψ . Finally, let S ≡ S1 × S2 be the state space. Each state in S summarizes the relevant information about the current economy. For k = 1, 2, let Sk−1 be the copy of Sk denoting the previous state. The following definition is a standard temporary equilibrium concept: Definition 3.1. A temporary equilibrium given (s1−1 , e, ψ) ∈ S1−1 × E × S2 is a triplet (a1∗ , a2∗ , p∗ ) ∈ Z satisfying the following conditions: (i) (ii) (iii) (iv)

a1i∗ ∈ ξ1i (p∗ , p−1 , ei1 , ψi ) for all i; i,−1 a2i∗ = ξ2i (p∗ , ei2 , a 1 ) for all i; i∗ i∗ i i i (x1i∗ + x2 ) = i (e1 + e2 ); i m1 = M.

A monetary temporary equilibrium given (s1−1 , e, ψ) is a temporary equilibrium given (s1−1 , e, ψ) with r∗ > 0. Let V (s1−1 , e, s2 ) ⊂ Z be the set of temporary equilibria given (s1−1 , e, s2 ). Define the temporary equilibrium correspondence: W : S1−1 × E × S2 → S1 × S2 by W1 (s1−1 , e, s2 ) ≡ (e, V (s1−1 , e, s2 )) ⊂ S1 and W2 (s1−1 , e, s2 ) ≡ s2 ∈ S2 . This correspondence captures the relation between each previous state and the current states consistent with the temporary equilibria. As in Grandmont and Hildenbrand (1974), we can prove the following proposition. A proof is omitted.

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Proposition 3.3. Under the assumptions in Sections 3.1–3.3, W is non-empty-valued, compactvalued and u.s.c. 3.7. Stochastic processes of temporary equilibria A triplet (Ω, B(Ω), π) is called a probability space if Ω is a complete separable metric space and π is a Borel probability measure on Ω. Definition 3.2. A stochastic process of temporary equilibria is a Markov process {st }∞ t=0 , where each st is a measurable function from a probability space (Ω, B(Ω), π) into S, such that the following two conditions hold almost surely: (i) The stochastic process {et , s2,t }∞ t=0 , where st = ((et , zt ), s2,t ), is consistent with the exogenous shocks, that is, the conditional distributions of et and of s2,t coincide with P(et−1 ) and with Q(pt−1 , et−1 ), respectively; and (ii) zt is a temporary equilibrium given (st−1 , et , s2,t ), that is, zt ∈ V (st−1 , et , s2,t ). If zt is a monetary temporary equilibrium given (st−1 , et , s2,t ) in addition, {st }∞ t=0 is called a stochastic process of monetary temporary equilibria. We are interested in a stochastic process of temporary equilibria satisfying a standard notion

of stationarity. A stochastic process {st }∞ t=0 is stationary if, for any t, t ≥ 0 and any h ≥ 0, the distribution of st , st+1 , · · · , st+h is the same as that of st , st +1 , · · · , st +h . Furthermore, a stationary process {st }∞ t=0 with a stationary distribution ν on S is called an ergodic process if the limiting time average of an integrable function f on (S, ν) exists and is equal to the space average, that is:
T −1 1  lim f (st ) = f (s) dν(s), T →∞ T S t=0

almost surely. Details are found in Rosenblatt (1962). Definition 3.3. A stationary (ergodic) process of temporary equilibria is a stochastic process of temporary equilibria satisfying stationarity (ergodicity). In addition, if such a stochastic process is monetary, it is called a stationary (ergodic) process of monetary temporary equilibria. 4. Existence theorems To show the existence of stationary or ergodic processes of temporary equilibria, we adapt the technique provided by DGMM. One of their requirements, the existence of non-empty compact self-justified sets, does not necessarily hold in our model because Ψ is not compact and hence Q(p, e) ∈ P(Ψ ) may have a non-compact support. As shown below, our main contribution is to extend DGMM’s existence theorem so as to admit non-compact self-justified sets.

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4.1. The intertemporal restriction and time-homogeneous Markov equilibria Recall the temporary equilibrium correspondence: W : S1−1 × E × S2  (s1−1 , e, s2 ) → (e, V (s1−1 , e, s2 ), s2 ) ⊂ S. Notice that W may not be single-valued because of multiplicity of temporary equilibria. A selection f of W is, therefore, one possible deterministic relation between each previous state and the current state consistent with the temporary equilibria. Each selection f of W is denoted by f = (f1 , f2 ), where fk is a selection of Wk , k = 1, 2. We construct the intertemporal restriction of the model from all the measurable selections of the m temporary equilibrium correspondence. Given s1 = (e, a1 , a2 , p) ∈ S1−1 and f ∼ W, a distribution over the current states can be obtained as the distribution of the product measure P(e) ⊗ Q(p, e) induced by the measurable mapping f (s1 , ·, ·) : E × S2 → S. Denote this induced distribution by P(e) ⊗ Q(p, e)f (s1 , ·, ·)−1 ∈ P(S). Define the intertemporal restriction G : S1 → P(S) by m

G(s1 ) ≡ {P(e) ⊗ Q(p, e)f (s1 , ·, ·)−1 |f ∼ W}.

(1)

That is, given the previous state s1 , G(s1 ) is the set of all possible distributions over the current states consistent with the requirements for temporary equilibria. Notice that G is independent of the previous state of expectations profile. This is because Q has, by assumption, no serial autocorrelation.2 For convenience, we hereafter regard G as a correspondence G : S → P(S) with assuming G(s1 , s2 ) = G(s1 , s2 ) for all s1 ∈ S1 and s2 , s2 ∈ S2 . Now we are ready to introduce an equilibrium concept for G. Definition 4.1. A time-homogeneous Markov equilibrium (THME) for G is a pair (J, Π), where J is a non-empty measurable subset of S and Π : J → P(J) is a measurable selection of G on J, that is, Π(s) ∈ G(s) for all s ∈ J. For any THME (J, Π), Π is regarded as a stochastic dynamic system on J satisfying the intertemporal consistency of the model. As is well-known, given Π and an arbitrary initial distribution ν ∈ P(J), we can construct a probability space (Ω, B(Ω), π) and a time-homogeneous Markov process {st }∞ t=0 , where st : Ω → J is a measurable mapping, such that the conditional distribution of st given s0 , · · · , st−1 is Π(st−1 ) almost surely.3 Hence a THME for G can generate a stochastic process of temporary equilibria. 4.2. Stationary Markov equilibria To guarantee stationarity or ergodicity of stochastic processes of temporary equilibria, we need to find a special kind of initial distribution over S. An invariant measure for (J, Π) is a measure 2 Recall that Q is a function from ∆ × E into P(S ). That is, the current distribution over S depends on components 2 2 in s1−1 but is independent of s2−1 . 3 See, for example, Rosenblatt (1962), Chapter 6.

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ν ∈ P(J) such that
ν(D) = Π(s)(D) dν(s), for all D ∈ B(J). An invariant measure is called also a stationary distribution. Given (J, Π) and an invariant measure ν, say that a measurable subset D ⊂ J is a ν-invariant set if Π(s) ∈ P(D) for ν-a.e. s ∈ D. If D is a ν-invariant set, J \ D is also a ν-invariant set. An invariant measure ν is ergodic if, for any ν-invariant set D, either ν(D) = 0 or ν(D) = 1. An ergodic measure is a natural stochastic analog of a steady state of a deterministic system. Definition 4.2. A stationary Markov equilibrium (SME) for G is a triplet (J, Π, ν) such that (J, Π) is a THME which has an invariant measure ν. In addition, if ν is ergodic, (J, Π, ν) is called an ergodic Markov equilibrium (EME). It is well-known that a stationary (ergodic) Markov equilibrium (J, Π, ν) for G generates a ∞ stationary (ergodic) Markov process {st }∞ t=0 . By construction of G, {st }t=0 is a stationary (ergodic) process of temporary equilibria as given by Definition 3.3. Turn to the existence of stationary Markov equilibria. A necessary condition for such equilibria is the existence of a non-empty measurable subset J of S satisfying G(s) ∩ P(J) = ∅, for all s ∈ J. Such a subset J is called a self-justified set for G. Under additional assumptions, DGMM show a sufficiency. Proposition 4.1 (DGMM). If (i) G is convex-valued with a closed graph and (ii) there exists a non-empty compact self-justified set J ⊂ S, then there exists an EME (J, Π, ν) for G. 4.3. An extension of DGMM’s existence theorem We have to verify whether G defined as (1) satisfies the assumptions of Proposition 4.1. Condition (ii), that is, the existence of a non-empty compact self-justified set is problematic in our model. Since S2 is not compact and hence the support of Q(p, e) ∈ P(S2 ) is not necessarily compact in S2 , the support of an element ν ∈ G(s) may not be compact, either. To admit noncompact self-justified sets, we extend Proposition 4.1 in such a manner as to be applicable to our model. Throughout this subsection, we use notation independent of that in the other sections. Let S1 and S2 be complete separable metric spaces. We call S ≡ S1 × S2 a state space. Consider a correspondence G : S1 → P(S). Thus, S2 is the set of variables without serial autocorrelation. In a natural way, G can be identified with a correspondence G : S → P(S). As a candidate for a non-compact self-justified set for G, we focus on a closed self-justified set J ⊂ S such that: J1 ≡ {s1 ∈ S1 |(s1 , s2 ) ∈ J

for some s2 }

(2)

is compact in S1 . That is, J1 is the image of J under the projection map from S onto S1 . Any compact self-justified set satisfies this condition. Any product self-justified set J1 × J2 , where J1 is compact in S1 and J2 is closed in S2 , also satisfies this condition. Now a question is whether this weaker requirement for self-justified sets ensures the existence of ergodic Markov equilibria together with condition (i) in Proposition 4.1. Since G is

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independent of s2 , the answer seems positive. The following example, however, shows that the weaker assumptions are not enough: Example 4.1. Let S1 ≡ [0, 1] and S2 ≡ R+ . Let:  {δ(s1 /2,− log (s1 /2)) } if s1 ∈ (0, 1], G(s1 ) ≡ {δ(1,0) } if s1 = 0, where δx is called the Dirac measure at x, which assigns a unit measure to the singleton set {x}. Obviously, G is convex-valued with a closed graph, and J ≡ S1 × S2 is a closed self-justified set for G such that J1 = S1 is compact. Since G(s1 ) is a singleton set:  δ(s1 /2,− log (s1 /2)) if s1 ∈ (0, 1], Π(s1 ) ≡ δ(1,0) if s1 = 0, is the only possible measurable selection of G on J. However, the THME (J, Π) has no invariant measure. To ensure the existence of stationary Markov equilibria, we need an additional assumption. For any self-justified set J ⊂ S, J is said to satisfy the marginal restriction if there exists a nonempty compact subset M ⊂ P(S2 ) such that, for any s ∈ J and any ν ∈ G(s) ∩ P(J), the marginal distribution of ν on S2 belongs to M. Under this assumption, the support of ν ∈ G(s) ∩ P(J) is not necessarily compact. Notice that any compact self-justified set J satisfies the marginal restriction. Notice also that, in Example 4.1, J does not satisfy the marginal restriction because the set of the marginals on S2 is {δ0 } ∪ {δ(− log(s1 /2)) |s1 ∈ (0, 1]}, which is a closed but non-compact subset of P(S2 ). We are ready to state the main theorem. Theorem 4.1. Assume that (i) G : S1 → P(S) is convex-valued with a closed graph, (ii) there exists a non-empty closed self-justified set J ⊂ S such that J1 defined as (2) is compact, and (iii) J satisfies the marginal restriction. Then: (1) there exists a non-empty compact subset N ⊂ P(J) such that, for any ν ∈ N, there exists a THME (J, Π) for G satisfying:
ν(D) = Π(s1 )(D) dµ(s1 ), for all D ∈ B(J), where µ is the marginal of ν on J1 , that is, the triplet (J, Π, ν) is an SME for G; (2) for any extreme point ν ∈ N, (J, Π, ν) can be taken to be an EME for G. Proof. For proof of the above theorem see Appendix A.




If S2 is a singleton set, G is regarded as a correspondence G : S1 → P(S1 ). Then, conditions (ii) and (iii) in Theorem 4.1 simply require the existence of a non-empty compact self-justified set J1 ⊂ S1 . Thus Proposition 4.1 is a special case of Theorem 4.1. Blume (1982) shows a similar result under a stronger assumption than the marginal restriction. Precisely, his result corresponds to the case where a self-justified set J satisfies the marginal restriction with respect to M = {λ} for some fixed λ ∈ P(S2 ), while Theorem 4.1 allows any compact subset M ⊂ P(S2 ). A detailed discussion can be found in Appendix B.

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4.4. Existence of ergodic processes of temporary equilibria Recall the intertemporal restriction G defined as (1). Now Theorem 4.1 applies to show the existence of ergodic processes of temporary equilibria. Theorem 4.2. Under the assumptions in Sections 3.1–3.4, there exists an EME for G. Proof. For proof of the above theorem see Appendix A.




As mentioned in Section 4.2, an EME for G generates an ergodic process of temporary equilibria. Theorem 4.2 is an extension of results by Grandmont and Hildenbrand (1974) and Blume (1982). In Grandmont and Hildenbrand (1974), an expectations profile, say ψ ∈ Ψ , is constant across time. This corresponds to the case where Q(p, e) = δψ for all (p, e) and δψ denotes the Dirac measure at ψ. By adapting the technique in Blume (1982), the existence of a stationary process can be shown under any general but fixed distribution over expectations profiles, say λ ∈ P(S2 ). This assumption corresponds to the case where Q(p, e) = λ for all (p, e), that is, an expectations profile follows an i.i.d. distribution λ over time. In Theorem 4.2, Q can be any continuous conditional probability system. For some expectations profile s2 ∈ S2 , the set of temporary equilibria V (s1 , e, s2 ) may contain a non-monetary temporary equilibrium. Hence a stationary distribution associated with an SME may assign positive probability to the event consisting of non-monetary temporary equilibria. An SME (J, Π, ν) for G is said to be monetary if ν assigns probability one to the event consisting of monetary temporary equilibria. To show the existence of a monetary EME, we focus on the restricted class of expectations profiles. Let: Ψmi ≡ {ψi ∈ Ψ i |ψi (p, p−1 , ei1 )(∆++ × E2i ) > 0

for all (p, p−1 , ei1 )}.

If ψi ∈ Ψmi , agent i always anticipates that money has positive value in the future. From Proposition 3.2 (iii-a), a temporary equilibrium given (s1−1 , e, s2 ) is monetary if s2i ∈ Ψmi for some agent i. Let Ψˆ i be a complete separable subspace of Ψ i such that Ψˆ i ⊂ Ψmi . Since: S2,m ≡ {s2 = (ψ1 , · · · , ψi , · · · , ψI ) ∈ S2 |ψi ∈ Ψˆ i

for some i}

is a complete separable metric space, the following theorem is obtained as an immediate consequence of Theorem 4.2. Theorem 4.3. Under the assumptions in Sections 3.1–3.4, if Q(p, e) ∈ P(S2,m ) for all (p, e) in addition, then there exists a monetary EME for G. A Markov process {st }∞ t=0 generated by a monetary EME is an ergodic process of monetary temporary equilibria. Acknowledgements I would like to thank Ken Urai for invaluable suggestion and constant support. I am grateful also to Atsushi Kajii, Kazuya Kamiya, Kiyoshi Kuga, Hiroaki Nagatani, and the audiences at the 1999 Mathematical Economics Conference in Kyoto and the 2001 Annual Meeting of the Japanese Economic Association in Hiroshima for helpful comments. Detailed suggestions by a referee led to substantial improvements. I retain responsibility for the remaining errors and

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omissions. The main part of the paper is my master’s thesis at the Graduate School of Economics, Osaka University. I gratefully acknowledge the hospitality of the school. Appendix A. Proofs Proof of Theorem 4.1 (1). We adapt the argument in the proof of Theorem 1.1 of DGMM. Since J is a self-justified set, the graph of G can be restricted on J1 × P(J). Let GJ1 : J1 → P(J) be the restricted correspondence and Gr(GJ1 ) denote the graph. Let m1 : P(Gr(GJ1 )) → P(J1 ) be the restriction of the function on P(J1 × P(J)) that assigns the marginal distribution on J1 . Similarly, m2 : P(Gr(GJ1 )) → P(P(J)) is defined by the restriction of the function on P(J1 × P(J)) assigning the marginal distribution on P(J). Since J satisfies the marginal restriction, there exists a compact subset M ⊂ P(S2 ) such that, for all s1 ∈ J1 and ν ∈ GJ1 (s1 ), the marginal distribution of ν on S2 belongs to M. By the Mazur theorem (Dunford and Schwartz (1958), p. 416, Theorem 6), the closed convex hull of M, denoted by co(M), is compact. Let L ⊂ P(J) be the set of ν such that the marginal of ν on S1 belongs to P(J1 ) and the marginal of ν on S2 belongs to co(M). Lemma A.1. L is (i) compact and (ii) convex. Proof. (i) From Theorem 6.7 (p. 47) of Parthasarathy (1967), compactness under the weak convergence topology is equivalent to uniform tightness, that is, for any ε > 0, there is a compact subset Kε ⊂ J such that ν(Kε ) > 1 − ε for all ν ∈ L. Since co(M) ⊂ P(S2 ) is compact, the same theorem ensures the existence of a compact subset Kε2 ⊂ S2 such that λ(Kε2 ) > 1 − ε for all λ ∈ co(M). Let Kε ≡ J1 × Kε2 . For any ν ∈ L, let ν2 be the marginal of ν on S2 . Since ν2 ∈ co(M), we have ν(J1 × Kε2 ) = ν2 (Kε2 ) > 1 − ε. Therefore, L is compact. (ii) To show convexity, take any ν, ν ∈ L and α ∈ [0, 1]. Let ν1 and ν1 be the marginal distributions of ν and of ν on J1 , respectively. By definition, ν1 and ν1 belong to P(J1 ). For any A ∈ B(J1 ), (αν + (1 − α)ν )(A × P(J)) = αν(A × P(J)) + (1 − α)ν (A × P(J)) = αν1 (A) + (1 − α)ν1 (A) = (αν1 + (1 − α)ν1 )(A). Since P(J1 ) is convex, αν1 + (1 − α)ν1 ∈ P(J1 ). Thus the marginal of αν + (1 − α)ν on J1 belongs to P(J1 ). By the similar way, the marginal of αν + (1 − α)ν on S2 belongs to co(M) because co(M) is convex. Therefore, αν + (1 − α)ν ∈ L.
Since Gr(GJ1 ) ⊂ J1 × L ⊂ J1 × P(J), Lemma A.1 implies that Gr(GJ1 ) is compact. Lemma A.2. For each µ ∈ P(J1 ), there exists θ ∈ P(Gr(GJ1 )) such that m1 (θ) = µ. m

Proof. There exists a measurable selection g ∼ GJ1 by the Kuratowski–Ryll–Nardzewski theorem (Hildenbrand (1974), p. 55). Define the measurable function h : J1 → J1 × P(J) by h(s1 ) ≡ (s1 , g(s1 )). Let θ ≡ µh−1 be the distribution of µ induced by h. Then, θ ∈ P(Gr(GJ1 )). Since θ(A × P(J)) = µ(A) for all A ∈ B(J1 ), we have m1 (θ) = µ.


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From Lemma A.2, define the non-empty-valued correspondence m−1 1 : P(J1 ) → P(Gr(GJ1 )) by m−1 (µ) ≡ {θ | m1 (θ) = µ}. It can be shown that m−1 1 is convex-valued with a closed graph. Since P(Gr(GJ1 )) is compact, −1 m1 is u.s.c. Let C(J) be the set of all real-valued bounded continuous functions on J with the supnorm. Since L is compact, the argument in Lemma A.1 implies that there exists a sequence of compact n subsets of J, {J n }∞ n=1 , such that ν(J ) > 1 − 1/n for all ν ∈ L and n. For all n and η ∈ P(L), define the bounded linear functional n on C(J) by

n f (s1 , s2 ) dν(s1 , s2 ) dη(ν).  (f ) ≡ L

Jn

Since n can be regarded as a bounded linear functional on C(J n ), the Riesz representation theorem (Dunford and Schwartz (1958), p. 265) ensures that there exists a unique En η ∈ P(J n ) such that, for all f ∈ C(J):
f dEn η. n (f ) = Jn

The function : P(L) → P(J n ) is continuous and mixture linear. n Notice that E η ∈ L ⊂ P(J) for all n because L is convex. Since L is compact, we can assume without loss of generality that the sequence {En η}∞ n=1 converges to the limit point in L, denoted by Eη. This relation defines the function E : P(L) → L ⊂ P(J). En

Lemma A.3. (i) For all f ∈ C(J):


f dEη = f (s1 , s2 ) dν(s1 , s2 ) dη(ν). J

L

(3)

J

(ii) E is continuous and mixture linear. Proof. (i) By definition of En :


n f dE η = Jn

L

Jn

f (s1 , s2 ) dν(s1 , s2 ) dη(ν).

Since the left-hand side of (4) is equal to

n f dE η → f dEη



(4)

f dEn η, we have

Jn

as En η → Eη. On the other hand, since ν(J n ) > 1 − 1/n for all ν ∈ L, the right-hand side of (4) converges to that of (3). Hence, Eq. (3) holds. (ii) We show continuity of E. Let ηm → η with ηm , η ∈ P(L). We have a sequence {Eηm }∞ m=1 in L. Since L is compact, we can assume Eηm converges to the limit ν∗ ∈ L. It suffices to show that Eη = ν∗ .

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From (3), for any f ∈ C(J),


m f dEη = f dν dηm (ν). J

L

(5)

J

 By definition of the weak convergence, the left-hand side of (5) converges to f dν∗ as Eηm →  ν∗ . On the other hand, since f dν is continuous with respect to ν, the definition of the weak convergence implies that the right-hand side of (5) converges to:

f dν dη(ν), L

J

 which is equal to f dEη by (3). Thus, for all f ∈ C(J):

∗ f dν = f dEη. From Theorem 5.9 (p. 39) of Parthasarathy (1967), we have Eη = ν∗ . Next we show mixture linearity of E. Take any η, η ∈ P(L) and α ∈ [0, 1]. From (3),




f dE(αη + (1 − α)η ) = f dν d(αη + (1 − α)η )(ν). J

L

(6)

J

Again by (3), the right-hand side of (6) is equal to







α f dν dη + (1 − α) f dν dη = α f dEη + (1 − α) f dEη

L

J

L

J


=

J

J

f d(αEη + (1 − α)Eη ).

J

Thus, for all f ∈ C(J),

f dE(αη + (1 − α)η ) = f d(αEη + (1 − α)Eη ). J

J

From Theorem 5.9 (p. 39) of Parthasarathy (1967), E(αη + (1 − α)η ) = αEη + (1 − α)Eη , and hence E is mixture linear.
Let m0 : P(J) → P(J1 ) be the mapping assigning ν ∈ P(J) to the marginal distribution on J1 . It is easy to see that m0 is continuous and mixture linear. Now define the correspondence ϕ : P(J1 ) → P(J1 ) by ϕ(µ) ≡ m0 ◦E◦m2 ◦m−1 1 (µ).

(7)

Notice that ϕ is convex-valued because functions m2 , E and m0 are mixture linear and m−1 1 is convex-valued. Moreover, since m2 , E and m0 are continuous and since m−1 is u.s.c., ϕ is u.s.c. Thus we have a non-empty-valued, convex-valued and u.s.c. correspondence from a compact convex metric space P(J1 ) into itself. The Fan-Glicksberg fixed point theorem ensures that ϕ has a fixed point. Let R denote the set of fixed points of ϕ. It is easy to see that R is convex and compact. Define the non-empty set N ≡ {ν ∈ P(J)|µ = m0 (ν), ν ∈ E◦m2 ◦m−1 1 (µ),

for some µ ∈ R}.

(8)

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Since R is compact, it is easy to verify that N is closed. Furthermore, since m−1 1 is u.s.c. and R is −1 compact, m1 (R) is compact. Since E and m2 are continuous, we know E(m2 (m−1 1 (R))) ⊂ P(J) is compact. Consequently, N ⊂ E(m2 (m−1 (R))) is also compact. 1 We will claim that N as given by (8) is the required object in Theorem 4.1 (1). Take any ν¯ ∈ ¯ and m0 (¯ν) = m1 (θ) ¯ = µ. N. There exist θ¯ ∈ P(Gr(GJ1 )) and µ ¯ ∈ P(J1 ) such that ν¯ = E◦m2 (θ) ¯ From Theorem 10.2.1 (p. 269) of Dudley (1989), there exists a conditional probability system P : J1 → P(L) satisfying
¯ × B) = θ(A P(s1 )(B) dµ(s ¯ 1 ), A

for all A ∈ B(J1 ) and B ∈ B(P(J)). Define Π : J1 → P(J) by Π(s1 ) ≡ E◦P(s1 ). Lemma A.4. For all D ∈ B(J),
ν¯ (D) = Π(s1 )(D) dµ(s ¯ 1 ). J1

Proof. Step 1: For any real-valued measurable function g on L:





g dP(s1 ) dµ(s ¯ 1) = gd P dµ ¯ . J1

L

(9)

L

It is easy to see that Eq. (9) holds for all characteristic functions and simple functions. In case of a general measurable function, take an increasing sequence of simple functions that converges pointwise to g. The monotone convergence theorem (Dudley (1989), p. 100) ensures Eq. (9). Step 2: For any f ∈ C(J):





f dE◦P(s1 ) dµ(s ¯ 1) = f dE P dµ ¯ . J1

J

J

From Lemma A.3 (i), for any f ∈ C(J):




f dE◦P(s1 ) dµ(s ¯ 1) = f dν dP(s1 )(ν) dµ(s ¯ 1) J1

and

J

J1





f dE J

 P dµ ¯

L






=

f dν d L

(10)

J

 P dµ ¯ .

(11)

J

 Since J f dν is a measurable function with respect to ν, Step 1 implies that the right-hand side of Eq. (10) coincides with that of Eq. (11). Step 3: For any closed set F ⊂ J,



E◦P(s1 )(F ) dµ(s ¯ 1) = E P dµ ¯ (F ). (12) J1

Let Bn (F ) ≡ {s ∈ J|d(F, s) < n1 } for all n ≥ 1, where d is the metric on J. There is a continuous function fn : J → [0, 1] such that fn is zero on J \ Bn (F ) and one on F. Clearly, fn converges pointwise to the characteristic function associated with F. By Step 2 and the dominated

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convergence theorem (Dudley (1989), p. 101),





lim fn dE◦P(s1 ) dµ(s ¯ 1 ) = lim fn dE P dµ ¯ , n→∞ J J n→∞ J 

 1  ⇒ J1 J limn→∞ fn dE◦P(s1 ) dµ(s ¯ 1 ) = J limn→∞ fn dE P dµ ¯ , 

 ¯ 1 ) = E P dµ ¯ (F ). ⇒ J1 E◦P(s1 )(F ) dµ(s Thus, Eq. (12) holds for any closed set F. Since every Borel probability measure on a metric space is regular (Parthasarathy (1967), p. 27), Eq. (12) holds for any D ∈ B(J). Taking this fact and the condition:


 ¯ = ν¯ E P dµ ¯ = E(m2 (θ)) together, we obtain the required result.




Lemma A.5. Π(s1 ) ∈ GJ1 (s1 ), µ ¯ -a.e. Proof. For any f ∈ C(J) and c ∈ R, let       A ≡ s1 ∈ J1 maxν∈GJ1 (s1 ) J f dν ≤ c < J f dE◦P(s1 ) ,    B ≡ ν ∈ L  J f dν > c , and      A ≡ s1 ∈ J1 maxν∈GJ1 (s1 ) J f dν ≤ c, P(s1 )(B) > 0 . Step 1: A ⊂ A . Take any s1 ∈ A. Suppose P(s1 )(B) = 0. Since




f dE◦P(s1 ) = f dν dP(s1 )(ν) = f dν dP(s1 )(ν) c< J

L




J

B




+

J





f dν dP(s1 )(ν) ≤ L\B




J





c dP(s1 )(ν) =

+ L\B

f dν dP(s1 )(ν) B

J

f dν dP(s1 )(ν) + cP(s1 )(L \ B), B

J

we have

f dν dP(s1 )(ν) > (1 − P(s1 )(L \ B))c. B

(13)

J

Taking P(s1 )(L \ B) = 1 and (13) together, we have

f dν dP(s1 )(ν) > 0. 0= B

J

This is a contradiction. Thus, we must have P(s1 )(B) > 0 and hence s1 ∈ A . Step 2: For all f ∈ C(J),

f dE◦P(s1 ) ≤ max f dν, µ-a.e. ¯ ν∈GJ1 (s1 )

(14)

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It suffices to show that µ(A) ¯ = 0 for any f ∈ C(J) and c. Suppose otherwise. Then, µ(A) ¯ >0 for some f ∈ C(J) and c. From Step 1, µ(A ¯ ) > 0. Since θ¯ ∈ P(Gr(GJ1 )) and GJ1 (s1 ) ∩ B = ∅ for any s1 ∈ A ,
¯ × B) = 0 = θ(A P(s1 )(B) dµ(s ¯ 1 ) > 0. A

This is a contradiction. Step 3: E◦P(s1 ) ∈ GJ1 (s1 ), µ-a.e. ¯ ¯ > 0 such that Suppose otherwise. Then, there exists a measurable subset A ⊂ J1 with µ(A) E◦P(s1 ) ∈ GJ1 (s1 ) for all s1 ∈ A. Fix an arbitrary s1 ∈ A. Since GJ1 (s1 ) ⊂ P(J) is a convex set, the separating hyperplane theorem (Schaefer (1999), p. 65, Theorem 9.2) ensures that there exist a linear functional Γ on P(J) and a constant c ∈ R such that Γ (E◦P(s1 )) > c > Γ (ν) for all ν ∈ GJ1 (s1 ). Since C(J) is a weak∗ dense subset of the dual space of P(J) (Dunford and Schwartz (1958), p. 425, Corollary 6), there exists f¯ ∈ C(J) satisfying:

f¯ dE◦P(s1 ) > c > f¯ dν, for all ν ∈ GJ1 (s1 ), and hence:

f¯ dE◦P(s1 ) > c ≥ max f¯ dν. ν∈GJ1 (s1 )

This contradicts (14).




Finally, choose any measurable selection Π of GJ1 . Define:  ¯ Π(s1 ) if s1 belongs to the support of µ, Π(s1 ) ≡

Π (s1 ) otherwise. From Lemmas A.4 and A.5, the triplet (J, Π, ν¯ ) is an SME.




Proof of Theorem 4.1 (2). The argument is similar to that in Corollary 1.1 of DGMM. Let ν be an extreme point of the set N as given by (8). Since N is non-empty and compact, extreme points exist by the Krein–Milman theorem (Dunford and Schwartz (1958), p. 440). By Theorem 4.1 (1), m there exists a THME Π ∼ GJ1 satisfying:
ν(D) = Π(s1 )(D) dµ(s1 ), for all D ∈ B(J), where µ is the marginal of ν on J1 . To show that ν is ergodic, take a ν-invariant set C ⊂ J. We want to show that either ν(C) = 1 or ν(C) = 0. Suppose ν(C) ∈ (0, 1). Then, we can define the probabilities conditional on C and on J \ C, respectively. Formally, for each D ∈ B(J), let: η(D) ≡

ν(C ∩ D) ν(C)

and

ρ(D) ≡

ν((J \ C) ∩ D) . ν(J \ C)

By definition, ν = ν(C)η + (1 − ν(C))ρ. We will claim η, ρ ∈ N. Step 1: Let µη denote the marginal of η on J1 . Then,
η(D) = Π(s1 )(D) dµη (s1 ).

(15)

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 Since ν(C ∩ D) = Π(s1 )(C ∩ D) dµ, we have:
Π(s1 )(C ∩ D) dµ(s1 ). η(D) = ν(C)

(16)

Since C is ν-invariant, the iterated integral formula implies:


ν(C ∩ D) = Π(s1 )(C ∩ D) dµ(s1 ) = Π(s1 )(C ∩ D)Π(s1 )(ds1 , ds2 ) dµ(s1 )

= C



= Hence,

Π(s1 )(D)Π(s1 )(ds1 , ds2 ) dµ(s1 ) Π(s1 )(D)Π(s1 )(C ∩ (ds1 , ds2 )) dµ(s1 ).





Π(s1 )(C ∩ (ds1 , ds2 )) dµ(s1 ) ν(C)



Π(s1 )(C ∩ (·))

= Π(s1 )(D) d dµ(s1 ) . ν(C)

ν(C ∩ D) = ν(C)

Π(s1 )(D)

(17)

Taking (16) and (17) together,

η(D) = Π(s1 )(D) dη = Π(s1 )(D) dµη (s1 ). Step 2: µη ∈ ϕ(µη ), where ϕ = m0 ◦E◦m2 ◦m−1 1 is defined as (7). We know that Π(s1 ) can be written as E◦P(s1 ), where P : J1 → P(P(J)) is a conditional probability system such that E◦P(s1 ) ∈ GJ1 (s1 ), µ-a.e. Let θ ∈ P(Gr(GJ1 )) be a unique probability measure satisfying
θ(A × B) = P(s1 )(B) dµη (s1 ), A

 for all A ∈ B(J1 ) and B ∈ B(P(J)). Since m2 (θ) = P dµη , Step 3 of Lemma A.4 implies 




P dµη = E◦P dµη = Π dµη . E(m2 (θ)) = E From (15), m0 (E(m2 (θ))) = µη . Thus, µη ∈ ϕ(µη ). Since η ∈ E(m2 (m−1 1 (µη ))) and m0 (η) = µη , η ∈ N. Similarly, we can show ρ ∈ N. This is a contradiction because ν is an extreme point of N, but ν = ν(C)η + (1 − ν(C))ρ. Therefore, we must have either ν(C) = 1 or ν(C) = 0.
Proof of Theorem 4.2. First of all, since W is u.s.c., Lemma 1 (p. 55) of Hildenbrand (1974) ensures the existence of a measurable selection of W. Hence, G is non-empty-valued. Next we will claim that there is a non-empty closed self-justified set J for G such that J1 as defined by (2) is compact. To this end, we show the following lemma: Lemma A.6. For any state s1 ∈ S1 , there is a compact subset J1 ⊂ S1 containing s1 such that W(J1 , E, S2 ) ⊂ J1 × S2 .

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Proof. Suppose otherwise. There exists sˆ 1 = (ˆe, aˆ 1 , aˆ 2 , p) ˆ ∈ S1 such that, for every compact ˆ Let Un (p) ≡ subset J1 containing sˆ 1 , W(J1 , E, S2 ) ⊂ J1 × S2 . Choose p ∈ ∆ \ ∆+ with p = p. {z ∈ RL+1 | p − z < n1 }. For all n sufficiently large, pˆ ∈ Un (p). Let ∆n ≡ ∆ ∩ (RL+1 \ Un (p)) ˆ 1 , aˆ 2 )} × ∆n }∞ for all such n. Since ∆n is a compact subset of ∆, {J1n }∞ n=1 ≡ {{(ˆe, a n=1 is a sequence of compact subsets of S1 containing sˆ 1 . By assumption, there exist sequences (s1−1,n , en , s2n ) ∈ J1n × E × S2 and s1n ∈ S1 satisfying s1n ∈ W1 (s1−1,n , en , s2n ) and s1n ∈ J1n . The condition s1n ∈ J1n implies that pn → p ∈ ∆ \ ∆+ . From Proposition 3.1 (ii), a2n diverges to infinity. This contradicts the fact that {a2n }∞
n=1 is a bounded sequence. Take any compact subset J1 ⊂ S1 ensured by Lemma A.6. Let J ≡ J1 × S2 . For any s1 ∈ J1 m and f ∼ W, Lemma A.6 implies that f (s1 , ·, ·) is a measurable function from E × S2 into J. Hence, P(e) ⊗ Q(p, e)f (s1 , ·, ·)−1 ∈ P(J). Therefore, J is a non-empty closed self-justified set such that J1 is compact. Lemma A.7. J satisfies the marginal restriction. Proof. Let M be the range of Q : ∆ × E → P(S2 ). Since Q is continuous and since ∆ × E is a non-empty compact set, M is a non-empty compact subset of P(S2 ). m For any s1 ∈ J1 , take ν ∈ G(s1 ). There exists f ∼ W satisfying ν = P(e) ⊗ −1 Q(p, e)f (s1 , ·, ·) . For any B ∈ B(S2 ), ν(J1 × B) = P(e) ⊗ Q(p, e)f (s1 , ·, ·)−1 (J1 × B) = Q(p, e)f2 (s1 , ·, ·)−1 (B) = Q(p, e)(B). Since Q(p, e) ∈ M by assumption, J satisfies the marginal restriction.




The remaining task is to verify condition (i) of Theorem 4.1. For the purpose, we rely on results ˆ : J1 × S2 → P(J1 ) by in Blume (1982). Define the correspondence G m ˆ 1 , s2 ) ≡ {P(e)f1 (s1 , ·, s2 )−1 ∈ P(J1 )|f = (f1 , f2 ) ∼ G(s W}.

ˆ is convex-valued with a closed graph Blume (1982) shows in Theorems 3.1 and 3.2 (p. 65) that G whenever: (i) J1 and E are compact metric spaces and S2 is a complete separable metric space, (ii) W1 : J1 × E × S2 → J1 is non-empty-valued and u.s.c., and (iii) P : E → P(E) is continuous ˆ defined as above is and P(e) is atomless. Since all the assumptions are satisfied in our model, G actually convex-valued with a closed graph. Lemma A.8. G has a closed graph. m

Proof. Let s1n → s1 and νn → ν with νn ∈ G(s1n ). There exists f n ∼ W satisfying νn = P(en ) ⊗ Q(pn , en )f n (s1n , ·, ·)−1 . For any s2 , since the sequence {P(en )f1n (s1n , ·, s2 )−1 }∞ n=1 is contained in P(J1 ), without loss of generality, it can be assumed to converge to the limit µ ∈ P(J1 ). Since ˆ n , s2 ) and since G ˆ has a closed graph, µ ∈ G(s ˆ 1 , s2 ). Hence, there P(en )f1n (s1n , ·, s2 )−1 ∈ G(s 1 m −1 exists f1 ∼ W1 satisfying µ = P(e)f1 (s1 , ·, s2 ) . For any A ∈ B(J1 ) and B ∈ B(S2 ), νn (A × B) = P(en ) ⊗ Q(pn , en )f n (s1n , ·, ·)−1 (A × B)
= P(en )f1n (s1n , ·, s2 )−1 (A) dQ(pn , en )(s2 ) B


=

χB P(en )f1n (s1n , ·, s2 )−1 (A) dQ(pn , en )(s2 ),

(18)

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where χB is the characteristic function associated with B. Define gn : S2 → R and g : S2 → R by gn (s2 ) ≡ χB P(en )f1n (s1n , ·, s2 )−1 (A),

g(s2 ) ≡ χB P(e)f1 (s1 , ·, s2 )−1 (A).

and

n ∞ n Since {gn }∞ n=1 is uniformly bounded, {g }n=1 is uniformly integrable. Furthermore, g converges to g in distribution. The generalized Lebesgue’s theorem (Hildenbrand (1974), p. 52) says that term (18) converges to
χB P(e)f1 (s1 , ·, s2 )−1 (A) dQ(p, e)(s2 ),

which is equal to P(e) ⊗ Q(p, e)f (s1 , ·, ·)−1 (A × B). Since νn → ν, we have ν(A × B) = P(e) ⊗ Q(p, e)f (s1 , ·, ·)−1 (A × B), for all A ∈ B(J1 ) and B ∈ B(S2 ). Therefore, ν = P(e) ⊗ Q(p, e)f (s1 , ·, ·)−1 ∈ G(s1 ).




Lemma A.9. G is convex-valued. m

Proof. Fix s1 ∈ J1 and take any ν, ν ∈ G(s1 ) and α ∈ [0, 1]. There are f, f ∼ W satisfying ν = P(e) ⊗ Q(p, e)f (s1 , ·, ·)−1 and ν = P(e) ⊗ Q(p, e)f (s1 , ·, ·)−1 . Define Σ(s1 , s2 ) ≡ αP(e)f1 (s1 , ·, s2 )−1 + (1 − α)P(e)f1 (s1 , ·, s2 )−1 . m ˆ 1 , s2 ) is convex, Σ(s1 , s2 ) ∈ G(s ˆ 1 , s2 ). There exists f ∗ ∼ Since G(s W such that Σ(s1 , s2 ) = P(e)f1∗ (s1 , ·, s2 )−1 . For any A ∈ B(J1 ) and B ∈ B(S2 ),

αν(A × B) + (1 − α)ν (A × B) = αP(e) ⊗ Q(p, e)f (s1 , ·, ·)−1 (A × B) + (1 − α)P(e) ⊗ Q(p, e)f (s1 , ·, ·)−1 (A × B)
= α P(e)f1 (s1 , ·, s2 )−1 (A) dQ(p, e)(s2 ) B




+(1 − α)
=

B

P(e)f1 (s1 , ·, s2 )−1 (A) dQ(p, e)(s2 )


Σ(s1 , s2 )(A) dQ(p, e)(s2 ) = B

B

P(e)f1∗ (s1 , ·, s2 )−1 (A) dQ(p, e)(s2 )

= P(e) ⊗ Q(p, e)f ∗ (s1 , ·, ·)−1 (A × B). Thus, αν + (1 − α)ν must coincide with P(e) ⊗ Q(p, e)f ∗ (s1 , ·, ·)−1 . Hence, αν + (1 − α)ν ∈ G(s1 ).
We have verified all the assumptions of Theorem 4.1. The existence of an EME for G is ensured by Theorem 4.1 (2).


N. Takeoka / Journal of Mathematical Economics 42 (2006) 269–290

289

Appendix B. The Relation between Blume (1982) and Theorem 4.1 ˆ : S1 × S2 → P(S1 ), where S1 is a compact metric Blume (1982) considers a correspondence G ˆ is convex-valued with a space and S2 is a complete separable metric space. He shows that, if G m ˆ closed graph, then, for any λ ∈ P(S2 ), there exist Σ ∼ G and µ ∈ P(S1 ) such that
µ(A) = Σ(s1 , s2 )(A) dµ ⊗ λ, for all A ∈ B(S1 ). His result can be connected with Theorem 4.1 as follows: let S ≡ S1 × S2 . For any fixed λ ∈ P(S2 ), define G : S1 → P(S) by   m ˆ 1 , ·) . G(s1 ) ≡ λ ⊗ Σ(s1 , ·)|Σ(s1 , ·) ∼ G(s Let (Σ ∗ , µ) be an object ensured by his result. Define Π ∗ (s1 ) ≡ λ ⊗ Σ ∗ (s1 , ·) ∈ P(S) and
ν ≡ Π ∗ dµ ∈ P(S). Proposition B.1. (S, Π ∗ , ν) is an SME for G. Proof. By construction, (S, Π ∗ ) is a THME for G. The remaining task is to verify that ν is an invariant measure for Π ∗ . We will claim that the marginal distribution of ν on S1 is µ. For all A ∈ B(S1 ), we have

ν(A × S2 ) = Π ∗ (A × S2 )(s1 ) dµ(s1 ) = λ ⊗ Σ ∗ (s1 , ·)(A × S2 ) dµ(s1 )

=

Σ ∗ (s1 , s2 )(A) dλ(s2 ) dµ(s1 )=




Σ ∗ (s1 , s2 )(A) dµ ⊗ λ(s1 , s2 ) = µ(A).

Thus the claim holds. For all D ∈ B(S),

∗ Π (s1 )(D) dν(s1 , s2 ) = Π ∗ (s1 )(D) dµ(s1 ) = ν(D). Hence ν is an invariant measure for Π ∗ .




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