Statistical Equilibrium in One-Step Forward Looking Economic Models

Journal of Economic Theory  2231 journal of economic theory 73, 365394 (1997) article no. ET962231

Statistical Equilibrium in One-Step Forward Looking Economic Models* Tom Krebs Department of Economics, University of Illinois, 117 David Kinley Hall, 1407 W. Gregory Drive, Urbana, Illinois 61801 Received November 15, 1994; revised May 23, 1996

This paper combines statistical with economic equilibrium analysis in the context of one-step forward looking economic models. For a given state space, the economic analysis determines a set of Markov processes consistent with economic equilibrium, the set of expectational equilibria. The concept of statistical equilibrium rationalizes the introduction of a probability measure on the set of expectational equilibria. In the infinite observations limit, there exists a unique expectational equilibrium which is realized with probability one, the statistical expectational equilibrium. The statistical method is applicable whenever the set of expectational equilibria is a compact and convex set. Statistical expectational equilibrium is characterized as the maximum entropy expectational equilibrium. An exponential representation of statistical expectational equilibrium is derived for a large class of one-step forward looking economic models. Journal of Economic Literature Classification Numbers: D50, C60, E10.  1997 Academic Press

1. INTRODUCTION This paper investigates the relationship between statistical equilibrium analysis, a standard method of statistical physics [20, 21], and the economic concept of expectational equilibrium [26]a (statistical) state of economic affairs in which agents' expectations are fulfilled. 1 The formal analysis is conducted for one-step forward looking economic models, a class of * I am grateful to Richard Ericson, Duncan Foley, and Paolo Siconolfi for encouragement and valuable advice. I also wish to thank for valuable comments Omar Azfar, Evan Kalimtgis, Pravin Krishna, Joerg Oechsler, Nathalie Rey, Nicholas Yannelis, an associate editor of this journal, and seminar participants at Columbia University, at the North American Meeting of the Econometric Society, Summer 1994, and the Meeting of the Society for the Advancement of Economic Theory, Summer 1995. I am solely responsible for all remaining errors. 1 There are recent applications of the statistical method to economic theory [17, 36], but none of those studies uses expectational equilibrium as the unifying economic concept. For applications of the maximum entropy formalism to econometrics, see [38].

365 0022-053197 25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.

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economies introduced in [10, 11]. 2 These economies are open ended (no final period) and display a sequential market structure (markets reopen). The mathematical branch of economic theory dealing with competitive markets formalizes the idea of an expectational equilibrium in terms of an equilibrium price function 3 which associates with each realization of the vector of exogenous variables one equilibrium price vector. Although different economic theories allow for different variables to enter as arguments into the equilibrium price function, they all use the mathematical concept of a function. In [27] equilibrium prices depend on signals correlated with fundamentals, [4, 9, 29] allow signals uncorrelated with fundamentals (sunspots) to influence equilibrium prices, and [23] confines attention to fundamentals themselves as the only source of price fluctuations. Viewed as a solution concept, the price function approach maps exogenous probability arrays on equilibrium price functions. It takes probabilities as given and determines a part of the state space of the economic system endogenously, namely the set of possible equilibrium price vectors. In this paper, the basic logic is reversed: for a given state space, probabilities are determined endogenously. If these probabilities describe the statistical properties of privately observed signals, then this change in modeling technique merely reflects the informational status of an outside observer (econometrician). We might have a good idea of the set of possible price constellations, but a very limited knowledge of the statistical properties of the signal variables observed by market participants. As it is well known [10], the fixed state space approach is observationally equivalent to the price function approach in the sense that to each equilibrium price process we can construct a signal process and a corresponding equilibrium price function inducing the equilibrium price process. Despite the mathematical equivalence, the two approaches differ in at least two important respects. First, if probabilities are determined endogenously, it is tempting to interpret the associated random fluctuations as randomness created within the economic system itself. The economy not only propagates exogenous shocks, but also creates endogenous fluctuations. Second, the price function approach passes the minimum test for determinacythe number of equations is equal to the number of unknowns. 4 2 In [13] a related abstract framework is used to prove existence of stationary Markov equilibria for a large class of economies. The analysis conducted here is equally applicable to the framework developed in [13]. The set of expectational equilibria then becomes the set of stationary Markov equilibria. 3 More generally, the standard approach uses the concept of an equilibrium function. 4 In the following discussion, we assume a finite number of states. In this case, equality of the number of equations and the number of unknowns holds for one-step forward looking economic models [10], Arrow-Debreu economies [12], and Radner's rational expectations equilibrium [27], but does not hold for incomplete market economies with nominal asset payoffs [5, 18].

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Hence, the ``typical'' equilibrium, if it exists, is locally unique. On the other hand, when the probabilities are endogenous and the state space is fixed, there is no economic mechanism linking the number of equations to the number of unknowns (which are now probabilities). Since the number of unknowns can be arbitrarily increased by increasing the number of states (the cardinality of the state space), one typically faces a huge indeterminacy problem: there is a continuum of equilibria. 5 In this paper, the indeterminacy problem is resolved by combining economic with statistical equilibrium analysis. Attention is confined to one-step forward looking economic models with a finite state space. For a given state space, the economic analysis determines a set of Markov processes consistent with economic equilibrium, called here the set of expectational equilibria. In the terminology of the sunspot literature, the set of expectational equilibria is the set of all stationary sunspot equilibria generating the same state space (support of the induced price process) plus the set of stationary perfect foresight equilibria. This set is assumed to be non-empty (existence of at least one equilibrium), compact, and convex. For a finite number of observations, the concept of statistical equilibrium rationalizes the introduction of a probability measure on the set of expectational equilibria. The statistical method is based on an elementary combinatorial argument. Some empirical transition matrices can be realized in a larger number of ways than others, and therefore some expectational equilibria are more likely than others. The probability measure on the set of expectational equilibria is uniquely defined by combinatorics as soon as a prior measure on the set of sample points is specified. In this paper, the uniform distribution assigning equal probability weight to each sample point consistent with expectational equilibrium is chosen as the prior measure. It is then shown that when the number of observations goes to infinity, there is a unique expectational equilibrium which is realized with probability one, the statistical expectational equilibrium. This paper also provides a characterization of statistical expectational equilibrium. Statistical expectational equilibrium is the maximum entropy expectational equilibrium. Since the entropy is a measure of uncertainty of an outside observer, 6 an economy in statistical expectational equilibrium displays a maximum of endogenous random fluctuations consistent with expectational equilibrium. This result contrasts sharply with the perfect foresight solution, which is the minimum entropy expectational equilibrium. If we do not rule out endogenous randomness by assumption (degenerate prior measure on the set of sample points), then endogenously created random fluctuations become a probability one event. 5 In the macroeconomic literature, this indeterminacy of rational expectations equilibrium was first pointed out in [31, 37]. 6 Entropy is potential information, that is, absence of actual information.

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Finally, this paper derives an exponential representation of statistical expectational equilibrium valid for a large class of economies. In this exponential formula, also known as Gibbsian canoncial distribution, any restriction imposed upon the statistical outcome by rational choice of forward looking agents is represented by one term. This particular representation of statistical expectational equilibrium might help bridge the gap between economic theory and econometric testing. Indeed, for the case of the linear expectational difference equation the exponential representation is used to show that the conditional variance is increasing in levels. This statistical property is frequently found in the data and has been extensively used in the econometric literature on autoregressive conditional heteroscedastic models [7, 14]. Here, it is derived from a general statistical principle applied to the linear expectational difference equation. At this stage, three general comments are in order. The first comment regards the assumption of a finite state space. This paper confines attention to the finite state space case in order to emphasize general principles rather than mathematical technicalities. Moreover, finite state space models remain an important workhorse in the literature and examples of such models are numerous. 7 Continuous state spaces, however, naturally arise in several sunspot models [15, 16, 24, 35] and, more importantly, are needed to avoid non-existence problems in multi-agents, multi-goods overlapping generations models [33, 34]. 8 The extension of the statistical expectational equilibrium approach to more general state spaces is an important and mathematically demanding topic for future research. 9 The assumption of equal a priori probability adopted here also deserves a comment. This assumption has given rise to many ``paradoxes'' when distributions with continuous support are considered [21, 22]. If the state space is finite, however, assigning equal probability to each sample point consistent with expectational equilibrium seems still the most convincing choice of a prior measure. The postulate of fulfilled expectations partitions the set of sample points into two subsets, the set of economically possible 7

See, for example, [10, 19] for an exhaustive survey. The importance of extending of the analysis to more general state spaces was brought to my attention by an associate editor of this journal. 9 The physics and engineering literature usually considers the special case of linear constraints (Sect. 6). The typical approach taken in this literature is to disregard all mathematical and conceptual problems and to replace summation by integration in the exponential formula (canonical distribution) [20, 21]. This procedure is unsatisfactory since it does not address the two main issues, namely: (i) (easy) in what sense is the canonical distribution the solution to a maximum entropy program and (ii) (hard) in what sense can one think of the maximum entropy distribution of a continuous state space economy as the statistical equilibrium of the limit of a sequence of finite state space economies? But see also [8] for an attempt of providing a rigorous answer to these questions for the linear constraint case. 8

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sample points (sample points consistent with expectational equilibrium) and the set of economically impossible sample points (sample points inconsistent with expectational equilibrium). Absent any further information about the economy, the ``principle of insufficient reason,'' also known as ``principle of indifference,'' suggests that possible sample points should be judged equally likely. The final comment regards the range of applications of statistical expectational equilibrium. The class of economies studied here, namely one-step forward looking economic models, was first introduced in [10] in order to discuss various contributions to the sunspot literature within a unified framework. Since these sunspot models assume non-random fundamentals, the random fluctuations displayed by these models are necessarily endogenous. However, the definition of one-step forward looking economic models is general enough to deal with the case in which endogenous and exogenous fluctuations coexist. 10 Thus, the concept of statistical expectational equilibrium is by no means restricted to pure sunspot models. Indeed, instead of thinking of statistical expectational equilibrium as a formal selection criterion helping resolve a theoretical indeterminacy problem, the interpretation favored in this paper, one could also view statistical expectational equilibrium as an econometric alternative to the usual i.i.d. normality assumption, an alternative that has the advantage of providing a close link to economic theory.

2. AN EXAMPLE This section illustrates the basic ideas underlying the concept of statistical expectational equilibrium by means of an example. Consider the following version of a simple overlapping generations model studied in detail in [4, 32]. There is a perishable consumption good which is produced from labor under a constant returns to scale production function (one unit of labor produces one unit of the consumption good). Agents live for two periods and the number of agents per generation is constant and normalized to one. An agent born in period t supplies labor in quantity y t when young and consumes c t+1 of the consumption good when old. An agent born in period t finances consumption c t+1 by the money wage received for work in period t. The money is ``injected'' into the economy in period one by a generation of one-period lived old agents who inelastically supply a certain quantity M>0 of money. 10 This is demonstrated in [13] by showing that the framework encompasses a generalized version of the Lucas asset pricing model [23].

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All agents have the same preferences. In the certainty case, preferences over laborconsumption bundles can be represented by a time additive utility function U( y t , c t+1 )=&u( y t )+v(c t+1 ). The functions u(}) and v(}) are assumed to be twice continuously differentiable with u$(})>0, v$(})>0, u"(})>0, v"(})<0, and the boundary conditions u$(0)v$(). These assumptions ensure the existence of a unique stationary perfect foresight equilibrium in which money is valued. If the economic environment is uncertain, preferences are assumed to be time and state additive (expected utility representation). Given their expectations, agents maximize (expected) utility subject to their budget constraint. First, suppose that agents expect the money price p t of the consumption good, which is equal to the money wage in this economy, to be constant over time: p et = p e # R ++ . Thus, agents have stationary point expectations. The maximization problem of an agent born in period t reads: max [&u(y t )+v(c t+1 )]

yt , ct+1

s.t. p e c t+1 = p e y t .

(2.1)

The associated first-order conditions are u$(y t )=v$(y t ).

(2.2)

Given our assumptions on u(}) and v(}), (2.2) is a necessary and sufficient condition for utility maximization if p e < (monetary equilibrium). Further, (2.2) has a unique solution y, the optimal labor supply. For any optimal labor supply y there is only one price level, p =My, which clears markets (recall that the number of agents per generation is normalized to one). If agents' expectations are fulfilled, we must also have p e = p. The stationary equilibrium p is an expectational equilibrium in the following sense. If utility maximizing agents expect the price level p to prevail forever, they will choose actions which lead to an economic outcome consistent with their expectations. Since agents have point expectations, we call it a stationary perfect foresight equilibrium (this terminology follows [10]). Let us now replace the assumption of point expectations by a more general assumption, but still uphold the assumption of fulfilled expectations. More specifically, consider an economy (system) consisting of a large number of structurally identical islands (subsystems). Each island is an economy in itself. 11 The islands are identical in the sense that the fundamentals (including money supply) are the same for all islands. Further, fundamentals are still non-random and known to all agents. However, we 11 We choose the cross-sectional interpretation only for convenience. The statistical method as presented here is equally applicable to time-series analysis. In this case, the focus is on aggregate fluctuations.

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assume that there are now statistical fluctuations which cause different islands to experience a different fate in terms of finally observed price level. Since the fundamentals are certain, this randomness is best interpreted as being created by the economic systems itself. In this sense, it is endogenous randomness. Since this paper focuses attention on formal models of market economies, it is also randomness created by the market process, called market uncertainty in [25, 30]. In this paper, we attempt to make precise statements about the statistical properties of the economic system without explicitly modeling the ``ultimate source'' of the randomness. For our example economy, we proceed as follows. We place ourselves at one particular date, today. Consider all those islands on which the currently observed price level is p= p. Suppose further that all agents expect the future price level on their island to be uncertain. More precisely, they believe that tomorrow's price p$ can take on three distinct values: p$ # [ p &=, p, p +=]. Agents' common expectations about the uncertain future is represented by a conditional probability distribution ?(} | p ) # 2 with components ?(p$ | p ) # [0, 1]. Here 2 is the unit simplex in R 3. ?(p$ | p ) is the probability of observing price level p$ tomorrow if today's price level is p, that is, the transition probability of moving from p to p$. A young agent on an island with current price p now chooses a plan (y, c) consisting of labor supply y and contingent consumption c p$ , p$ # [ p &=, p, p +=], which solves max : ?( p$ | p )[&u( y)+v(c p$ )] y, c

s.t. : p$c p$ = py \p$.

(2.3)

p$

Using the budget constraint, the first-order conditions are 1 1 u$( y )=: v$ p p$ p$

p

\ p$ y+ ?( p$ | p).

(2.4)

Assuming market clearing of current markets on each island, p y =M, we can rewrite (2.4) as M 1 M 1 u$ v$ =: p p p$ p$ p$

\ +

\ + ?( p$ | p).

(2.5)

Let us introduce the functions g : g( p)=

1 M u$ p p

\ +

and

f : f (p)=

1 M v$ , p p

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\ +

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which are the marginal disutility of earning an additional dollar and the marginal utility of spending an additional dollar, respectively. Using the new notation, Eq. (2.5) can be written in compact form: g(p )&: f (p$) ?(p$ | p )=0.

(2.6)

p$

Note that (2.6) is linear in probabilities ?( p$ | p ). However, Eq. (2.6) only becomes a version of the linear expectational difference equation frequently encountered in applied macroeconomics [31, 37] if the functions g(}) and f (}) are also linear. As already mentioned in the Introduction, in this paper we take the state space [ p &=, p, p +=] as given and determine the probabilities ?( p &= | p), ?(p | p ), and ?( p += | p ) endogenously. Eq. (2.6) defines two equations (one maximization condition and one normalization constraint) in the three unknowns ?(p &= | p ), ?( p | p ), and ?( p += | p ). It is straightforward to show that there exists a non-degenerate (conditional) probability distribution ?(} | p ) # 2 with components ?(p &= | p ), ?(p | p ), and ?( p += | p ) which solves (2.6), implying the existence of a continuum of solutions to (2.6) (for a given price level p ). If the stationary perfect foresight equilibrium is stable, then the same applies to the price levels p &= and p += for sufficiently small = [4], i.e., there is a continuum of conditional probability distributions ?(} | p &=) solving g(p &=)&: f (p$) ?(p$ | p &=)=0,

(2.7)

p$

and a continuum of conditional probability distributions ?(} | p +=) solving g(p +=)&: f (p$) ?(p$ | p +=)=0.

(2.8)

p$

For a given state space [ p &=, p, p +=], there is an indeterminacy problem in the sense that the economic theory of fulfilled (rational) expectations only determines a continuum of solutions. In this paper, the indeterminacy problem is resolved by combining economic with statistical equilibrium analysis. The following heuristic argument conveys the basic intuition underlying the concept of statistical equilibrium. Consider T islands with current price level p. In the next period (tomorrow), unmodeled economic interaction will evolve and generate a new price level on each island, which is not necessarily equal to p. Put differently, we imagine a random experiment consisting of T trials. The result of this random experiment is summarized by a sample of size T : | p =( p 1 , ..., p t , ..., p T ), p t # [ p &=, p, p +=]. Each component of | p is the observed ``future'' price level on an island currently in state p. Notice

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that we do not make the assumption that individual observations are realizations of T independently distributed random variables. On the contrary, the postulate of expectational equilibrium introduced below necessitates a certain statistical dependence between individual trials (islands). Let us now take the position of an outside observer (the econometrician) whose complete knowledge of the economy is represented by Eq. (2.6), the set of possible ``future'' price levels [ p &=, p, p +=], and the additional information that agents' expectations entering into (2.6) coincide with the finally observed frequency distribution (expectational equilibrium). For simplicity, assume also that the function f (}) satisfies f (p &=)& f (p )= f (p )& f (p +=).

(2.9)

The outside observer, not knowing what economic agents ``really think,'' will consider a sample |^ p =(p, p, ..., p ) to be consistent with expectational equilibrium, since this sample is associated with an empirical distribution, ?^(} | p )=(0, 1, 0), which satisfies (2.6) (this follows from direct substitution using the steady state condition g(p )= f ( p )). Evidently, ?^(} | p )=(0, 1, 0) is the stationary perfect foresight equilibrium; all islands starting with the same price level p experience the same fate, namely having a price level p tomorrow as well. There are no endogenous random fluctuations. p &=, p, p +=, p &=, p, p +=, ..., p &=, p, p +=) However, the sample | *=( p generates an empirical distribution, namely ?*(} | p )=(13, 13, 13), which also satisfies (2.6) (assuming that T is divisible by 3 and using (2.9)). In this case, structurally identical islands starting from the same initial position (price level p ) can experience a different fate. Endogenous random fluctuations are possible. Both distributions, (0, 1, 0) and (13, 13, 13), are consistent with expectational equilibrium, but they differ from each other in the following way: there is only one sample associated with the degenerate distribution (0, 1, 0), namely |^ p , but there are T!((T3)!) 3 r3e T (using Stirling's formula) samples generating the uniform distribution (13, 13, 13), namely all those samples | p which differ from | * p merely by a permutation of components (islands). If we are not willing to exclude endogenous randomness by assumption, then the point-solution (0, 1, 0) becomes extremely unlikely (for T=100 we have 3e T =8.06_10 43 ). Hence, the stationary perfect foresight equilibrium is a very rare event. The combinatorial argument can be generalized. For current price level p define the set of temporary expectational equilibria of a T-observations economy as 6 epT r[?(} | p ) # 2 T | ?(} | p ) solves (2.6)].

(2.10)

Here 2 T denotes the set of empirical distributions which can be generated by a sample of size T. In (2.10) we use the approximation sign r for the

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following reason. The concept of expectational equilibrium is inherently a limiting concept. If for a finite number of observations the observed distribution deviates from the expected distribution, but this deviation is ``not too large,'' then it seems reasonable to say that agents' expectations are fulfilled in the sense of not being falsified. Thus, the set of expectational equilibria of a T-observations economy 6 epT is larger than the right-hand side of (2.10). Of course, the deviations must disappear in the limit T   in order to speak of an expectational equilibrium. The mathematical details can be found in Section 5. The following analysis is facilitated by denoting the transition probabilities ?(p$ | p) # [0, 1] by ? pp$ and the conditional probability distribution ?(} | p) # 2 by ? p . This convention will be followed throughout the remainder of the paper. Each empirical distribution ? p # 2 T is associated with T! (2.11) > p$(T? pp$ )! samples | p , where ? pp$ is the proportion of the T islands on which the price level p$ is realized in the ``next period'' (all T islands started from current price level p ). If we assign equal probability to each sample generating an empirical distribution ? p # 6 T , then the likelihood of each ? p # 6 peT is proportional to the number of ways this distribution can be realized + T (? p ) B

T! , > p$(T? pp$ )!

(2.12)

where the constant of proportion is determined by normalization. The most likely temporary expectational equilibrium is the solution to maxe

?p # 6 pT

T! . > p$(T? pp$ )!

(2.13)

Take the logarithm of the maximand in (2.13) and use Stirling's formula ln(n!)rn ln(n)&n, an approximation with vanishing relative error in the limit T  . If we also replace 6 epT by the set 6 ep . [? p # 2 | ? p solves (2.6)], then (2.13) becomes max &: ? pp$ ln ? pp$ .

?p # 6 ep

(2.14)

p$

The strictly concave function H(? p )= & p$ ? pp$ ln ? pp$ is called the (informational) entropy of the (conditional) probability distribution ? p

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[28]. The preceding argument demonstrates that for finite sample size the most likely temporary expectational equilibrium is approximately the maximum entropy temporary expectational equilibrium. In this paper we show that this approximation becomes exact, in a sense to be defined, in the infinite observations limit T  . Hence, we can define the temporary statistical expectational equilibrium (for price level p ) as the most likely equilibrium of the limit economy and calculate it as the solution of the concave programming problem (2.14). Calculation shows that the unique solution to (2.14) is the uniform distribution (13, 13, 13). 12 The temporary statistical expectational equilibrium is the ``broadest'' distribution consistent with temporary expectational equilibrium as expressed by (2.6). This result contrasts sharply with the degenerate distribution (0, 1, 0) which corresponds to the stationary perfect foresight equilibrium. The statistical analysis suggests that non-negligible endogenous random fluctuations are a very likely event. The temporary statistical expectational equilibrium for price level p, (13, 13, 13)), is a conditional probability distribution. For price levels p= p &= and p= p += the constraint set in (2.14) will change and so will the solution. Although we cannot explicitly solve for the temporary statistical expectational equilibrium for price level p &= and p +=, an exponential representation is available (see Sect. 6). All three temporary statistical expectational equilibria taken together constitute a statistical expectational equilibrium. The three conditional probability distributions (one for each price level) are the row vectors of a 3_3 stochastic matrix (transition matrix). Therefore, the statistical expectational equilibrium is a Markov process with stationary transition probabilities ?* pp$ and state space [p &=, p, p +=]. Its transition matrix is one of the many transition matrices satisfying \p : g

M M &: ? pp$ f =0. p p$ p$

\ +

\ +

(2.15)

Note also that the statistical expectational equilibrium is only unique for a given state space [ p &=, p, p +=]. As we vary the state space, we change the statistical expectational equilibrium. 3. THE ECONOMIC MODEL Following [10, 11] we consider a one-step forward looking economic system. The vector of state variables, denoted by x, belongs to a time 12 This can be easily seen as follows. The uniform distribution maximizes the entropy function H(}). Further, the uniform distribution is an element of the constraint set in (2.14) (because of the symmetry assumption (2.9)).

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invariant set X/R q, called the state space. Real spaces are always endowed with the canonical metric. The formal analysis is conducted under the following assumption on X: Assumption 1. The state space X/R q is a non-empty, finite set of cardinality |X| =S. The common expectations of agents about the future state of their island, given that the present state is x, is represented by a conditional probability distribution ? x . ? x is an element of the unit simplex in R S which we denote by 2. The common theory of agents, specifying their expectations about the future for each current state x, is an array of conditional probability distributions, called a transition matrix (stochastic matrix, Markov matrix). Let 6=(2) S stand for the set of all transition matrices with generic element ?. The conditional probability distribution ? x is simply a row vector of the matrix ?. Denote a typical component of ? by ? xx$ # [0, 1]. ? xx$ is the conditional probability of x$ given x, that is, the transition probability of moving from state x to state x$. Following [10, 11], we assume that the equilibrium dynamics of the economic system can be characterized as the zeros of a function 13 F : X_2  R n (x, ? x ) ~ F(x, ? x ). The function F is time invariant and the same for all states x. In many applications, the state x is the price vector and F the intertemporal excess demand function. However, as the example of Section 2 shows, F can also stand for the first-order conditions of expected utility maximizing agents. 14 If expectations ? x induce agents to take actions consistent with the occurrence of the state x, the economy is said to be in a state of temporary expectational equilibrium: Definition 1. A temporary expectational equilibrium for state x is a conditional probability distribution ? x satisfying F(x, ? x )=0.

(3.1)

13 We deviate from [10, 11] by considering the more general case in which the dimension q of the state space is not necessarily equal to the number of equations n. 14 In the example of Section 2, the state x is the price level p and the function F is defined by F( p, ? p )= g( p)& p$ f ( p$) ?( p$ | p).

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If (3.1) holds for all states x, the economy is said to be in a statistical state of expectational equilibrium: 15 Definition 2. An expectational equilibrium is a Markov process with state space X and stationary transition matrix ? satisfying \x # X : F(x, ? x )=0.

(3.2)

The set of temporary expectational equilibria (for state x) will be denoted by 6 ex . [? x # 2 | F(x, ? x )=0] and the set of expectational equilibria by 16 6 e . [? # 6 | F(x, ? x )=0 \x # X]= _ 6 ex . x#X

In this paper, we think of the state space X as given and fixed. The equilibrium conditions (3.2) then define a system of Sn equations in the S(S&1) unknowns ? xx$ . In many applications the number of states S is large compared to the number of equations n. For instance, in our introductory example n=1 and S is the number of possible price levels: S=3. Thus, the case S(S&1)>Sn seems to prevail and we face, at least ``typically,'' a serious indeterminacy problem. Statistical equilibrium provides for a general selection criterion in situations where the set of expectational equilibria is compact and convex: Assumption 2. The set of expectational equilibria 6 e is a compact and convex set. Remark. Since 6 e is the S-times Cartesian product of the sets 6 ex , it is compact and convex if and only if the sets 6 ex are all compact and convex. Compactness of 6 ex /2 is ensured for closed 6 ex , since 2 is bounded. Therefore, the first part of Assumption 2 only rules out weak inequality constraints and a discontinuous function F(x, } ). Quasi-concavity of F(x, } ) implies the convexity of 6 ex . In the example of Section 2 the function F(x, } ) is affine linear and therefore satisfies Assumption 2. 15

The economic equilibrium conditions define a set of Markov processes with stationary transition matrix (the initial distribution is still not specified). Since the state space is finite, an invariant distribution for each stationary transition matrix exists. Thus, we can always work with a stationary Markov process by choosing as initial distribution one of the invariant distributions. Although the subsequent mathematical analysis does not hinge at all on the assumption of stationarity, the interpretation of economic equilibrium in terms of expectational equilibrium suggests the focus on situations displaying some sort of stationarity. On this point, see also [13, 19]. 16 In [13] the economic model is defined by an expectations correspondence which assigns to each state x the set 6 ex .

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So far we have not addressed the issue of existence. Since the focus of this paper is on the indeterminacy problem, we will assume the existence of at least one expectational equilibrium. In other words, we presuppose that an internally consistent economic model has been constructed. Assumption 3. The set of expectational equilibria 6 e is non-empty.

4. THE MATHEMATICAL MODEL OF RANDOMNESS In a statistical state of expectational equilibrium, the probability distribution ? x represents both agents' probabilistic expectations about the uncertain future and the frequency distribution of the observed past. The mathematical model of randomness developed in this section focuses on the frequency interpretation of ? x . Let there be T islands currently in state x 1 # X, T islands currently in state x 2 # X, ..., T islands currently in state x S # X. 17 Unmodeled economic interaction will produce an economic outcome x$ # X on each island in the next period. Thus, there are in total TS observations. For each state x, we observe a subsample | x # (X) T. The T-dimensional vector is a list of T observations of the ``future'' state of each of the islands whose current state is x. The entire sample |=(..., | x , ...) is a TS-dimensional vector which completely specifies the outcome of the random experiment. We denote the set of all sample points (sample space) by 0 T =(X) TS. Each subsample | x is associated with one, and only one, empirical distribution which we construct by drawing a histogram. Let

{

}

2 T . ? x # 2 ? xx$ =

t x$ \x$ ; T

=

t x$ # [0, 1, 2, ..., T ]

be the set of empirical distributions of a T-observations experiment. An empirical distribution of a T-observations experiment, ? x # 2 T , will also be called a T-observations distribution. If we draw the histogram for each state x by using the entire sample |, we construct a collection of S empirical distributions. This array of empirical distributions, denoted by ?, is the empirical transition matrix of a T-observations experiment, or T-observations transition matrix. Evidently, each sample | is associated with one, and only one, empirical transition matrix ?=?(|), namely that ? being produced by drawing S histograms. Denote the set of all empirical 17 We could introduce T1 , T2 , ..., TS observations without changing anything substantial. In particular, we could choose T1 , T2 , ..., TS such that (T1 T, ..., TS T), T= S TS , converges to one of the invariant distributions associated with the transition matrix of the statistical expectational equilibrium as the number of observations goes to infinity.

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transition matrices of a T-observations experiment by 6 T =(2 T ) S. Clearly, the set 2 T is a strict subset of 2 and the set 6 T is a strict subset of 2 and the set 6 T is a strict subset of 6. 18 We formalize the relationship between sample points and empirical transition matrix by means of the mapping 6 T : 0 T  6 T | ~ ?(|). 6 T is a surjective but in general not injective function since there are many subsamples | x # (X) T associated with the same empirical distribution ? x # 2 T , and thus there are also many sample points | # 0 T associated with the same empirical transition matrix ? # 6 T . This property of the mapping 6 T is the key to the limit results derived below. Combinatorics, i.e., the rule for the change of variables, implies an extremely sharply peaked probabilistic ordering of the set 6 T even if the prior measure on the set 0 T is very broad (see below). Both sets, 0 T and 6 T , are finite sets. Therefore, the power set F0T and F6T are surely _-algebras and we can construct the canonical measurable spaces (0 T , F0T ) and (6 T , F6T ). Since 6 T is a measurable function with respect to these two measurable spaces, we have a random variable 6T : (0 T , F0T )  (6 T , F6T ). As we vary T, we define a sequence of random variables [6 T ] T . This sequence of random variables is not the sequence of random variables which is commonly used to define a Markov process. Each 6 T maps samples on Markov processes (more precisely, Markov matrices ?there is still an undetermined initial distribution). The remainder of this section deals with the introduction of a prior measure on (0 T , F0T ) in order to complete our mathematical model of randomness. Our objective is to define a prior measure on (0 T , F0T ) which properly reflects the idea that all sample points consistent with expectational equilibrium are equally likely, and all others are impossible. As already mentioned in Section 2, it seems natural to require that expectational equilibrium holds exactly only in the limit T  , that is, for finite T there are always some deviations of observed distributions from expected distributions, but these deviations should disappear in the limit. Thus, we are faced with the problem of defining a concept for finite T which is properly defined only in the limit T  . Our approach is as follows. Recall that the set of expectational equilibria 6 e is the Cartesian product of the S sets 6 ex : 6 e =_x # X 6 ex . Similarly, the sample space 0 T is the S-times Cartesian product of the set (X) T. Thus, it makes sense to separate 18 More precisely, 2 T =([0, 1T, 2T, ..., 1]) S & 2. The set   T=1 2 T is dense in 2 and the set   T=1 6 T is dense in 6.

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the following analysis into S identical problems. For any state x, we consider a sequence of sets [6 exT ] T , 6 exT /2 T . A typical term 6 exT of the sequence is a set of T-observations distributions. We require that 6 exT becomes arbitrarily close to 6 ex as T increases, that is [6 exT ] T converges to 6 ex . Here convergence is defined in terms of the Haussdorf metric, where all sets are viewed as subsets of the underlying metric space ( 2, d ) (d the canonical metric). This convergence requirement formally expresses the idea of an expectational equilibrium in the infinite observations limit. In addition, we require non-emptiness for all T in order to provide a consistent definition of a prior measure (see below). Definition 3. For any given state x, a sequence of sets of temporary expectational equilibria is a sequence [6 exT ] T , 6 exT /2 T , with the following properties: (i)

Each 6 exT is non-empty.

(ii)

The sequence [6 exT ] T converges to 6 ex .

We show that a sequence of sets of temporary expectational equilibria exists by construction: Proposition 1. For any x # X a sequence of sets of temporary expectational equilibria [6 exT ] T exists. Proof. Let B = (? x ) be the open ball in 2 with radius =>0 and center ? x # 2. Here open always means open relative to 2. Define B = (6 ex ) . . B = (? x ). e

?x # 6 x

Recall that there are S states and T observations. Choose ==ST and define the set of temporary expectational equilibria of a T-observations economy as 6 exT . B ST (6 ex ) & 2 T . Evidently, [6 exT ] T converges to 6 ex . It remains to show that B ST (6 ex ) & 2 T is non-empty for all T. To see this, take ``the worst'' case in which 6 ex is a singleton: 6 ex =[? ex ]. Then B ST (6 ex )=B ST (? ex ). Since we can ``walk'' a distance of 1T in each of the S directions, this ball always contains at least one T-observations distribution. K Having formally defined the concept of a temporary expectational equilibrium in the limit, the concept of an expectational equilibrium in the limit falls into place:

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Definition 4. Take S sequences of sets of temporary expectational equilibria [6 exT ] T , one for each x # X. A sequence of sets of expectational equilibria is a sequence [6 eT ] T with terms 6 eT /6 T defined as \T : 6 eT = _ 6 exT . x#X

Remark. Evidently, Proposition 1 also implies that a sequence of sets of expectational equilibria exists. Fix now S sequences [6 exT ] T and the corresponding sequence [6 eT ] T . We use the sequences [6 exT ] T to define, for each state x, a sequence of sets of subsamples consistent with temporary expectational equilibrium [0 exT ] T . For any state x, the terms 0 exT of the sequence [0 exT ] T are defined as \T : 0 exT =[| x # (X) T | ? x (| x ) # 6 exT ] .

(4.1)

Since 6 exT contains at least one empirical distribution, the set 0 exT is nonempty for all T. The non-emptiness of the set 0 exT is of importance for the consistent definition of a prior measure. A sequence [0 eT ] T of sets of samples consistent with expectational equilibrium is defined in the obvious way by forming the S-times Cartesian product for each T : 0 eT = _x # X 0 exT . The sequence [0 eT ] T constitutes the basis for defining a prior measure [P T ] T which formalizes the idea of equal a priori probability of all sample points consistent with expectational equilibrium. More precisely, we define for each T a prior measure P T defined on (0 T , F0T ) as follows: \| # 0 T : P T (|)= ` P xT (| x ) x#X

1 e |0 P xT (| x )= xT |

{

0

if

| x # 0 exT

(4.2)

otherwise.

The definition of the prior measure amounts to the introduction of a metamathematical assumption: Postulate. A priori probabilities are given by the prior measure PT defined in (4.2). The definition of the sequence [PT ] T completes the description of our mathematical model of randomness. To summarize, we have constructed a sequence of probability spaces [(0 T , F0T , PT )] T and a sequence of random variables [6 T ] T , 6 T : (0 T , F0T )  (6 T , F6T ). Of course, the form of these two sequences will depend on the particular sequence [6 eT ] T we

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choose. However, for the results we are about to prove only the properties of [6 eT ] T entering into its definition are relevant.

5. STATISTICAL EXPECTATIONAL EQUILIBRIUM Fix a sequence of random variables [6 T ] T , 6 T : (0 T , F0T )  (6 T , F6T ), and a sequence of prior measures [PT ] T . For each T, the prior measure PT orders the set of sample points 0 T . The next proposition shows the ordering of the set of empirical transition matrices 6 T induced by the random variable 6 T . Proposition 2. For any T, prior measure PT in conjunction with random variable 6 T induce the following measure + T on (6 T , F6T ): \? # 6 T : + T (?)= ` + xT (? x ) x#X

[> x$ # X (T? xx$ )!] &1 &1 + xT (? x )=  ?x # 6 exT [> x$ # X (T? xx$ )!]

{

if ? x # 6 exT

(5.1)

otherwise.

0

Proof. Since both sets, 0 eT and 6 eT , are the Cartesian product of the sets 0 and 6 exT , respectively, the problem factorizes + T (?)= _x # X + xT (? x ). Hence, it suffices to show that the marginal measure + xT (? x ) is given by (5.1). To this end, note that each temporary expectational equilibrium ?x # 6 exT is associated with e xT

T! > x$ # X (T? xx$ )!

(5.2)

subsamples | x # 0 exT . The multinomial factor (5.2) is also called the multiplicity of a temporary expectational equilibrium. Since all | x # 0 exT have equal probability weight (postulate of equal a priori probability), this yields ? xT (? x ) B

_

` (? xx$ T)! x$ # X

&

&1

.

(5.3)

Normalization leads to (5.1). K In the infinite observations limit T  , the ordering of the set of expectational equilibria becomes degenerate in the sense that any set of transition matrices not contained in an arbitrary small ball around a uniquely

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determined ?* is a measure zero event. Neighborhoods, defining a measure of closeness, are introduced as follows. Let B e (?)/6 be the open ball in 6 with radius =>0 and center ?. The set 6 T"B = (?) is then the set of all T-observations transition matrices not contained in an =-ball around ?. 19 We have: 20 Theorem. Suppose that Assumptions 13 hold. Then the sequence of random variables [6 T ] T converges in probability to a unique expectational equilibrium ?* # 6 e : \=>0 : lim + T (6 T"B = (?*))=0. T

Proof.

(5.4)

Appendix. K

The transition matrix ?* # 6 e is called the statistical expectational equilibrium. Definition 5. The statistical expectational equilibrium is the unique limit ?* # 6 e of the sequence of random variables [6 T ] T . Statistical expectational equilibrium is the unique transition matrix consistent with both statistical and expectational equilibrium. Define the following function H which assigns to each (conditional) probability distribution ? x (the row vectors of the matrix ?) a positive real number: Definition 6. The entropy function is defined as H : 2  R+ ? x ~ & : ? xx$ ln ? xx$ . x$ # X

The proof of the theorem yields as a corollary an interesting characterization of statistical expectational equilibrium. 19

More precisely, we should write 6 T"(B = (?) & 6 T ). An alternative approach would be to show that the sequence of probability measures [+ T ] T converges weakly to the delta measure $ ?* . The technical difficulty with this approach is the dependence of the measurable space (2 T , F0T ) on T. In order to apply the standard definition of weak convergence of measures, the measure + T has to be extended to a measurable space independent of T (a natural choice is (2, B 2 ), where B 2 is the Borel _-algebra in 2). A proof along these lines is available on request. 20

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Corollary. The statistical expectational equilibrium is the maximum entropy expectational equilibrium, that is, the row vectors ? * x are solutions to the following maximization problems max H(? x ). \x : ? *=arg x e

?x # 6 x

Proof.

Follows from the proof of the theorem. K

It was first shown [28] that the entropy is a measure of uncertainty (absence of information) contained in a distribution. In this sense, temporary statistical expectational equilibrium is the boradest distribution consistent with temporary expectational equilibrium. It contains a maximum amount of uncertainty as measured by the entropy H. The case 6 e =6 illuminates the basic premise behind the concept of statistical expectational equilibrium. It is an immediate consequence of the corollary that in this case the temporary statistical expectational equi..., 1S) librium is a uniform distribution on the state space X : ? *=(1S, x for all states x # X. In the absence of economic restrictions, the theory predicts the absence of statistical regularity. 6. STATISTICAL EXPECTATIONAL EQUILIBRIUM WITH LINEAR CONSTRAINTS The theorem only refers to the convexity of the set of expectational equilibria 6 e, but does not mention the specific representation of 6 e in Eq. (3.2). The system of functional relations (3.2) gains special importance once we are willing to add the further assumption of an affine-linear function F(x, } ). In many applications, the affine-linearity of F(x, } ) is a direct consequence of the existence of an expected utility representation of individual preferences (example of Sect. 2). The problem of calculating the statistical expectational equilibrium ?* # 6 e reduces to an ordinary concave programming problem with linear constraints and strictly concave objective function. The first-order conditions of this specific maximization problem can be solved for an explicit solution. An additional technical assumption of the non-emptiness of the relative interior of the set of expectational equilibria 6 e is needed to ensure the existence of a vector of KuhnTucker multipliers: Assumption 4. The set of expectational equilibria 6 e is a convex polyhedral, that is, the functions F(x, } ): 2  R n are affine-linear: \j=1, ..., n : Fj (x, ? x )=a jx + : ? xx$ f jxx$ ;

a jx , f jxx$ # R.

x$ # X

Moreover, 6 e has a non-empty relative interior.

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(6.1)

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Assumption 4 implies Assumption 2 and Assumption 3. Assumption 1 and Assumption 4 suffice to derive an explicit representation of the statistical expectational equilibrium: Proposition 3. If Assumption 1 and Assumption 4 hold, then the statistical expectational equilibrium ?* is of the exponential class: n

\ + exp : * f \ +;

&1 exp \x, x$ : ?* xx$ =Z x

: * jx f jxx$

j=1

(6.2)

n

Z x := : x$ # X

Proof. tion 4): max e

?x # 6 x

jx

jxx$

* jx # R.

j=1

For each state x, ? * x is the solution to (theorem and Assump-

{& :

? xx$ ln ? xx$

x$ # X

=

s.t. : a jx + : ? xx$ f jxx$ =0;

j=1, ..., n.

x$ # X

(6.3)

In (6.3) we omitted the non-negativity constraints ? xx$ 0 and the normalization constraints  x$ ? xx$ =1. The next step in the proof will show that the non-negativity constraints are never binding; hence this omission is justified. Normalization will be taken care of by introduction of a normalization constant Z &1 x . Because the objective function for each of the maximization problems (6.2) is strictly concave, we know that ? * x achieves n * *), * * # R , is a saddle point of the maximum in (6.2) if and only if (? *, x x x the Lagrangian n

\

+

L x = & : ? xx$ ln ? xx$ + : * jx a jx + : f jxx$ ? xx$ . x$ # X

j=1

x$ # X

(6.4)

First-order conditions for the characterization of a saddle point of L x yield n

\x, x$ : &1&ln ?* xx$ + : ** jx f jxx$ =0.

(6.5)

j=1

Solving for ?* xx$ in (6.5) and normalizing leads to (6.2), where the asterisk at the multipliers is omitted. K The representation (6.2) of the statistical expectational equilibrium has two remarkable properties. First, the probability of moving from state x to state x$ is strictly positive for all pairs of states. Statistical expectational equilibrium does not rule out the occurrence of any transition deemed to

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be possible. Second, to each constraint j corresponds one term * jx f jxx$ in (6.2) carrying its specific economic meaning. The representation (6.2) establishes the link between economic theory and econometric testing alluded to in the Introduction. Let us return to our introductory example of Section 2. Suppose that the marginal disutility of working is a constant, u, and the marginal utility of consumption is a constant, v. 21 Further, let the set of possible price vectors (state space) be a finite set of positive real numbers. The system of equations (2.15) then reads u v \p : =: ? pp$ . p p p$

(6.6)

Denote the marginal utility ratio vu by : and introduce the purchasing power x=1p. Equation (6.6) then becomes a version of the linear expectational difference equation frequently encountered in macroeconomics: \x : x=: : ? xx$ x$.

(6.7)

x$

Let us assume that :>1. Further, assume that the state space X (set of possible values of purchasing power) contains the state x 1 =0 : X= [0, x 2 , ..., x S ] with 01 ensures that no absorption occurs when the largest value of X is realized. Clearly, for all states x # X except x=0 Assumption 4 holds. Hence, Proposition 3 is applicable and transition probabilities of the statistical expectational equilibrium are given by:22 &1 &*x x$ \x, x$ : ?* xx$ =Z x e

Z x =: e &*x x$.

(6.8)

x$

In (6.8) we introduced a minus sign in front of the ``multiplier'' * x for reasons which will become apparent below. For each state x, the real number * x is 21

Although this specification does not satisfy the conditions stated in Section 2, an equilibrium still exists and is given by Eq. (6.6). With linear utility, all agents are indifferent between all levels of labor supply in equilibrium. The equilibrium value of employment is determined by the exogenous money stock M in the economy. 22 The following formulas are valid for all ? xx$ but ? 0x$ , for which we have: ? 00 =1 and ?0x$ =0 otherwise (the state 0 is an absorbing state).

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implicitly defined by Eq. (6.7). This number * x uniquely defines the conditional distribution ? x , x{0. If the cardinality of X is large, the discrete distribution ? x can be approximated by a density function ?( } | x) defined on the interval [0, x ]. In this case, the normalization constant in (6.8) can be calculated by straightforward integration and we get: ?(x$ | x)=

*(x) e &*(x) x$. 1&e &*(x) x

(6.9)

In the limit x   (6.9) approaches the density function of an exponential distribution with parameter *(x)>0: ?(x$ | x)=*(x) e &*(x) x$.

(6.10)

For given state x, 1*(x) is then the (conditional) mean and the (conditional) variance of the distribution (6.10). Substituting the expression (6.10) for ?( } | x) into the equilibrium conditions (6.7) shows that 1*(x)=x. Hence, the conditional variance of purchasing power is increasing in the current level of purchasing power x. If the linear expectational difference equation is an exhaustive description of our knowledge of the economy, statistical equilibrium predicts conditional variances increasing in levels. This property seems to be a characteristic of the typical economic time series and has given rise to a large body of econometric work on autoregressive conditional heteroscedastic models [7, 14]. The analysis conducted here suggests that this relationship between levels and conditional variances can be understood as a purely statistical effect. Notice also that, in contrast to [7, 14], this relationship is derived from a fundamental statistical principle; statistical equilibrium, or equivalently, the maximum entropy formalism, applied to Eq. (6.7) implies a strong connection between level and conditional variance. 23

7. CONCLUDING REMARKS This paper combined statistical and economic equilibrium analysis. We assumed that the entire economic analysis can be summarized by a set of expectational equilibria, 6 e. The statistical analysis introduced a mapping, 6 T , from sample space, 0 T , to the set of empirical transition matrices, 6 T . 23 If we work with the original Eq. (6.6) and first calculated the statistical expectational equilibrium for prices p, the equilibrium distribution for purchasing power x changes; the conditional distribution becomes a gamma distribution with parameters 3 and * x . Even in this case, however, the conditional variance is still increasing in level x.

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Given a prior measure on the sample space, 0 T , combinatorics allowed us to derive a measure on the set of expectational equilibria of a T-observations economy which provided a powerful equilibrium selection criterion: in the infinite observations limit, there exists only one equilibrium, ?*, with non-vanishing probability of occurrence (theorem). The maximum entropy characterization of statistical expectational equilibrium (corollary) summarizes the main implication of the statistical analysis. If we do not rule out endogenous randomness by assumption, then endogenously created random fluctuations should not come as a surprise to the rational researcher. The perfect foresight equilibrium is a highly unlikely event. The exponential representation (Proposition 3) indicates the usefulness of the theory by providing a mechanical way of calculating the statistical expectational equilibrium. Only future research can demonstrate the practical merits of the statistical method, but its success in statistical physics justifies a certain degree of optimism. In this paper we confined attention to one-step forward looking economic models, but the statistical method is more general. Whenever the economic analysis pins down a compact and convex set of probability distributions, the concept of statistical equilibrium can be applied. Aumann's game-theoretic solution concept of correlated equilibrium [1, 2] determines a compact and convex set of probability distributions (strategy profilespure, mixed, and correlatedmixed). The issue of endogenous versus exogenous state space does not arise in this application, since game theory traditionally takes the state space (set of pure strategy profiles) as exogenously given. If we assume that Aumann's concept of correlated equilibrium is a complete representation of the economic analysis, then the statistical expectational equilibrium is the maximum entropy correlated equilibrium. The application of statistical expectational equilibrium to game theory is an interesting subject for future research.

APPENDIX Proof of the Theorem. Since the measure + T (?) factorizes and the number of states, S, is a finite number, (5.4) holds if and only if the S sequences of marginal measures [+ xT ] satisfy \x>0 : lim + xT (2 T"B = (? *))=0, x T

(A.1)

where B = (? x*)/2 is the open ball in 2 with radius = and center ? x*. Hence, we will proceed by showing that (A.1) holds for any state x, and then conclude that the theorem is true.

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Let ? xx$ # [0, 1] be a typical component of the conditional probability distribution ?x . Define the entropy function H: 2  R + ; ? x ~ & x$ # X ? xx$ ln ? xx$ which assigns a positive real number to each distribution ? x (the row vectors of the transition matrix ?). Define the maximum entropy distribution as ?~ x =arg max H(? x ).

(A.2)

e

?x # 6 x

Clearly, the maximum entropy distribution exists and is unique, since it is defined as the solution to a maximization problem with strictly concave objective function H(? x ) and compact, convex constraint set 6 ex . We will show that (A.1) is true and that ? * x is identical to ?~ x . The idea of the proof is to show that (i) the most likely distribution converges to the maximum entropy distribution in the limit T  ; (ii) the most likely distribution becomes ``extremely likely'' in the sense that all distributions not contained in B = (?~ x ) are a probability zero event in the limit T  . The proof is conducted by finding, for finite but arbitrary T, an upper bound to the probability + xT (2 T"B = (?~ x )). This upper bound is equal to a factor times the ratio of the multiplicity of two distributions: the most likely T-observations distribution not contained in the ball B = (?~ x ), and the maximum entropy T-observations distribution contained in the ball B = (?~ x ). By using Stirling's formula we can show that this upper bound converges to zero as T goes to infinity. The key to this result is that the term dominating the limiting behavior of this upper bound is the entropy difference of both distributions. The proof is conducted in three steps. Step 1. For finite T, we show the existence of the most likely T-observations distribution not contained in the =-ball around ?~ x and the existence of the maximum entropy T-observations distribution contained in the =-ball around ?~ x . We need two lemmas. Lemma 1. For any =>0 there exists a positive integer T1 =T1 (=) such that

Proof.

\T>T1 : 6 exT & B = (?~ x ){<.

(A.3)

6 exT & B sT (?~ x ){<.

(A.4)

We have:

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For given =>0 there exists a positive integer T1 =T1 (=) such that \TT1 : B sT (?~ x )/B = (?~ x ).

(A.5)

The conjunction of (A.4) and (A.5) implies (A.2). K Lemma 2. There exists an = >0 and T2 such that \TT2 , == : 6 exT"B = (?~ x ){<.

(A.6)

Proof. Since 6 ex has at least two elements, there exists a ? x # 6 e, ? x {?~ x . Clearly, we can find an = >0 such that ? x # 6 ex"B = (?~ x ).

(A.7)

Therefore, there also exists an =^ such that B =^ (? x )/6 ex"B = (?~ x ).

(A.8)

Moreover, we can always find a T2 such that \TT2 : B =^ (? x ) & 2 T {<.

(A.9)

Statements (A.8) and (A.9) in conjunction with (6 ex"B = (?~ x )) & 2 T /6 exT"B = (?~ x )

(A.10)

imply statement (A.6). K Without loss of generality, assume that Tmax[T1 , T2 ] and == . Define the maximum entropy T-observations distribution contained in an =-ball around ?~ x as ?~ Tx =arg

H(? x ).

max e

(A.11)

?x # (6 xT & Be (?~x ))

Because of the first lemma, the constraint set in (A.11) is non-empty. The constraint set is also finite. Therefore, the maximization problem (A.11) is well-defined. If here are multiple solutions to (A.11), pick one according to an arbitrary rule. Define the most likely T-observations distribution not contained in an =-ball around ?~ x as ?~ Tx =arg

max e

+ xT (? x ).

?x # (6 xT" Be (?~x ))

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(A.12)

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STATISTICAL EQUILIBRIUM

Again, the maximization problem (A.12) is well-defined because of Lemma 2 and the finiteness of the constraint set. If there are multiple solutions, pick one according to an arbitrary rule. Step 2. For finite T, we establish an upper bound to the probability +xT (2 T" B = (?~ x )). We use the following version of Stirling's formula: n # N : - 2?nn n e &n
(A.13)

We also have the obvious fact |2 T | T S. Putting everything together, + xT (2 T" B = (?~ x )) as follows:

(A.14)

we establish

an upper bound to

+ xT (2 T" B = (?~ x )) T S + xT (?^ Tx ) =T S

[> x$ # X (T?^ Txx$ )!] &1  ?x # 6 exT [> x$ # X (T? xx$ )!] &1

T S

[> x$ # X (T?^ Txx$ )!] &1 [> x$ # X (T?~ Txx$ )!] &1



T S > x$ # X - 2? exp((T?~ Txx$ +12) ln(T?~ Txx$ )&T?~ Txx$ ) > x$ # X - 2? exp((T?^ Txx$ +12) ln(T?^ Txx$ )&T?^ Txx$ +1(12(T?^ Txx$ &1)))

\

\:

& :

1 12(T?^ Txx$ &1)

=exp S ln T&T

x$ # X

?^ Txx$ ln ?^ Txx$ & : ?~ Txx$ ln ?~ Txx$

x$ # X

x$ # X

+

+

\

=exp S ln T&T(H(?~ Tx )&H(?^ Tx ))& : x$ # X

1 . 12(T? Txx$ &1)

+

(A.15)

The proposition is proved if we can show that lim (H(?~ Tx )&H(?^ Tx ))>0.

T

(A.16)

Step 3. Show that the upper bound vanishes as T goes to infinity, i.e., show that (A.16) holds.

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TOM KREBS

First we show that lim H(?~ Tx )=H(?~ x ).

(A.17)

T

To see that (A.17) holds relabel the constraint sety in the maximum entropy problem (A.11) as follows. Define the set 3=[0, ST, S(T +1), S(T +2), ...]R with T =max[T1 , T2 ]. Denote a generic element of 3 by % T =ST for all TT and % T &1 =0. The correspondence 1: 3  2 % T ~ 1(% T )=

{

6 exT & B = (?~ x ), 6 ex & B = (?~ x ),

TT T=T &1

is non-empty (lemma) and compact valued (obvious). Since 3 is a discrete set and the range of the correpondence 1 is a discrete set for % T {0, it is continuous at all points % T {0. Continuity at 0 follows from the fact that e e e (  T=1 6 xT ) & 6 x is dense in 6 x . Thus, 1 is a non-empty and compactvalued continuous correspondence. The maximization problem (A.11) has a continuous objective function H( } ) and constraint set which is represented by a non-empty and compactvalued, continuous correspondence 1. Hence, we can apply the maximum theorem [6]. Using ?~ x =arg max H(? x )=arg ?x # 6 ex

max ?x # (6 ex & B= (?~x ))

H(? x )

(A.18)

we conclude that lim H(?~ Tx )=H(?~ x ).

T

(A.19)

Define H .

max e

H(? x ).

?x # (6 x & B= (?~x ))

An argument similar to the previous one shows that lim H(?^ Tx )=H.

T

(A.20)

Since H =H(?~ x ) is the unique maximum of the strictly concave function H( } ) on the domain 6 ex , (A.17) in conjunction with (A.20) implies (A.16). This completes the proof of the theorem. K

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393

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