Rational Expectations Equilibria in Dynamic

Rational Expectations Equilibria in Dynamic Economies with Interacting Agents∗ Alberto Bisin†

Ulrich Horst‡

¨ ur§ Onur Ozg¨

November 2002

Abstract We consider general economies in which rational agents interact locally. The local aspect of the interactions is designed to capture in a simple abstract way social interactions, that is, socio-economic environments in which markets do not mediate all of agents’ choices, and each agent’s choice might be in part determined, for instance, by family, peer group, or ethnic group effects. We study static as well as dynamic infinite horizon economies; we allow for economies with incomplete information, and we consider jointly global and local interactions, to integrate e.g., markets and global externalities with peer and group effects. We provide conditions under which such economies have equilibria with rational expectations which depend in a Lipschitz continuous manner on the parameters. Journal of Economic Literature Classification Numbers: C62, C72, Z13 Keywords: rational expectations, local interactions, existence of equilibria

Preliminary; Comments Welcome

∗

This research was supported by the German Academic Exchange Service and the C.V. Starr Center for Applied Economics. It was carried out while the second author visited Princeton University. We wish to thank Jos´e Scheinkman for support and valuable comments. † New York University, Department of Economics, 269 Mercer St., New York, NY, 10003; e-mail: [email protected] ‡ Humboldt University of Berlin, Department of Mathematics, Unter den Linden 6, D-10099 Berlin, e-mail: [email protected] § New York University, Department of Economics, 269 Mercer St., New York, NY, 10003; e-mail: [email protected]

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1

Introduction

We consider general economies with local interactions. Agents interact locally when each agent interacts with only a finite (small) set of other agents in an otherwise large economy. The local aspect of interactions is designed to capture in a simple abstract way a socio-economic environment in which markets do not exist to mediate all of agents’ choices. In such an environment each agent’s ability to interact with others might depend on the position of the agent in a predetermined network of relationships, e.g., a family, a peer group, or more generally any socio-economic group. Local interactions represent an important aspect of several socio-economic phenomena. For instance, the decision of a teen to commit a criminal act or to drop out of school is often importantly influenced by the related decisions of peers, as documented by ([12], [22]) and ([13]), respectively. Other phenomena for which relevant peer effects have been identified include out-ofwedlock births ([13]), and smoking habits ([26]). More generally, local interactions occur not only between peers but also between family members, ethnic groups, neighbors in a geographical space. For example, neighborhood effects are important determinants of employment search ([37]), [29]), of the pattern of bilateral trade and economic specialization ([27]), and of local technological complementarities ([16], [14]); while ethnic group effects play a fundamental role in explaining urban agglomeration, segregation ([4], [35]), income inequality and stratification ([15]). 1 The documented empirical evidence of peer and neighborhood effects in socio-economic phenomena has spurred an interesting theoretical literature studying formally economies with local interactions. This literature has generated existence and characterization results for important but special classes of static economies: for instance economies with additive quadratic preferences, extreme value distributed shocks, and symmetric interaction effects, introduced by [6] and [9] (see also [10]); or general economies with a finite number of agents, studied by [20]. In this paper we contribute to this literature by extending the classes of economies under study in various dimensions. First of all we do not impose any specific functional form assumption on the agents’ preferences or on the nature of the stochastic elements of the economy, and we study economies with a (countable) infinity of agents. Most importantly, we look at equilibria of both static and dynamic economies with local interactions. Also, we study economies in which the interactions as well as the distribution of information across the agents is local (economies of incomplete information). Finally, we introduce global interactions together with local interactions to integrate e.g., markets and global externalities with peer and group effects. The study of equilibrium dynamics of economies with local interactions is our major contri1

Finally, local interactions control the dynamics and spread of ideas and beliefs, and therefore might in particular play an important role in the microstructure of financial markets; see ([9], [23]) for theoretical applications to financial markets. See [11], [8], [20] and [21] for good surveys of the theory of social interactions, and its applications.

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bution. The theoretical literature studying local interactions is in fact not yet fully integrated into the mainstream dynamic economic analysis of equilibrium. When dynamic economies are studied, in fact, the analysis is only confined to the case of backward looking myopic dynamics, either as a simple explicit dynamic process ([10]) with random sequential choice or as an equilibrium selection procedure for static economies ([20], [7]); the only exception is a simple example in [21]. It is our contention that the study of equilibrium dynamics of economies with local interactions, by allowing for rational expectations of forward looking agents, may elucidate several important aspects of social interactions. An example of a specific socio-economic environment might be helpful to illustrate the usefulness of the proper explicit equilibrium analysis of dynamic economies in the presence of local interactions. Consider a teenager evaluating the opportunity of dropping out of high school. His decision will depend on the conditions of the labor market, and in particular on the relevant wage differentials, which requires him to form expectations about the wage and labor conditions he will face if he graduates from high school. The teenager’s decision might depend also on the school attendance of a restricted circle of friends and acquaintances: dropping-out is generally made simpler if one’s friends also drop-out (local interactions). But as the decision of dropping out depends on the teenager’s expectations of the wage differential, it will also in part depend on his consideration of the possibility that, for instance, while his friends have not yet dropped out of school, they soon will, perhaps even motivated by his own decision of dropping out. Similarly, our teenager might decide to stay in school even if most of his friends dropped out, if he has reason to expect their decision to be soon reversed. The teenager will form expectations about his friends’ future behavior as well as about the future wage rate. As we noted, we extend our analysis of static and dynamic economies with local interactions by considering also economies of incomplete information, and economies with both local and global interactions. Economies in which information is incomplete and is distributed locally allow to study environments in which, for instance, only the agents directly interacting with each other observe each other’s private characteristics. This extension is important because the restriction to complete information, that is to economies in which each agent observes all the other agents’ private characteristics, is hard to justify in the socio-economic environments we study in which strategic interactions are instead only local.2 Finally, allowing for both local and global interactions is important to integrate the standard economic analysis of competitive markets and externalities, in which interactions are mediated by global components like prices and aggregate externalities, with the analysis of peer and social group effects captured by local interactions. In other words, the extension to local and global interactions allows us to consider economies in 2

The restriction to economies with complete information is in fact implicitly adopted by the literature only for simplicity: it allows the direct use of mathematical techniques and results from the physics of interacting particle systems and statistical mechanics ([6], [7], [9], [10]; see [30] for a presentation of such techniques and results).

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which agents interact locally and, at the same time, act as price takers in competitive markets, or take as given global externalities, such as average per-capita human or physical capital effects on aggregate productivity ([34], and [31]). The paper proceeds as follows. We first study static economies with local interactions with complete and incomplete information. We then study dynamic economies with local interactions and incomplete information (the results adapt simply to the complete information case). An example with quadratic preferences is studied in detail to illustrate the analysis and the conditions required by the general results for dynamic economies. Finally, we introduce global as well as local interactions in the dynamic analysis. For all the distinct economies we study, static economies with complete and incomplete information, as well as dynamic infinite horizon economies, we provide conditions under which such economies have rational expectations equilibria which depend in a Lipschitz continuous manner on the parameters. For each of these economies, we show that such conditions impose an appropriate bound on the strength of the interactions across agents. 3 They exclude in particular economies in which strategic coordination gives rise to multiple equilibria; for instance economies in which it is always optimal for each agent to match the action of a neighboring agent.

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Static Economies with Local Interactions

In this section we introduce our concept of a static economy with local interactions. We consider economies with a countable set A of agents. To each agent a ∈ A is associated a type, the realization of a random variable θ a taking values in a set Θ ⊂ R. Types are independent and identically distributed across agents with law ν, and with no loss of generality we assume that the random variable θ := (θ a )a∈A is defined on the canonical probability space (Ω, F, P). That is, Ω = Θ := {(θ a )a∈A : θa ∈ Θ}, and θ a (ω) = ω a for any ω = (ω a )a∈A ∈ Ω. The utility of an agent a ∈ A depends on his type θ a , and on an action xa he chooses from a common compact and convex action set X ⊂ R. The agent interacts locally with the agents b in some small neighborhood N (a) ⊂ A, and his utility also depends on the actions xb taken by the agents b in his neighborhood N (a). Remark 2.1 We assume that X and Θ are subsets of the real line. This assumption is made primarily for notational convenience. Our results carry over to models where X,which is closed and compact and Θ are subsets of some Hilbert space. With regards to the interaction structure in our economy, we first assume that the agents are disposed on the integer lattice Z, i.e., we assume that A = Z. We also assume that each agent 3

Similar conditions for related economies have appeared in the literature; [21] refers to them as conditions of Moderate Social Influence.

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a ∈ A only interacts with the agent a + 1, i.e., N (a) = {a + 1}. Such a system of local interactions has two properties: (i) neighborhoods never overlap; that is, if b ∈ N (a), then b ∈ / N (c) for any c 6= a; (ii) interactions are one-sided; that is, if b ∈ N (a), then a 6∈ N (b). While the first property is not crucial for our analysis, the second has an important technical role in the analysis. Furthermore, assuming interactions are one-sided is quite special in that it excludes any strategic interaction inside groups. Instead, the focus is on the analysis of interactions across groups. Economies which allow for more complex interaction patterns are analyzed in [25], but under the assumption that each agent observes the realizations of the types of all the agents in the economy, an assumption we will relax in this analysis. Future work might succeed in joining these two strands of the literature. Any heterogeneity across agents can be incorporated in the probabilistic structure of the types a θ . Agents can therefore be considered identical ex-ante without loss of generality. Thus, the preferences of each agent a ∈ A are described by a utility function u of the form ¡ a a+1 a ¢ ¡ ¢ x , x , θ 7→ u xa , xa+1 , θa .

We assume throughout that u : X 2 × Θ → R is continuous and strictly concave in its first argument. We can now be specific about the information structure of our economy. Prior to his choice, each agent a ∈ A observes the realization of his own type θ a as well as the realizations of the types θ b of the agents b ∈ {a + 1, a + 2, ..., a + N }. Here N ∈ N∗ ∪ {∞}.4 If N = ∞ each agent has complete information about the current configuration of types when choosing his action. In particular each agent a ∈ A observes the types of all the agents b ∈ {a + 1, a + 2, . . .} with whom he is directly or indirectly linked. When instead N ∈ N, an agent only has incomplete information about the types of the other agents. In particular, if N = 1, agents only observe the type of the agent with whom they directly interact. Remark 2.2 Observe that actions and types of agents ‘to his left’ have no intrinsic effect on the choice of agent a ∈ A. Definition 2.3 A static economy with local interactions is a tuple S = (X, Θ, u, ν, N ). A static economy with local interactions is an economy with complete information if N = ∞, and with incomplete information if N ∈ N. In order to introduce our notion of an equilibrium for a static economy with local interactions S = (X, Θ, u, ν, N ), we need some notation. The vector of types whose realization is observed by ¡ ¢ the agent a = 0 is denoted θN := {θ 0 , θ1 , . . . , θN }; by analogy T a θN := θa , . . . , θa+N denotes 4

If N = ∞, it is understood that {a + 1, . . . , a + N } = {a + 1, a + 2, . . .}. The case N = 0 corresponds to {a + 1, a + 2, ..., a + N } = ∅.

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the vector of types whose realization is observed by the agent a ∈ A.5 In case N = ∞, we put θN = {θ 0 , θ1 , θ2 , . . .} and T a θN = {θ a , θa+1 , θa+2 , . . .}. Finally, the set of possible configurations of types of all agents a ≥ 0 is given by Θ0 := {(θ a )a≥0 : θa ∈ Θ}. We first focus on the simpler case of economies with complete information. In such a situation the choice of the agent a ∈ A is represented by a function g a that maps any T a θN ∈ Θ0 into an element of X. Since the agent observes T a θN = {θ a , θa+1 , . . .}, his optimization problem takes the form ¡ a a+1 ¡ a+1 ¢ a ¢ max u x ,g T θN , θ . a x ∈X

If S is an economy with incomplete information, by observing the realization of the types T a θN := ¡ a ¢ θ , . . . , θa+N , agent a cannot determine his neighbor’s optimal choice. The choice depends on the realization of the random variable θ a+N +1 and this information is not available to the agent a ∈ A. The choice of each agent is represented by a function mapping g a which associates to each T a θN ∈ ΘN +1 the action g a (T a θN ) ∈ X. Thus, in a situation with incomplete information an agents’ optimization problem is given by Z ¡ ¡ ¢ ¢ max u xa , g a+1 θa+1 , . . . , θa+N , θ , θa ν(dθ). a x ∈X

Since the utility functions are strictly concave with respect to an agents’ own action, the maps Z ¡ ¡ ¢ ¢ ¡ ¡ ¢ ¢ xa 7→ u xa , g a+1 T a+1 θN , θa and xa 7→ u xa , g a+1 θa+1 , . . . , θa+N , θ , θa ν(dθ)

are strictly concave, too. Thus, the conditional choice of agent a ∈ A given the types of all the agents b ∈ {a + 1, a + 2, . . . , a + N } is uniquely determined both for economies with complete and with incomplete information. We are now ready to introduce our notion of an equilibrium for static economies with locally interacting agents.

Definition 2.4 Let S = (X, Θ, u, ν, N ) be a static economy with local interactions. (i) If S is an economy with complete information, then an equilibrium is a family (g ∗a )a∈A of measurable mappings g ∗a : Θ0 → X such that ¡ a ∗a+1 ¡ a+1 ¢ a ¢ g ∗a (T a θN ) = arg max u x ,g T θN , θ P-a.s. (1) a x ∈X

for all a ∈ A.

(ii) If S is an economy with incomplete information, then an equilibrium is a family (g ∗a )a∈A of measurable mappings g ∗a : ΘN +1 → X such that Z ¡ ¡ ¢ ¢ ∗a a u xa , g ∗a+1 θa+1 , . . . , θa+N , θ , θa ν(dθ) P-a.s. (2) g (T θN ) = arg max a x ∈X

Θ

for all a ∈ A.

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Formally, T a : Ω 7→ Ω (a ∈ A) is the a-fold iteration of the canonical right shift operator T on Ω; that is, ¡ ¢ ¡ ¢ T ((ωb )b∈A ) = (ωb+a )b∈A ; furthermore, T a θN := θ0 (T a ω), . . . , θN (T a ω) = θa , . . . , θa+N . a

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An equilibrium (g ∗a )a∈A for an economy S is symmetric if g ∗a (T a θN ) = g ∗ (T a θN )

P-a.s.

(3)

for some mapping g ∗ and each a ∈ A. In what follows we shall restrict our analysis to symmetric equilibria. Remark 2.5 (i) In our setting, the conditional optimal choice of an agent a ∈ A, given T a θN , is uniquely determined. Thus, the complete information case is essentially equivalent to a situation in which each agent a ∈ A directly observes and reacts to his neighbor’s action. In this sense, if the economy is one of complete information, a symmetric equilibrium g ∗ : Θ → X can be identified with the mapping G∗ : Θ × X → X defined by ¡ ¢ G∗ (θa , xa+1 ) = arg max u xa , xa+1 , θa . a x ∈X

In fact, the previous literature has only studied this case, that is environments where each agent reacts to the actions taken by his neighbors; see, e.g. [6], [7], [9], [10], [20], [22] or [25]. In this sense we conclude that the previous literature has implicitly restricted attention to complete information economies in which each agent a ∈ A observes the whole configuration of types (θ b )b≥a . (ii) Let us assume that an agent’s utility function takes the additive form u(xa , xa+1 , θa ) = v(xa ) + xa xa+1 + g(xa , θa ). In such a situation, the agent may as well maximize his utility with respect to the expected action Ea xa+1 of his neighbor. Here, Ea denotes the conditional expectation operator of the agent a ∈ A. Thus, we may as well assume that the utility functions are given by u(xa , xa+1 , θa ) = v(xa ) + xa Ea xa+1 + g(xa , θa ). Such preferences are analyzed in, e.g., [10]. In this sense, our framework can be viewed as an extension of the model by [10] to situations in which the agent’s utility functions are more complex.

2.1

Existence, Uniqueness, and Lipschitz Continuity of Equilibrium

In order to guarantee the existence and uniqueness of an equilibrium for static economies with local interactions, we need to impose a form of strong concavity on the agents’ utility functions. To this end, we recall the notion of an α-concave function.

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Definition 2.6 Let α ≥ 0. A real-valued function f : X → R is α-concave on X if the map x 7→ f (x) + 12 α|x|2 from X to R is concave. This definition is first due to Rockafellar (1976), and is used for purposes related to ours by Montrucchio (1987). Remark 2.7 Observe that a twice continuously differentiable map f : X → R is α-concave, if and only if the second derivative is uniformly bounded from above by −α. For a more detailed discussion of the properties of α-concave functions we refer the reader to [32] and references therein. We will also require any agent’s marginal utility with respect to his own action to depend in a Lipschitz continuous manner on the action taken by his neighbor. In this sense we impose a qualitative bound on the strength of local interactions between different agents. More precisely, we assume that the following condition is satisfied. Assumption 2.8 The utility function u : X × X × Θ → R satisfies the following conditions: (i) The map x 7→ u(x, y, θ) is continuous and uniformly α-concave for some α > 0. (ii) The map u is differentiable with respect to its first argument, and there exists a map L : Θ → R such that ¯ ¯ ¯∂ ¯ ¯ u(x, y, θ 0 ) − ∂ u(x, yˆ, θ 0 )¯ ≤ L(θ 0 )|ˆ y − y| and such that EL(θ 0 ) < α. (4) ¯ ∂x ¯ ∂x Let us consider a simple example where our Assumption 2.8 can indeed be verified.

Example 2.9 Let α1 α2 ≥ 0 and consider a utility function of the form u(xa , xa+1 , θa ) := −α1 (xa − θa )2 − α2 (xa − xa+1 )2 .

(5)

It is easily seen that the map xa 7→ u(xa , xa+1 , θa ) is α-concave for all α ≤ 2(α1 + α2 ). Moreover, Assumption 2.8 is satisfied with L(θ 0 ) = L := 2 max{α1 , α2 } and with α := 2(α1 + α2 ). Remark 2.10 The interpretation of the condition EL(θ 0 ) < α is of interest. In the differentiable 2 case, the quantity L(θ 0 ) puts a bound on ∂ u(x,y,θ) ∂x∂y , whereas α may be viewed as a bound on ∂ 2 u(x,y,θ) . ∂x2

Thus, EL(θ 0 ) < α means that, on average, the marginal effect of the neighbor’s action on an agent’s marginal utility is smaller than the marginal effect of the agent’s own choice. It is in this sense that (4) imposes a bound on the strength of the interactions between different agents. A similar condition has also been employed to study uniqueness of equilibria in related environments by [2] (see also [3]). The Moderate Social Influence condition in [20] corresponds to the stronger contraction condition L(θ 0 ) < α. 7

We are also interested in deriving conditions which guarantee that the economy admits a Lipschitz continuous equilibrium. Lipschitz continuity of the equilibrium map may be viewed as a minimal robustness requirement on equilibrium analysis. In particular it justifies comparative statics analysis. We therefore introduce the notion of Lipschitz continuity we shall apply in our analysis. For an arbitrary constant η > 0 we define a metric dη on the product space Θ0 by ˆ := dη (θ, θ)

X

2−η|a| |θa − θˆa |

(θ = (θ a )a∈N , θˆ = (θˆa )a∈N )

(6)

a≥0

and denote by Lipη (1) the class of all continuous functions f : Θ0 → X which are non-expanding with respect to the metric dη , i.e., ˆ ≤ dη (θ, θ)} ˆ Lipη (1) := {f : Θ0 → X : |f (θ) − f (θ)| We are now ready to formulate the main result of this section. Its proof can be found in Appendix B. Theorem 2.11 Let S = (X, Θ, u, ν, N ) be a static economy with local interactions and complete information, that is with N = ∞. (i) If the utility function u : X 2 × Θ → R satisfies Assumption 2.8, then S admits a unique symmetric equilibrium g ∗ . (ii) If, instead of (4), the utility function u satisfies the stronger condition, ¯ ¯ ³ ´ ¯ ¯∂ ˆ ˆ ¯ ≤ L |ˆ ¯ u(x, y, θ) − ∂ u(x, yˆ, θ) y − y| + |θ − θ| with L < α, ¯ ∂x ¯ ∂x

(7)

then there exists η ∗ > 0 such that the unique symmetric equilibrium g ∗ is almost surely Lipschitz continuous with respect to the metric dη∗ : ˆ ≤ L dη∗ (θ, θ) ˆ |g ∗ (θ) − g ∗ (θ)| α

P-a.s..

Establishing the existence of a symmetric equilibrium is equivalent to proving the existence of a measurable function g ∗ : Θ0 → X which satisfies g ∗ (θ) = arg max u(xa , g ∗ ◦ T (θ), θ 0 ) a x ∈X

P-a.s.

(8)

Observe that each such a map is a fixed point of the operator V : B(Θ0 , X) → B(Θ0 , X) which acts on the class B(Θ0 , X) of bounded measurable functions f : Θ0 → X according to V g(θ) = arg max u(xa , g ◦ T (θ), θ 0 ). a x ∈X

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(9)

On the other hand, each fixed point of V is a symmetric equilibrium. It is therefore enough to show that V has an almost surely uniquely defined fixed point. The existence and uniqueness of equilibrium for economies of incomplete information, where an individual agent only observes a finite number N < ∞ of types, requires an additional continuity assumption on the utility function. The proof is analogous to the proof of Theorem 2.11 and is given in Appendix B. Theorem 2.12 Let S = (X, Θ, u, ν, N ) be a static economy with local interaction and incomplete information, that is with N ∈ N. (i) If the utility function u : X 2 × Θ → R satisfies Assumption 2.8 and if it is continuously differentiable with respect to its first argument, then S admits a unique symmetric equilibrium g∗. (ii) If u satisfies condition (7), then g ∗ is almost surely Lipschitz continuous: L |g ∗ (θN ) − g ∗ (θˆN )| ≤ |θN − θˆN | α

P-a.s..

Example 2.9; continued If the preferences of the agents can be described by a simple quadratic utility function of the form (5) then, under each of the different information structures we study, the existence and uniqueness of a symmetric equilibrium can easily be established. In fact, it can 2 1 and β2 := α1α+α . If the agents be solved for in closed form. To this end, we put β1 := α1α+α 2 2 have complete information, i.e., if N = ∞, then the equilibrium takes the form g ∗ (T a θN ) = β1

∞ X

β2i−a θi .

i=a

P i−a Observe that β1 ∞ = 1. Thus, in equilibrium, the action of an agent a ∈ A is given by a i=a β2 convex combination of the types θ b of the agents b ∈ {a, a + 1, a + 2, . . .}. If the agents only have incomplete information, that is, if N is finite, then ! Ãa+N X g ∗ (T a θN ) = β1 β2i−a θi + β2N +1 Eθa . i=a

We close this section with a remark on the structure of the distribution of equilibrium action profiles in economies with complete information. Remark 2.13 For a given configuration θ = (θ a )a∈A of types, let the action profile (g ∗ (T a θN ))a∈A be called an (symmetric) equilibrium action profile. Let X := X A denote the compact set of

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all action profiles and µ∗ the law of the random variable g ∗ (θ). For any continuous function F : X → R, we have Z Z ¡ ¢Y ∗ F (x)µ (dx) = F {g ∗ (θa , . . . , θa+N )}a∈A ν(dθ a ). X

Θ

a∈A

We already argued that in situations where each agent a ∈ A has complete information about the actual realization of the types θ a , a symmetric equilibrium g ∗ : Θ → X can be identified with the mapping G∗ : Θ × X → X defined by ¡ ¢ G∗ (θa , xa+1 ) = arg max u xa , xa+1 , θa . a x ∈X

ˆ We can now The map G∗ is continuous by Assumption 2.8. We denote its compact range by X. explicitly specify the conditional distribution π ∗ (xa+1 ; ·) of an agent’s equilibrium action xa , given the equilibrium action xa+1 of his neighbor. We have Z ∗ a+1 π (x ; ·) := δG∗ (θ,xa+1 ) (·)ν(dθ) (10) ˆA In particular, (10) specifies a family of conditional probability distributions (π ∗a (x; ·))a∈A on X by π ∗a (x; ·) = π ∗ (xa+1 ; ·). The law µ∗ of the action profile g ∗ (θ) may therefore be viewed as a Gibbs measure specified by the family of conditional probability distributions (10).6

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Dynamic Economies with Local Interactions

The theoretical literature on dynamic economies with local interactions has so far concentrated on models with ad hoc myopic dynamics in which an agent a ∈ A chooses his new state x at+1 in reaction to the choices xbt of some neighbors b ∈ A in the previous period. In the dynamic extension of the static economy with complete information analyzed in Section 2.1 the dynamics of the processes {xat }t∈N can be described by deterministic difference equations of the form a ) xat+1 = G∗ (θt+1 , xa+1 t

(11)

in the random environments {θta }t∈N . These dynamics describe a backward looking behavior of the agents since the new configuration (xat+1 )a∈A only depends on the current configuration 6

In his pioneering contribution, [17], F¨ ollmer considers an economy where the law of the configuration of agents’ exogenous types is a Gibbs measure. In our setup of the complete information economies with local interactions, it is instead the law of the configuration of the endogenous choices of agents that is a Gibbs measure, as e.g., in [6], [7], [10], [20].

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of types and on the previous action profile (xat )a∈A . The solution to the stochastic difference equation (11) may also be viewed as the out-of-equilibrium dynamics in a static economy if each agent myopically maximizes his individual utility under the assumption that all the other agents’ actions remain constant. One of the main reasons for the widespread use of myopic dynamics of the form (11) is that such processes have been intensively investigated in the mathematical literature on interacting particle systems. Conditions for asymptotic stability of the processes {xat }t∈N have been established under suitable weak interaction and average contraction conditions; see e.g., [30], [28] or [18]. In this paper, we instead study economies with forward looking agents and consider rational expectations equilibrium dynamics. In our economy, therefore, an agents’ actual action typically depends on his current type, on his past choices, on the present states of all the other agents and on the expected future behavior of his neighbors. His expectations with respect to the future actions of his neighbors are assumed to be rational, that is, the expectations are assumed to be consistent with the equilibrium dynamics of the neighbor’s action at each time in the future. We are now going to introduce our notion of a dynamic economy with locally interacting agents as well as our equilibrium concept. As in the case of static economies, we consider a countable set A of agents. Each agent is infinitely lived, and is of type θta ∈ Θ at time t ∈ N. Types are assumed to be distributed independently and identically across agents and time. The law of the random variables θta is denoted by ν. The instantaneous utility of agent a ∈ A at time t ∈ N depends on his current type θta , on the action xat he chooses from a common compact and convex action set X ⊂ R, and on his action xat−1 in the previous period t − 1. We maintain the simple interaction structure introduced in the previous section, and assume that the agents’ momentary utility also depends on the current action xa+1 of his neighbor, agent a + 1. His t instantaneous preferences at time t are described by a bounded and continuous utility function ¡

¢ ¡ ¢ xat , xat−1 , xa+1 , θta → 7 u xat , xat−1 , xa+1 , θta t t

(12)

¡ ¢ where the map (xat , xat−1 ) 7→ u xat , xat−1 , xa+1 , θta is assumed to be strictly concave. An agent’s t overall utility is the sum of future expected utilities, discounted at a rate β < 1. Prior to his choice at time t, the agent a ∈ A observes the realization of his own type θ ta , and the entire action profile xt−1 = (xat−1 )a∈A of the previous period t − 1. In this sense, we focus on economies with incomplete information, in which each agents do not observe the type of any other agent except their own. This is just for notational and analytical convenience. Definition 3.1 A dynamic economy with local interactions is a tuple S = (X, Θ, u, ν, β). In order to describe an individual agents’ optimization problem, we need to introduce some notation. We denote by X := {x = (xa )a∈A : xa ∈ X} the space of all configurations of individual actions and put X0 := {x = (xa )a≥0 }. We equip the configuration spaces X and X0 with the 11

product topologies, so compactness of the individual action spaces implies compactness of X and X0 . At each time t, the agent a ∈ A observes his present type θta as well as the entire action profile xt−1 ∈ X of the previous period. In particular, he observes the past actions xbt−1 of all the agents b > a with whom he interacts either directly or indirectly. Even though his instantaneous utility only depends on present and expected future actions of his neighbor, the information contained in all the choices (xa+c t−1 )c≥1 is relevant for the solution to his decision problem. In fact, the choice a+c xt−1 of the agent a + c at time t − 1 effects the action of agent a + c − 1 in period t; this has an impact on the action of agent a + c − 2 in period t + 1, and so on. A rational agent a ∈ A anticipates these effects, and so he bases his current decision on all the states (x bt−1 )b≥a . As in the static case we shall focus on symmetric situations. Thus, we may assume that the optimal action of an economic agent a ∈ A is determined by a choice function g : X 0 × Θ → X in the sense that xat = g(T a xt−1 , θta ) where T a xt−1 = {xbt−1 }b≥a . In a symmetric situation, it is thus enough to analyze the optimization problem of a single reference agent, say of the agent 0 ∈ A. Given the action profile xt−1 = (xbt−1 )b≥0 ∈ X0 of the agents b ≥ 0 in the previous period and a continuous choice function g : X0 × Θ → X, the agent a ≥ 0 takes as given his neighbor’s current choice g(T a xt−1 , θta ). We denote by πg (T a xt−1 ; ·) the conditional law of the action xat , given the previous configuration xt−1 , and so the choice function g : X0 × Θ → X induces the Feller kernel Πg (x; ·) :=

∞ Y

πg (T a x; ·).

(13)

a=1

If the agent 0 ∈ A believes that the agents a > 0 choose their actions according to g, the kernel Πg describes the stochastic evolution of the process of individual states {(x at )a>0 }t∈N . In this case, for any initial configuration of individual states x ∈ X0 and for each initial type θ10 , agent 0’s optimization problem is given by   Z Z  X u(x01 , x0 , x11 , θ10 )πg (T x; dx1 ) + β t−1 u(x0t , x0t−1 , x1t , θt0 )Πtg (T x; dxt )ν(dθt0 ) max (14)  {x0t }t∈N  t≥2

The value function associated with this dynamic choice problem is defined by the fixed point of the functional equation ½Z ¡ ¢ 0 0 0 Vg (xt−1 , θt ) = Vg (xt−1 , T xt−1 , θt ) = max u x0t−1 , x0t , yt1 , θt0 πg (T xt−1 ; dyt1 ) (15) x0t ∈X ¾ Z 0 1 1 +β Vg (xt , x ˆt , θ )Πg (T xt−1 ; dˆ xt )ν(dθ ) X0 ×Θ

The following is a well known result from the theory of dynamic programming, see e.g., [36]. 12

Lemma 3.2 Assume that the choice map g is continuous. Under our assumptions on the utility function u, the functional fixed point equation (15) has a unique bounded and continuous solution Vg on X0 × Θ. Moreover, the map Vg (·, T xt−1 , θt0 ) is strictly concave on X and there exists a unique continuous policy function gˆg : X0 × Θ → X that satisfies ½Z ¢ ¡ ¢ ¡ u x0t−1 , x0t , yt , θt0 πg (T xt−1 ; dyt ) gˆg xt−1 , θt0 = arg max x0t ∈X ¾ Z 0 1 1 +β Vg (xt , x ˆt , θ )Πg (T xt−1 ; dˆ xt )ν(dθ ) . (16) We can now define a symmetric Markov equilibrium in a dynamic random economy with forward looking interacting agents. Definition 3.3 A symmetric Markov equilibrium of a dynamic economy with forward looking and locally interacting agents S = (X, Θ, u, ν, β), is a map g ∗ : X0 × Θ → X such that ½Z ¡ ¡ ¢ ¢ ∗ 0 g xt−1 , θt u x0t−1 , x0t , yt , θt0 πg∗ (T xt−1 ; dyt ) = arg max (17) x0t ∈X ¾ Z 1 0 1 ∗ ∗ xt )ν(dθ ) . ˆt , θ )Πg (T xt−1 ; dˆ +β Vg (xt , x By analogy to the static case, a symmetric equilibrium might be viewed as a fixed point of a certain operator. Indeed, every fixed point of the operator Vb that acts on the class of bounded measurable functions g : X0 × Θ → X by ½Z ¡ 0 0 ¢ Vb g(x, θ 0 ) = arg max u x ˆ , x , yt , θ0 πg (T x; dy) (18) x ˆ0 ∈X ¾ Z 0 1 1 +β Vg (ˆ x ,x ˆ, θ )Πg (T x; dˆ x)ν(dθ ) , defines a symmetric equilibrium; and every symmetric equilibrium g ∗ satisfies the fixed point relation Vb g ∗ (x, θ0 ) = g ∗ (x, θ0 ).

The analysis of equilibria of dynamic economies with local interactions is in general complex. In fact, it is a-priori unclear whether there is any compact set of functions which Vˆ maps continuously into itself. If such a set exists, the existence of a symmetric equilibrium follows from the theorem by Arzela and Ascoli along with Brower’s fixed point theorem. We proceed by establishing a series of general results, on existence and on the convergence of the equilibrium process. Such results require conditions on the policy function gˆg , and hence are not directly formulated as conditions on the fundamentals of the economy. In the next section we will then introduce an example economy with quadratic preferences which we are able to study in detail. For this economy we can show our general conditions are satisfied, and hence they are not vacuous. 13

3.1

Existence and Lipschitz Continuity of Equilibrium

In order to state a general existence result for equilibria in dynamic random economies with forward looking interacting agents we need to introduce the notion of a correlation pattern. Definition 3.4 For C > 0, let LC + := {c = (ca )a∈N : ca ≥ 0,

X

ca ≤ C}

a∈A

denote the class of all non-negative sequences whose sum is bounded from above by C. A sequence c ∈ LC + will be called a correlation pattern with total impact C. Each correlation pattern c ∈ LC + gives rise to a metric dc (x, y) :=

X

ca |xa − y a |

a∈N

that induces the product topology on X0 . Thus, (dc , X0 ) is a compact metric space. In particular, the class 0 LipC c := {f : X → R : |f (x) − f (y)| ≤ dc (x, y)} of all functions f : X0 → R which are Lipschitz continuous with constant 1 with respect to the metric dc is compact in the topology of uniform convergence. Remark 3.5 For a fixed θ 0 ∈ Θ, let g(·, θ 0 ) ∈ LipC c be the policy function of the agent 0 ∈ A. The constant ca may be viewed as a measure for the total impact the current action xa of the agent a ≥ 0 has on the optimal action of agent 0 ∈ A. Since C < ∞, we have lim a→∞ ca = 0. Thus, the impact of an agent a ∈ A on the agent 0 ∈ A tends to zero as a → ∞. In this sense we consider economies with weak social interactions. The quantity C provides an upper bound for the total impact of the configuration x = (xa )a≥0 on the current choice of the agent 0 ∈ A. We are now going to formulate a general existence result for symmetric Markov perfect equilibria in dynamic economies with local interaction. Theorem 3.6 Assume that there exists C < ∞ such that the following holds: C 0 0 (i) For any c ∈ LC + , for all θ ∈ Θ and for each choice function g(·, θ ) ∈ Lipc , there exists ˆg (·, θ 0 ) which solves (16), is Lipschitz F (c) ∈ LC + such that the unique policy function g continuous with respect to the metric dF (c) uniformly in θ 0 ∈ Θ. C (ii) The map F : LC + → L+ is continuous.

(iii) We have limn→∞ kˆ ggn (·, θ 0 ) − gˆg (·, θ 0 )k∞ = 0 if limn→∞ kgn − gk∞ = 0. 14

Then the dynamic economy with local interactions has a symmetric Markov perfect equilibrium g ∗ and the function g ∗ (·, θ 0 ) is Lipschitz continuous uniformly in θ 0 . Proof: Let us first show that, for any C < ∞, the convex set LC + may be viewed as a closed N subset of [0, C] with respect to the product topology. To this end, let {cn }n∈N be a sequence in N LC + that converges to c = (ca )a≥0 ∈ [0, C] in the product topology. Clearly, ca ≥ 0 for all a ∈ N, P e e := and so the sum C a≥0 ca exists by monotone convergence. If C > C, then there exists b ∈ N and ² > 0 such that b b X X X lim cna ≤ lim cna ca = C +²≤ a=0

a=0

n→∞

n→∞

a≥0

which contradicts cn ∈ LC +. Due to (i) and (iii) the operator Vˆ defined by (18) maps the compact and convex set LipC c∗ ∗ continuously into itself and therefore does also have a fixed point g . 2

3.2

Convergence to a Steady State

In the previous section we have formulated conditions on a dynamic economy with local interactions S = (X, u, β, Θ, ν) which guarantee the existence of a symmetric Markov perfect equilibrium g ∗ . In this section we study the asymptotic behavior of the process {xt }t∈N in equilibrium. To this end, we denote by Y πg∗ (T a x; ·) Πg∗ (x; ·) = a∈A

the stochastic kernel on X induced by the policy function g ∗ and by Πtg∗ its t-fold iteration. Given an initial configuration x ∈ X, the measure Πt (x; ·) describes the distribution of the configuration of individual states at time t. Let us introduce the vector r ∗ = (ra∗ )a∈A with components ra∗ := sup{kπg∗ (x; ·) − πg∗ (y; ·)k : x = y off a}.

(19)

Here, kπg∗ (x; ·) − πg∗ (y; ·)k denotes the total variation of the signed measure πg∗ (x; ·) − πg∗ (y; ·), and x = y off a means that xb = y b for all b 6= a. The next theorem gives sufficient conditions for convergence of the equilibrium process to a steady state. Its proof follows from a fundamental convergence theorem by Vasserstein (1969). P Theorem 3.7 If a∈A rga∗ < 1, then there exists a unique probability measure µ∗ on the infinite configuration space X such that, for any initial configuration x ∈ X, the sequence Π tg∗ (x; ·) converges to µ∗ in the topology of weak convergence for probability measures.

15

4

Dynamic Economies with Local Interactions: Quadratic Utilities

In this section we analyze an example with quadratic utility functions where the assumption of Theorem 3.6 and of Theorem 3.7 can indeed be verified. Theorem 4.1 Let X = Θ = [−1, 1], and assume that Eθt0 = 0, and that an agent a ∈ A only observes his own type θ a . If the instantaneous utility function takes the quadratic form ¢2 ¡ ¢2 ¡ ¢ ¡ − xat u xat−1 , xat , xa+1 , θta = −α1 xat−1 − xat − α2 (θta − xat )2 − α3 xa+1 t t

(20)

for positive constants α1 , α2 and α3 , then the economy has a symmetric Markov perfect equilibrium g ∗ . The policy function g ∗ can be chosen to be of the linear form g ∗ (x, θ0 ) = c∗0 x0 + γθ 0 +

X

c∗b xb

b≥1

for some positive sequence c∗ = (c∗a )a≥0 and some constant γ > 0. The proof of Theorem 4.1 will be carried out in several steps. In a first step, we are now going to establish the existence of an interior solution to an agent’s optimization problem in an economy with quadratic utility functions. Lemma 4.2 Let g : X0 × Θ → X be a continuous choice function for the agents a > 0. Under the assumptions of Theorem 4.1, the induced policy gˆg function of the agent 0 ∈ A is uniquely determined and ¡ ¢ P gˆg (xt−1 , θt0 ) ∈ {−1, 1} for some t ∈ N = 0. (21) Thus, we have almost surely an interior solution.

Proof: The existence of a unique policy function follows from continuity of g along with the quadratic form of the utility functions using standard arguments from the theory of discounted dynamic programming. In order to prove (21), we put © ª τ := inf t > 0 : gˆg (xt−1 , θt0 ) = 1

and

yt := gˆg (xt−1 , θt0 ).

It suffices to show that P[τ < ∞] = 0. Let us assume to the contrary that P[τ < ∞] > 0. In such a situation yτ = 1 is optimal and this means that ¡ ¢ −α1 (1 − yτ −1 )2 − β α1 (1 − yτ +1 )2 + α2 (1 − x1τ )2 + α3 (1 − θτ0 )2 ¡ ¢ ≥ −α1 (y − yτ −1 )2 − β α1 (y − yτ +1 )2 + α2 (y − x1τ )2 + α3 (y − θτ0 )2 16

for all y ∈ X. Otherwise yτ < 1 would lead to a higher payoff. This, however, requires θτ0 = yτ −1 = yτ +1 = 1. This shows that yt = 1 = θt0 for all t ∈ N. This, of course, contradicts Eθt0 = 0. Thus, P[τ < ∞] = 0. 2 Let us now establish a first representation of the agents’ policy function. To this end, we denote by M(X0 ) the class of all probability measures on X0 equipped with the topology of weak convergence. The utility of the agent 0 ∈ A at time t ∈ N depends on the actions x at taken by the agents a > 0 only through his neighbor’s expected action Z zt := y 1 Πg (T xt ; dy)

R and through (y 1 )2 Πg (T xt ; dy). We may thus view the agent’s dynamic problem as an optimization problem depending only on the stochastic sequence {θt0 }t∈N and on the deterministic © ª sequence Πtg (T x; ·) t∈N . In fact, in our present setting we can, for any initial configuration x ∈ X0 , put µ(·) := Πg (T x; ·) and rewrite his optimization (14) as   Z   X (22) U (x01 , x00 , θ10 , µ) + β t−1 U (x0t , x0t−1 , θt0 , µΠtg )ν(dθt0 ) max  {x0t }t∈N  t≥2

where

U (x01 , x00 , θ0 , µ)

:=

−α1 (x01

−

x00 )2

−

α1 (x01

0 2

− θ ) − α3

Z

(x01 − y 1 )2 µ(dy).

This allows us to show that the agent’s optimal action is given as a weighted sum of his present type, of his action taken in the previous period and of the expected future actions of his neighbor. Lemma 4.3 Assume that the assumptions of Theorem 4.1 are satisfied. Given an action profile x ∈ X0 and a choice function g : X0 × Θ → X for the agents a > 0, the policy function of agent 0 ∈ A is of the linear form Z X 0 0 gˆg (x, θ) = γ1 x + γ2 θ + δt−1 y 1 Πtg (T x; dy). (23) t≥1

With λ := α1 + α2 + α3 + α1 β the constants γ1 , γ2 , δ0 , δ1 , . . . are given by p λ − λ2 − 4α12 β α2 γ1 := , and γ2 := , 2α1 β λ − γ 1 α1 β and by δ0 :=

α3 λ − γ 1 α1 β

and

δt+1 =

α1 β δt λ − γ 1 α1 β

for t ≥ 1.

The constants in (24) and (25) do not depend on g and satisfy γ1 + γ2 +

17

(24)

P

t≥0 δt

(25) ≤ 1.

Proof: Let us fix an initial configuration x = (xa )a≥0 and put µ := Πg (T x; ·). The value function associated with the optimization problem (22) solves the functional fixed point equation ½ Z 0 0 2 0 0 2 0 0 Vg (x0 , θ1 , µ) = max −α1 (x0 − x1 ) − α2 (θ1 − x1 ) − α3 (y 1 − x01 )2 µ(dy) x01 ∈X

+β

Z

Vg (x01 , θ20 , µΠg )ν(dθ20 )

¾

.

(26)

In view of Lemma 4.2 the fixed point equation (26) has a unique solution Vg∗ : X ×Θ×M(X0 ) → R and the agent’s policy function gˆg : X × Θ × M(X0 ) → X is uniquely determined and the optimal solution is almost surely interior. Thus, the first order condition takes the form Z Z ∂ ∗ 0 0 1 0 0 0 0 0 2 −2α1 (x0 − x1 ) − 2α(θ1 − x1 ) − 2α3 (y − x1 )µ(dy) + β V (x , θ , µΠg )ν(dθ20 ) = 0, ∂x01 g 1 2 and the envelope theorem gives us

This yields x01

¢¢ ¡ ¡ ∂ V (x01 , θ20 , µΠg ) = −2α1 (x01 − x02 ) = −2α1 x01 − gˆg x01 , θ20 , µΠg . 0 ∂x1

1 = α1 + α 2 + α 3 + β 1 α1

µ

α1 x00

+

α2 θ10

+ α3

Z

1

y µ(dy) + α1 β

¡

gˆg x01 , θ20 , µΠg

(27)

¶ ¢

.

(28)

Let us now assume that we have the following alternative representation for the optimal path {x0t }t∈N : ∞ X x0t = γ1 x0t−1 + γ2 θt0 + δi zt+i ∈ (0, 1) (29) i=0

where zt denotes the expected action of the agent a = 1 at time t. Using Eθt0 = 0, it does then follow from the first order condition, from (27) and from (28) that ! Ã Z ∞ X 1 0 1 0 0 0 x1 = z2+i . (30) α1 x0 + α2 θ1 + α3 y Πg (x; dy) + α1 βγ1 x1 + α1 β α1 + α 2 + α 3 + β 1 α1 i=0

Now we need to find coefficients γ1 , γ2 , δ0 , δ1 , . . . such that the representations in (29) and in (30) coincide. This can be accomplished recursively and yields the constants in (24) and (25). 7 Notice, however, that we have not yet shown that the sum of the coefficients is bounded from above by 1. In order to prove this, we consider the situation in which the agents maximize the discounted sum of their expected utilities over the periods t ∈ {0, 1, . . . , τ } and denote by g τ (x, θ0 ) the optimal action of the agent 0 ∈ A. Using a cumbersome, but rather straightforward 7

The solution method for linear rational expectation models in [5] cannot be applied in a context, as ours, with an infinite dimensional state space.

18

induction argument along with an argument similar to the one given in the proof of Lemma 4.2 one can easily show that τ X τ 0 τ 0 τ 0 τ g (x, θ ) = γ1 x + γ2 θ + δi−1 zi . i=1

Here, the coefficients satisfy the recursive relations γiτ =

αi λτ

(i = 1, 2),

δ0τ =

α3 , λτ +1

δiτ =

α1 β τ −1 δ λτ i−1

(i = 1, 2, . . .)

and

λτ +1 = λ −

α12 β λτ

with λ0 = α1 + α2 + α3 . This shows that γiτ → γi and δiτ → δi for all i = 0, 1, 2, . . . as τ → ∞. Thus, X X γ1 + γ 2 + δi ≤ 1 because γ1τ + γ2τ + δiτ ≤ 1 for all τ. i≥0

i≥0

2

Our representation (23) of the policy function does not yet allow us to apply Theorem 3.6. For this we need a representation of gˆg in terms of the sequence (xa )a≥0 . This, however, can be 2 accomplished as follows: Let us fix a correlation pattern c = (ca )a≥1 ∈ L1−γ and assume for the + moment that the choice function of the agents a > 0 takes the linear form g˜(T a x, θa ) = c0 xa + γ2 θa +

X

cb xa+b .

(31)

b≥1

In view of (23), we have c0 = γ1 and that continuous choice function g˜ induces a Feller kernel Πg˜ on X0 . Thus, it follows from (31) and from Eθt0 = 0 that Z X y 1 Πg˜(x; dy) = ca xa+1 . a≥0

Thus, the expected action of the agent a = 1 in the second period is given by Z Z X X X ca2 xa1 +a2 +1 , ca 1 y 1 Πg2˜(x; dy) = ca1 y a+1 Πg˜(x; dy) = a1 ≥0

a1 ≥0

a2 ≥0

and a simple induction argument shows that     Z X X X  ca 1 ca2 · · · cat−1 y 1 Πgt˜(x; dy) = cat xa1 +···+at +1  · · ·  a1 ≥0

a2 ≥0

at ≥0

for all t ∈ N. This yields the following alternative representation for our policy function: gˆg˜(x, θ) = γ1 x0 + γ2 θ +

X b≥1

19

lb xb

(32)

where the strictly positive sequence (lb )b≥1 is given by ! ! b−1 Ã b−1 Ã b−1 X X X X c a t 1{ P t ca 2 · · · lb = Fb (c0 , c1 , . . . , cb−1 ) := δt−1 ca 1 t≥1

at =0

a2 =0

a1 =0

i=1

ai =b−1} .

(33)

We are now ready to prove Theorem 4.1. P Proof of Theorem 4.1: Since gˆg˜(x, θ0 ) ∈ X, we have b≥1 lb ≤ 1 − γ1 − γ2 . Thus, the map F defined by F (c) := (Fb (γ1 , c1 , . . . cb−1 ))b≥1 (34) 1−γ1 −γ2 maps the set L+ into itself. Since F is continuous in the product topology, it has a fixed ∗ ∗ point c = (ca )a≥1 and

lb = Fb (γ1 , c∗1 , . . . , c∗b−1 ) = c∗b

for all b ≥ 1.

Thus, the assumptions of Theorem 3.6 are satisfied and this proves the assertion.

2

We turn now to study the convergence to a unique steady state for the example economy with quadratic preferences. Consider the representation X g ∗ (x; θ0 ) = c∗0 x0 + γ2 θ0 + c∗a xa . a≥1

of the policy function g ∗ . For any two configurations x, y ∈ X0 which differ only at site a ∈ A we have |g ∗ (x, θ0 ) − g ∗ (y, θ0 )| ≤ c∗a |xa − y a |, Thus, assuming that the taste shocks are uniformly distributed on [−1, 1] we obtain |πg∗ (x; A) − πg∗ (y; A)| ≤ 2c∗a P P for all A ∈ B([−1, 1]), and so a≥0 rga∗ < 1 if a≥0 c∗a < 12 . Hence in our quadratic case study, we obtain convergence to a steady state whenever α1 is big enough and if α3 is small enough, i.e., if the interaction between different agents is not too strong.

5

Dynamic Economies with Local and Global Interactions

This section extends the analysis of dynamic economies with local interactions by adding a global interaction component in the utility functions of the agents. Our main goal is to demonstrate that, in a well-specified sense, the analysis of economies with locally and globally interacting agents can be reduced to the analysis of economies with purely locally interacting agents which we have developed in Section 3 and 4. 20

Let %(xt ) denote the average actions associated with the configuration x t : n X 1 %(x) := lim xa n→∞ 2n + 1 a=−n

To model global interactions, we shall assume that the instantaneous utility of an individual agent depends on his current type, on his previous action, on the present action taken by his neighbor, as in Section 3, and on the actual average action throughout the entire population: ¡ a a ¢ ¡ ¢ xt , xt−1 , xa+1 , θta , %(xt ) 7→ u xat , xat−1 , xa+1 , θta , %(xt ) (35) t t Note that the agent’s utility function depends on the action profile x ∈ X in a global manner through the average action %(x). Therefore, it will typically not be continuous in x in the product topology.8 Thus, standard results from the theory of discounted dynamic programming cannot be applied to solve the agent’s dynamic optimization problem induced by instantaneous preferences represented by (35). We will instead separate the impact of the global and local components on each agent’s preferences. If the interaction between different agents is not too strong, in fact, the aggregate dynamics of the configuration of actions evolve in a deterministic manner and can be described through a suitable deterministic difference equation. This allows us to reduce situations where we have an additional global component in the interaction to an inhomogeneous situation with purely locally interacting agents.9 In order to carry out our separation argument, we recall the notion of an ergodic empirical field. We also need to introduce the law of large numbers of empirical fields established by Horst (2002); see also F¨ollmer-Horst (2001). We will then proceed with the analysis of dynamic economies with local and global interactions for the special case in which each agent’s utility function is quadratic. While we have not attempted to derive more general results, we conjecture this to be possible along the lines of Section 3. Definition 5.1 A probability measure µ on the infinite product space X is called ergodic, if µ is invariant under the shift maps (T a )a∈A and if it satisfies a 0-1-law on the σ-field J := σ{B ∈ F : T a B = B

for all a ∈ A}

of all shift invariant events: µ(B) ∈ {0, 1}

for all B ∈ J .

The class of all ergodic probability measures µ on X is denoted by Me (X). 8

A function f : X → R is continuous in the product topology if and only if it can be uniformly approximated by continuous functions which depend on finitely many coordinates only. 9 Such separation arguments originally appeared in [18] and [24] where the long run dynamics of locally and globally interacting Markov chains is analyzed. Similar separation arguments have also been successfully applied in the context of a static equilibrium model with locally and globally interacting agents in [25].

21

We denote by Xe the set of all configurations x ∈ X such that the empirical field R(x), defined as the weak limit n X 1 δT a x (·), (36) R(x) := lim n→∞ 2n + 1 a=−n exists and belongs to Me (X). Thus, for any continuous function f : X → R we have Z

n X 1 1 f (T a x). f (y)R(x)(dy) = lim n→∞ 2n + 1 a=−n 2n + 1 X

(37)

In particular, choosing the function f (y) = y 0 in (37) we see that the average action associated to x is given by Z Z n X 1 a 0 x = y R(x)(dy) = y a R(x)(dy), %(x) := lim n→∞ 2n + 1 X X a=−n where we use spatial homogeneity of the empirical field. It is well known (see, e.g., [19]) that the empirical field R(x) associated with a configuration x ∈ Xe is concentrated on Xe . Remark 5.2 The empirical field R(x) will represent the aggregate state variable of our dynamic economy with local and global interactions. It might be viewed as a probability measure on the configuration space that carries more macroscopic information about the configuration x = (xa )a∈A ∈ Xe than just the associated empirical distribution or the average action. For instance, R(x) also contains information about the average behavior of k + 1 neighboring agents: Z n X 1 a a+1 a+k f (x , x , . . . , x ) = f (y 0 , y 1 , . . . , y k )R(x)(dy) lim n→∞ 2n + 1 a=−n whenever f is continuous. (Observe that this information is not contained in the average action %(x).) The dynamics of the process {R(xt )}t∈N can in our set-up be described by deterministic recursive difference equations; see (38). In models with local interactions, such a recursive relation typically fails to hold for average actions %(x). Thus, in contrast to the empirical field, the average action typically is not an appropriate state variable, a sufficient statistic, for the aggregate behavior of the configuration x. Only in binary choice mean field models with a countably infinite set of agents is the average action at time t + 1 a function of the average action in the previous period t. We refer the reader to [18] and [24] for a more detailed discussion of the role of empirical fields in the analysis of locally and globally interacting stochastic systems. We can now state the version of the law of large numbers for ergodic empirical fields which turns out to be the key to the analysis of dynamic economies with locally and globally interacting agents; see [24], and [18]. It might be viewed as a generalization of Kolmogorov’s law of large numbers for independent and identically distributed random variables and . 22

Lemma 5.3 For any ergodic probability measure µ on X let Πµ (x; ·) =

Y

πµ (T a x; ·)

a∈A

be a Feller kernel on the product space X. Then the following holds: (i) For all x ∈ Xe , the empirical field R(y) exists for Πµ (x; ·)-a.e. y ∈ X, is ergodic and satisfies Z R(y)(·) = R(x)Πµ = Πµ (y; ·)R(x)(dy) Πµ (x; ·)-a.s.

Thus, if {xt }t∈N denotes the Markov chain on X with transition kernel Πµ and if x0 ∈ Xe , then it holds almost surely that R(xt ) = R(xt−1 )Πµ = R(x0 )Πt

(38)

(ii) Let Π(x; ·) := ΠR(x) (x; ·). For x ∈ Xe we can define a sequence of empirical fields {Rtx }t∈N by x R0x = R(x) and Rt+1 := R0x Πt , and this sequence satisfies x Rt+1 = Rtx ΠRtx = R0x ΠR0x · · · ΠRtx

5.1

(t = 0, 1, 2, . . .)

(39)

Existence of an Equilibrium for the Example with Quadratic Utilities

We are now ready to state the following extension of Theorem 4.1 to situations in which we have an additional global component in the agents’ utility functions. Theorem 5.4 Let X = Θ = [−1, 1]. Assume that Eθ0 = 0 and that, for a configuration of individual actions xt ∈ Xe , the utility function u takes the quadratic form ¡ ¢ ¡ ¢2 ¢2 ¡ u xat−1 , xat , xt , θta = −α1 xat−1 − xat − α2 (θta − xat )2 − α3 xa+1 − xat − α4 (%(xt ) − xat )2 (40) t

for some positive constants α1 , . . . , α4 . If α4 is small enough, then the economy has a symmetric Markov perfect equilibrium g ∗ : Xe × Θ → X. The policy function g ∗ induces a stochastic kernel Π on Xe , and so the sequence of average actions %(xt ) exists almost surely.10 10

The notion of Markov perfect equilibrium we use for economies with local and global interactions is as in Definition 3.3, once it is postulated that an individual agent does not take,into account, his impact on the evolution of the average action throughout the entire population.

23

In our economy with local and global interactions, the instantaneous utility function of an individual agent depends on the entire action profile x ∈ X in a global manner through the average action %(x). As we argued earlier, it will thus typically not be continuous in the product topology, and so we proceed by separating the impact of the global components. To this end, we fix a continuous function g : Me (X) × X0 × Θ → X and, for any configuration x = (xa )a∈A and for each a ∈ A, we denote by πµ (T a x; ·) the law of the random variable g(µ, T a x, θa ). Since g is continuous, the stochastic kernel Y b µ (x; ·) := Π πµ (T a x; ·) a∈A

on X has the Feller property. For any ergodic probability measure µ ∈ Me (X) let us define the b tµ . By the law of large numbers for ergodic empirical deterministic sequence {µt }t∈N by µt := µΠ fields we have Z n X 1 0 xa µt -a.s. %(µt ) := x dµt = lim n→∞ 2n + 1 a=−n

and so %(µt ) may be viewed as the average action associated with the ergodic measure µ t . Moreover, the map µt 7→ %(µt ) from M(Xe ) to X is continuous. In a first step towards proving Theorem 5.4, we are going to prove the existence of a symmetric Markov perfect equilibrium in an auxiliary economy with purely locally interacting agents, whose instantaneous utility also depends on an exogenous deterministic process {%(µt )}. The following result will be proved below. Proposition 5.5 Consider an economy with purely local interactions in which X = Θ = [−1, 1] and where Eθ 0 = 0. Let α1 , . . . , α4 be positive constants and assume that the agents’ instantaneous preferences can be described by a continuous utility function Ug : Me (X) × X 3 × Θ → R of the form ¢2 ¡ ¢2 ¡ ¢ ¡ − xat −α4 (%(µt ) − xat )2 . , θta = −α1 xat−1 − xat − α2 (θta − xat )2 −α3 xa+1 Ug µt , xat−1 , xat , xa+1 t t Then the economy admits a symmetric Markov perfect equilibrium if α 4 is small enough. That is, for all sufficiently small α4 , there exists a continuous function g ∗ : Me (X) × X0 × Θ → X such that the policy function gˆg∗ of the agent 0 satisfies gˆg∗ (µ, x, θ 0 ) = g ∗ (µ, x, θ 0 ). The proof of Proposition 5.5 requires some preparation. We fix an initial configuration x = a≥0 and introduce the Feller kernel Y Πµ (T x; ·) := πµ (T a x; ·)

(xa )

a≥1

on X0 . In our present setting, the utility of the agent a = 0 depends on x = (xa )a≥0 only © ª through x0 and through the deterministic sequence Πtµ (T x; ·) . It also depends on the exogenous 24

deterministic sequence {%(µt )}. We may therefore proceed as in the purely local case in order to prove that the value function Vµ∗ associated with an agent’s optimization problem is uniquely determined, and that there exists a unique corresponding policy function gˆg : Me (X) × X0 × Θ → X. More precisely, the same arguments that lead to the proof of Lemma 4.3 yield the following result. Lemma 5.6 Assume that X = Θ = [−1, 1] and that Eθt0 = 0. Let µ ∈ Me (X) be an ergodic probability measure and let g : Me (X) × X0 × Θ → X be a continuous choice function. Then the policy function gˆg : Me (X) × X0 × Θ → X of the agent a = 0 takes the linear form ¾ Z X½ (41) gˆg(µ,·) (µ, x, θ) = γ1 x0 + γ2 θ0 + δt−1 y 1 Πtµ (T x; dy) + ηt−1 %(µt ) t≥1

for suitable strictly positive constants γ1 , γ2 , δt and ηt satisfying γ1 + γ2 +

P

t≥1 (δt

+ ηt ) ≤ 1.

Now our aim is to derive a representation of the policy function in (41) in terms of the actions 0 a≥0 , the agent’s type θ and in terms of the ergodic probability measure µ. To this end, we proceed by analogy to the local case analyzed in Section 3 with the sole difference that the policy function will contain an additional term that depends on the average action associated with the ergodic measure µ. Suppose that the agent 0 ∈ A assumes that X cb xa+b + B(µ0 ) g(µ, T a x, θa ) = γ1 x0 + γ2 θ0 + (xa )

b≥1

for some non-negative sequence {ca }a≥1 and for some constants B(µ0 ). Here µ0 := By (41) we have X 1 − γ2 > γ1 and so ca ≤ 1 − γ1 − γ2 − B(µ0 ).

R

x0 µ(dx).

a≥1

The policy function gˆg(µ,·) in (41) then takes the form X X gˆg(µ,·) (µ, x, θ) = γ1 x0 + γ2 θ0 + la xa + B(µ) δt C t + A(C, B(µ0 ), µ). a≥1

t≥1

P Here the constants la = Fa (γ1 , c1 , . . . , ca−1 ) are given by (33), C := γ1 + a≥1 ca , and Z X X ηt−1 C t . A(C, B(µ0 ), µ) := ηt−1 y 1 (µΠtµ )(dy) = (B(µ0 ) + µ0 ) t≥1

t≥1

The last equality follows from (32) because any ergodic probability measure µ on X satisfies R 0 R x µ(dx) = xa µ(dx) for all a ∈ A. Thus, in order to prove the existence of a symmetric Markov perfect equilibrium, we need to find a strictly positive correlation pattern c ∗ = (c∗a )a≥1 and a constant B(µ0 ) such that c∗a = Fa (γ1 , c∗1 , . . . , c∗a−1 ) 25

for all a ≥ 1

and such that B(µ0 ) = A(C ∗ , B(µ0 ), µ) + B(µ0 )

X

ηt−1 (C ∗ )t

t≥1

where C ∗ := γ1 +

X

c∗a .

a≥1

To this end, we need to prove the following result.

Lemma 5.7 If α4 > 0 is not too big, then, for all µ0 ∈ [0, 1], the system of equations ´X ³ X ˆ 0 )t = B(µ ˆ 0) ˆ 0 ) + µ0 ˆ 0 ) + γ2 + Cˆ = 1 and B(µ ˆ 0) ηt C(µ B(µ δt Cˆ t + B(µ

(42)

t≥1

t≥1

ˆ 0 ), C(µ ˆ 0 )) with C(µ ˆ 0 ) ∈ (γ1 , 1 − γ2 ], and C(µ ˆ 0 ) ≥ C(1) ˆ ˆ has a unique solution (B(µ =: C. ˆ 0 ), C(µ ˆ 0 )) with C(µ ˆ 0 ) ≤ 1 − γ2 follows from Proof: The existence of a unique solution (B(µ P ˆ 0 ) ≥ C(1), ˆ holds because 1 − t≥0 (δt + ηt ) > 0 along with µ0 ≥ 0. The estimate C(µ µ0 7→ µ0

1−

P

P

ˆt t≥0 ηt C ˆ t P δt Cˆ t t≥0 t≥0 ηt C −

ˆ is an increasing function for all Cˆ ≤ 1. In order to show that C(1) > γ1 observe that for α4 = 0 ˆ ˆ ˆ we have ηt = 0, and so (B(1), C(1)) = (0, 1 − γ2 ) solves (42). C(1) > γ1 for α4 > 0 follows from the continuous dependence of the constants ηt on the parameter α4 . 2 We are now ready to prove Proposition 5.5. We shall proceed very much by analogy with the proof of Theorem 4.1. Our proof is based on the general existence result for economies with purely locally interacting agents formulated in Theorem 3.6 and on Lemma 5.7. Proof of Proposition 5.5: In order to prove the existence of a symmetric Markov perfect equilibrium for our economy, we now need to find a correlation pattern c = (c a )a≥1 and a constant B(µ0 ) such that X ca = Fa (γ1 , c1 , . . . , ca−1 ) and such that B(µ0 ) = A(C, B(µ0 ), µ) + B(µ0 ) ηt C t . t≥1

To this end, we may with no loss of generality assume that µ(x0 ) ≥ 0. The case µ(x0 ) < 0 follows from similar considerations. (i) Let µ be an ergodic probability measure on X. By Lemma 5.7 there exists a unique pair ˆ 0 ), C(µ ˆ 0 )) that satisfies C(µ ˆ 0 ) ∈ (γ1 , 1 − γ2 ] as well as (B(µ P t ˆ t≥1 ηt−1 C(µ0 ) ˆ ˆ ˆ B(µ0 ) + γ2 + C(µ0 ) = 1 and B(µ0 ) = µ0 . P P t ˆ ˆ 0 )t − 1 − t≥1 ηt−1 C(µ t≥0 δt C(µ0 ) 26

ˆ 0 ), C(µ ˆ 0 )), we obtain For the pair (B(µ gˆg (µ, x, θ0 ) = γ1 x0 + γ2 θ0 +

X

ˆ 0 ). la xa + B(µ

a≥1

Since gˆg (µ, x, θ0 ) ∈ [−1, 1] almost surely, we also have ˆ 0 )−γ1 C(µ

F (c) = {Fa (γ1 , c1 , . . . , ca−1 )}a≥1 ∈ L+

ˆ 0 )−γ1 C(µ

if c = (ca )a≥1 ∈ L+

.

The same arguments as in the purely local case analyzed in Section 3 show that there exists ˆ C(µ )−γ c∗ = (c∗a )a≥1 ∈ L+ 0 1 such that Fa (γ1 , c∗1 , . . . , c∗a−1 ) = c∗a

for all a ≥ 1.

(43)

ˆ 1 ˆ 0 ) ≥ C(1) ˆ (ii) Since C(µ =: Cˆ ∈ (γ1 , 1 − γ2 ] for all µ0 ∈ [−1, 1], there exists c∗ ∈ LC−γ such + ∗ that the fixed point equation (43) is satisfied. For the correlation pattern c , we put P ∗ t X t≥1 ηt (C ) ∗ ∗ ∗ P P . C := γ1 + ca and B (µ0 ) := µ0 1 − t≥0 ηt (C ∗ )t − t≥0 δt (C ∗ )t a≥1

ˆ ˆ 0 ) because ηt , δt ≥ 0. This yields Now, C ∗ ≤ C(1) and B ∗ ≤ B(µ X gˆg (µ, x, θ 0 ) = γ1 x0 + γ2 θ0 + c∗a xa + B ∗ (µ0 ) = g(µ, x, θ 0 ),

(44)

a≥1

therefore g is an equilibrium. 2 We are now ready to establish the existence of Markov perfect equilibria in economies where we have both a local and a global component in the agents’ utility function. Proof of Theorem 5.4: Let g : Me (X) × X × Θ → X be the policy function defined in (44) and let πµ (T a x; ·) be the law of the random variable g(µ, T a x, ·). Since g(µ; ·, ·) is continuous, the stochastic kernel Y Πµ (x; ·) := πµ (T a x; ·) a∈A

on X has the Feller property. Thus, it follows from the law of large numbers for ergodic empirical fields that the stochastic kernel Π(x; ·) = ΠR(x) (x; ·)

on the configuration space Xe induces the almost surely deterministic sequence of average actions Z %t := y 0 (R(x)Π)t (dy) (t ∈ N). 27

Thus, the policy function g ∗ (x, θa ) := g(R(x), T a x, θa ), yields a symmetric equilibrium by Proposition 5.5.

2

Remark 5.8 For a configuration x ∈ Xe , we have R(x)(y 0 ) = %(x), hence the equilibrium action of the agent a ∈ A depends on the current configuration xt both locally through the present actions xbt taken by the agents b ≥ a and globally through the average action %(xt ) associated with xt . In particular, the policy function does not depend in a continuous manner on x t .

Appendix A

Maximizing α-Concave Functions

The following theorem plays a central role in our analysis. Even though its proof follows primarily from straightforward modifications of arguments given in the proof of Theorem 3.1 in [32], we prove it to keep the paper self contained. Theorem A.1 Let X ⊂ R be closed and convex set and let (Y, k · kY ) be a normed space. Let F : X × Y → R be a continuous function which satisfies the following conditions: (i) For each y ∈ Y , the map x 7→ F (x, y) is α-concave as a function on X. (ii) F is differentiable with respect to x and the derivative F1 (·) :=

∂ ∂x F (·)

satisfies

|F1 (x, y1 ) − F1 (x, y2 )| ≤ G(y1 , y2 ) for some function G : Y × Y → R. Then there exists a unique map f : Y → X such that supx∈X F (x, y) = f (y). Moreover, f satisfies |f (y1 ) − f (y2 )| ≤ In particular, |f (y1 ) − f (y2 )| ≤

L α ky1

1 G(y1 , y2 ). α

− y2 kY if G(y1 , y2 ) = Lky1 − y2 kY .

Proof: Our proof uses modifications of arguments given in [32]. It follows from Lemmas A1 A3 in [32] that, for any y1 ∈ Y , the α-concave function x 7→ F (x, y1 ) has a unique maximizer f (y1 ) which satisfies α|x − f (y1 )| ≤ F1 (x, y1 ). (45) 28

Thus, choosing x = f (y2 ) in (45) for some y2 ∈ Y , we obtain α|f (y1 ) − f (y2 )| ≤ F1 (f (y2 ), y1 ).

(46)

Since f (y2 ) maximizes the differentiable function x 7→ F (x, y2 ), we have F1 (f (y2 ), y2 ) = 0, so it follows from (ii) and (46) that α|f (y1 ) − f (y2 )| ≤ |F1 (f (y2 ), y1 ) − F1 (f (y2 ), y2 )| ≤ G(y1 , y2 ). This yields the assertion.

B

2

Proof of Theorems 2.11 and 2.12

This section gives the proofs of Theorems 2.11 and 2.12. While related, both proofs are reported for the sake of completeness.

B.1

Proof of Theorem 2.11

In this section we are going to prove Theorem 2.11. For this, it is enough to prove the existence of an almost surely uniquely defined fixed point of the operator V : B(Θ0 , X) → B(Θ0 , X) defined by (9). In a first step, we establish the following result. Lemma B.1 If the utility function u is uniformly α-concave in its first argument and if the contraction condition (7) holds, then the operator V satisfies the following conditions: (i) There exists η ∗ > 0 such that V maps the set Lip(η ∗ ) continuously into itself. (ii) The operator V has a unique fixed point g ∗ , and g ∗ ∈ Lip(η ∗ ). Proof: Let us denote by g(y, θ0 ) = arg max u(x, y, θ 0 ) x

(47)

the conditional optimal action of the agent 0 ∈ A, given his type θ 0 and given the action y ∈ X of his neighbor. It follows from Theorem A.1, from Assumption 2.8 (i) and from (7) that |g(y, θ 0 ) − g(ˆ y , θˆ0 )| ≤ γ(|y − yˆ| + |θ 0 − θˆ0 |) where γ :=

L α

(48)

< 1.

(i) Let us first show that V is a continuous operator on the Banach space (B(Θ 0 , X), k · k∞ ). Indeed, due to (48) we have |V g(θ) − V gˆ(θ)| ≤ γ|g ◦ T (θ) − gˆ ◦ T (θ)| ≤ γkg − gˆk∞ 29

(49)

for all g, gˆ ∈ B(Θ0 , X). Thus, kV g − V gˆk∞ ≤ γkg − gˆk∞ . Since L < α, the operator V is not only continuous, but also a contraction. In particular, it has a unique fixed point. (ii) Let us now choose η > 0 such that 2η γ < 1 and fix g ∈ Lip(η). It follows from (49) that   X ˆ ≤ γ |θ0 − θˆ0 | + |V g(θ) − V g(θ)| 2−η(a−1) |θa − θˆa | ≤ 2η γ

  

a≥1

|θ0 − θˆ0 | +

X

2−ηa |θa − θˆa |

a≥1

  

ˆ ≤ dη (θ, θ)

This shows that V maps Lip(η) into itself. Since Lip(η) is a closed subset of B(Θ0 , X) with respect to the topology of uniform convergence, (Lip(η), k · k∞ ) is a Banach space. Thus, V may also be viewed as a contraction on the Banach space (Lip(η), k · k ∞ ), and so the unique fixed point g ∗ of V belongs to Lip(η). This proves the lemma.

2

We are now ready to prove our Theorem 2.11.

Proof of Theorem 2.11: In view of Lemma B.1, it remains to prove part (i). Due to Theorem A.1 and Assumption 2.8, we have |g(y, θ 0 ) − g(ˆ y , θ 0 )| ≤

L(θ0 ) |y − yˆ| α

where the map g is defined by (47). This shows that the operator V satisfies |V g(θ) − V gˆ(θ)| ≤

L(θ0 ) |g ◦ T (θ) − gˆ ◦ T (θ)| α

for all g, gˆ ∈ B(Θ0 , X). Since the types are distributed independently across agents, the random variables L(θ 0 ) and |g ◦ T (θ) − gˆ ◦ T (θ)| are independent, and so E|V g(θ) − V gˆ(θ)| ≤ Here γ :=

EL(θ 0 ) α

EL(θ 0 ) E|g ◦ T (θ) − gˆ ◦ T (θ)| ≤ γE|g ◦ T (θ) − gˆ ◦ T (θ)|. α

< 1. As the types are also identically distributed, we obtain E|V g(θ) − V gˆ(θ)| ≤ γE|g(θ) − gˆ(θ)|. 30

Thus, denoting by V n the n-fold iteration of the operator V , we see that lim E|V n g(θ) − V n gˆ(θ)| = 0.

n→∞

(50)

In particular, the sequence (V n g)n∈N satisfies E|V n+m g(θ) − V n g(θ)| = E|V n (V m g)(θ) − V n g(θ)| ≤ γ n E|V m g − g| ≤ Cγ n for some C < ∞. This shows that (V n g)n∈N is a Cauchy sequence in L1 (P). Since L1 (P) is a complete space, there exists an almost surely uniquely determined random variable G ∗ [g] such that L1 V n g −→ G∗ [g]. Due to (50), the L1 -limit G∗ [g] does almost surely not depend on g. In other words, there exists an almost surely uniquely defined random variable g ∗ such that L1

V n g −→ g ∗

for all g ∈ B(Θ0 , X).

Recall now that L1 (P)-convergence of the sequence (V n g)n∈N to the random variable g ∗ implies almost sure convergence along a subsequence (nk )k∈N . Thus, it follows from the continuity of the choice correspondence that V g∗ = g∗ P-a.s. and so g ∗ is a symmetric equilibrium. Since any symmetric equilibrium g satisfies P[V g = g] = 1, it follows that g = lim V nk g = g ∗ P-a.s. k→∞

This shows almost sure uniqueness of the symmetric equilibrium.

B.2

2

Proof of Theorem 2.12

This section proves Theorem 2.12. For notational convenience, we restrict our attention to the case N = 1; the more general case N ∈ N can easily be established using similar arguments. We proceed by analogy with the proof of Theorem 2.11. In the present setting, it is enough to show that there exists a measurable function g ∗ : Θ2 → X which satisfies Z ∗ 0 1 g (θ , θ ) = arg max u(xa , g ∗ (θ1 , θ2 ), θ0 )ν(dθ 2 ). (51) a x ∈X

Each such function is a fixed point of the operator Ve : B(Θ2 , X) → B(Θ2 , X) which acts on the class B(Θ2 , X) of bounded measurable functions from θ 2 to X according to Z 0 1 e V g(θ , θ ) = arg max u(xa , g(θ1 , θ2 ), θ0 )ν(dθ 2 ). (52) x0 ∈X

31

In order to prove that this operator has a Lipschitz continuous fixed point if the utility function satisfies (7), we introduce the class Lip := {g : Θ2 → R : |g(θ 0 , θ1 ) − g(θ˜0 , θ˜1 )| ≤ |θ 0 − θ˜0 | + |θ1 − θ˜1 |} of all Lipschitz continuous functions on Θ2 with Lipschitz constant 1. The following lemma establishes some basic properties of the operator Ve .

Lemma B.2 If the utility function u is Lipschitz continuous in the sense of (7), then the operator Ve defined by (52) satisfies the following conditions: (i) Ve maps the set Lip continuously into itself.

(ii) Ve has a unique fixed point g ∗ and g ∗ ∈ Lip.

Proof: We introduce the continuous mapping U : Lip × X × Θ2 → R by Z 0 1 U (g, x, θ , θ ) := u(x, g(θ 1 , θ2 ), θ0 )ν(dθ 2 ).

(53)

Under the assumptions of Theorem 2.12, the map x 7→ u(x, y, θ 0 ) is uniformly α-concave, and so the mapping x 7→ U (g, x, θ 0 , θ1 ) is α-concave. (i) Let us first prove that Ve maps the class Lip into itself. To this end, we fix g ∈ Lip. Since ∂ the map (x0 , θ0 , θ1 , θ2 ) 7→ ∂x u(x0 , g(θ1 , θ2 ), θ0 ) is uniformly continuous, it follows from Assumption 2.8 (ii) that ¯ ¯ ¯ ¯∂ ∂ 0 0 1 0 0 1 ¯ U (g, x , θ , θ ) − U (g, x , θˆ , θˆ )¯¯ ¯ ∂x ∂x ¯ ¯ ¯ ¯∂ ∂ 1 2 0 1 2 ˆ0 ¯ ˆ ¯ u(x, g(θ , θ ), θ )¯ ≤ sup ¯ u(x, g(θ , θ ), θ ) − ∂x θ 2 ∂x n o (54) ≤ sup L |g(θ1 , θ2 ) − g(θˆ1 , θ2 )| + |θ 0 − θˆ0 | θ2 n o ≤ L |θ1 − θˆ1 | − |θ0 − θˆ0 | . Here, the last inequality follows from Lipschitz continuity of g. This shows that, for any fixed g ∈ Lip, the map (x, θ 0 , θ1 ) 7→ U (g, x, θ 0 , θ1 ) satisfies the assumption of Theorem A.1 with Y := Θ2 . Thus, L |Ve g(θ 0 , θ1 ) − Ve g(θˆ0 , θˆ1 )| ≤ {|θ0 − θˆ0 | + |θ1 − θˆ1 |}, α

and so Ve g ∈ Lip because L < α.

32

Let us now show continuity of Ve with respect to the sup-norm. To this end, we fix θ 0 , θ1 ∈ Θ. For any two functions g, gˆ ∈ B(Θ2 , X) we have ¯ ¯ ¯∂ ¯ 0 1 ¯ ¯ U (g, x, θ 0 , θ1 ) − ∂ U (ˆ g , x, θ , θ )¯ ≤ Lkg − gˆk∞ , ¯ ∂x ∂x

due to (54). Thus, for each pair (θ 0 , θ1 ), the mapping (g, x) 7→ U (g, x, θ 0 , θ1 ) satisfies the assumptions of Theorem A.1 with Y := B(Θ2 , X). Hence, ¯ ¯ ¯ L ¯ 0 1 0 1 g , x, θ , θ )¯¯ ≤ kg − gˆk∞ . kVe g − Ve gˆk∞ = sup ¯¯arg max U (g, x, θ , θ ) − arg max U (ˆ x∈X x∈X α θ 0 ,θ 1 This proves continuity of the operator Ve in the topology of uniform convergence.

(ii) Since L < α, it follows from (i) that Ve is a contraction on the Banach space B(Θ2 , X). Thus, Ve has a unique fixed point g ∗ . Since Ve maps the closed set Lip into itself, g ∗ ∈ Lip.

This completes the proof.

2

We are now ready to prove Theorem 2.12. Proof of Theorem 2.12: In view of Lemma B.2 it remains to show the existence of a unique symmetric equilibrium if the utility function satisfies Assumption 2.8. To this end, we apply Theorem A.1 to the function U defined by (53). We have ¯ ¯ ¯ Z ¯ ¯∂ ¯∂ ¯ ¯ ∂ 0 1 ¯ 1 2 0 1 2 0 ¯ ¯ ¯ U (g, x, θ 0 , θ1 ) − ∂ U (ˆ g , x, θ , θ )¯ ≤ u(x, g(θ , θ ), θ ) − u(x, gˆ(θ , θ ), θ )¯ ν(dθ 2 ) ¯ ¯ ∂x ∂x ∂x ∂x Z ¯ ¯ ≤ L(θ 0 ) ¯g(θ1 , θ2 ) − gˆ(θ 1 , θ2 )¯ ν(dθ 2 ).

It then follows from Theorem A.1 that ¯ ¯ L(θ0 ) Z ¯ ¯ ¯e 0 1 0 1 ¯ e ¯g(θ1 , θ2 ) − gˆ(θ 1 , θ2 )¯ ν(dθ 2 ). ¯V g(θ , θ ) − V gˆ(θ , θ )¯ ≤ α

Since the random variables θ 0 , θ1 and θ2 are independent and identically distributed ¯ ¯ EL(θ 0 ) Z Z ¯ ¯ ¯e 0 1 0 1 ¯ e ¯g(θ1 , θ2 ) − gˆ(θ 1 , θ2 )¯ ν(dθ 2 )ν(dθ 1 ) ≤ γE|g − gˆ|. E ¯V g(θ , θ ) − V gˆ(θ , θ )¯ ≤ α 0

) Since γ := EL(θ < 1 it follows by an argument similar to the one spelled out in detail in the α proof of Theorem 2.11 that the operator has a fixed point g ∗ which is uniquely defined up to a set of measure 0. 2

33

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