Multiple equilibria in dynamic rational expectations models: A critical review

Multiple equilibria in dynamic rational expectations models: A critical review

ARTICLE IN PRESS European Economic Review 50 (2006) 171–210 www.elsevier.com/locate/eer Multiple equilibria in dynamic rational expectations models:...

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ARTICLE IN PRESS

European Economic Review 50 (2006) 171–210 www.elsevier.com/locate/eer

Multiple equilibria in dynamic rational expectations models: A critical review Robert Driskill Department of Economics, Vanderbilt University, Nashville, TN, USA Received 12 February 2004; accepted 10 March 2005 Available online 4 May 2005

Abstract Multiple equilibria are a ubiquitous feature of dynamic rational expectations models. Researchers have been divided on the implications of this phenomenon. Some have viewed this as a reflection of reality and a possible explanation of a wide range of economic phenomena. Others have suggested various selection criteria for choosing one among the many equilibria. This paper reviews the major selection criteria that have been proposed, and through application to three well-known models shows under what circumstances one might expect them to choose the same or different equilibria. In addition, this paper proposes a new criteria based on the limit of finite-horizon equilibria and investigates its relation to the other criteria. r 2005 Elsevier B.V. All rights reserved. JEL classification: D840; E000 Keywords: Multiple equilibria; Rational expectations; Adaptive learning; Selection criteria; Indeterminacy; Minimal state variable

1. Introduction Multiple equilibria in rational expectations models are problematic because completion of such models as equilibrium models requires some specification about how people coordinate their beliefs. Without such a specification, if people differ in their expectations, not all of them can be right. Tel.: +1 615 343 1516.

E-mail address: [email protected] 0014-2921/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.euroecorev.2005.03.006

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Researchers have taken a variety of perspectives on the existence of multiple equilibria in rational expectations models. Some have viewed the existence of multiple equilibria as a reflection of reality and a possible explanation of phenomena ranging from speculative booms to the Great Depression to U.S. macroeconomic behavior in the era preceding the appointment of Paul Volker as Fed Chairman.1 In contrast, others have looked for reasons to focus on only one equilibrium. Broadly speaking, they have looked for conditions or reasons that might govern whether economic agents in a multiple equilibria environment will coordinate their beliefs (or behave as if they coordinate their beliefs) on one equilibrium. This paper focuses on this branch of analysis. Within this branch of analysis, there are two strands of research. In one strand, investigators have added to static multiple-equilibria models explicit dynamic processes that make multiple equilibria in the dynamic version a ‘‘less-likely’’ event. For example, the model of Eden (1983), many of the ‘‘coordination game’’ examples discussed in Cooper and John (1988), the models discussed in Morris and Shin (1998, 2001), and the migration-development models that form the basis of Matsuyama (1991), Krugman (1991), Frankel and Pauzner (2000), and Karp (1999) may have multiple equilibria even in their one-period or n-period versions. By adding dynamic ‘‘frictions’’ to some of these models, the range of initial conditions that are consistent with multiple equilibria can be shrunk.2 For the purposes of this paper, the hallmark of the models used in this research is existence of multiple equilibria in their finitehorizon manifestations. For many other models, though, the assumption of an infinite time horizon is a necessary condition for the existence of multiple equilibria. These models are the focus of analysis of this paper. For these models, the search for some sort of coordination-of-beliefs mechanism has been motivated by the assumption that some equilibria are more ‘‘natural’’ than others.3 That is, such equilibria can be thought of as satisfying a ‘‘selection criterion’’ with certain desirable properties.

1

See Hahn (1966) for an early example in which existence of both a non-stationary and stationary solution was viewed as a potential explanation of speculative booms. This approach is also found in Blanchard (1979), in which short-lived probabalistic bubbles are a result of existence of two solutions. See Bryant (1981, 1983) for an example in which existence of both a degenerate and non-degenerate solution is viewed as a potential explanation of bank collapse and depression. See Diamond (1982), Howitt and McAfee (1988) and Cooper and John (1988) for examples in which non-degenerate multiple equilibria are viewed as potential explanations of ‘‘good times’’ and ‘‘bad times.’’ See Flood and Marion (1998, 2000) for examples of the use of multiple equilibria in explaining currency crises. See Clarida et al. (1999, 2000) for the argument that multiple equilibria can explain pre-Volker macroeconomic behavior. 2 See Karp and Paul (2003) for an overview. For examples, Matsuyama (1991) has a dynamic model that reduces the range of initial conditions (relative to an analogous static model) over which there can be multiple equilibria. Frankel and Pauzner (2000) add Brownian motion shocks to Matsuyama’s model and reduce to zero the range of initial conditions over which there can be multiple steady states. 3 We put quotation marks around the word natural to emphasize this is not a well-defined concept, but more like U.S. Supreme Court Justice Potter’s concept of pornography: he could not define it but knew it when he saw it.

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Unfortunately, these criteria vary, model to model, both in terms of whether they select a unique equilibria and in terms of whether they select the same equilibrium. This paper will analyze under what conditions the most popular of these various selection criteria restrict equilibria of the above infinite-horizon models to a unique equilibrium. When they do select a unique equilibrium, this paper will analyze under what conditions, i.e., in which types of models and for what set of parameter values, and why, they restrict them to the same, or to a different, equilibrium. Perhaps the most frequently invoked criterion for selecting a unique equilibrium is that of stability. Stability usually means that, for arbitrary initial conditions, the model’s variables converge through time to a stationary state. For many models, imposition of this requirement suffices to eliminate all but one equilibrium.4 But there are at least two types of models for which the stability criterion is unsuitable. First, some models have multiple stable equilibria.5 Clearly, for such models the stability criterion does not select out a unique equilibrium. Second, some models with multiple equilibria have none that are stable. We might still be interested in the characteristics of these models, and consequently might still want to have some criterion for choosing one of the possible equilibria. Again, in such a case the stability criterion does not help.6 A second criterion, first recommended by Wallace (1980, p. 55), requires that a solution be derived under the assumption that agents’ (rational) expectations are a function of a minimal set of state variables. Wallace based his recommended selection criterion on the idea that, in a model in which the forecasting environment was stationary (in the sense that time did not appear explicitly in describing the environment), forecasting rules should also be stationary.

4

See Blanchard and Kahn (1980), Obstfeld and Rogoff (1986), Blanchard and Fischer (1989, Chapter 5), Turnovsky (1995, Chapter 5), McCallum (1999), Benhabib and Farmer (1999, pp. 392–393). Frequently, the appeal to stability as a selection criterion is based on existence of transversality conditions for infinitehorizon optimizing models that, if interpreted as necessary conditions for optimality, require convergence to a steady state. Such transversality conditions thus act as boundary conditions for the optimal dynamic equations of the models. But oftentimes the stability criterion is used even with models for which there are no transversality conditions. Furthermore, transversality conditions— ‘‘boundary conditions arising from optimality considerations’’ (Kamien and Schwartz, 1981, p. 49)—in infinite-horizon models are usually expressed as limits of finite-horizon terminal conditions. Thus the stability criterion as enforced by transversality conditions is really a criterion based on the limit of a finite-horizon first-order condition that is only sometimes, i.e., under some conditions, a necessary condition. See Halkin (1974) for the classic example of an optimal solution to an infinite horizon problem that does not satisfy the infinite-horizon transversality condition. 5 See, for example, Calvo (1978), McCafferty and Driskill (1980), and more recently, the macroeconomic literature focussed on choices of Taylor rules that eliminate multiple equilibria, such as Carlstrom and Fuerst (2000) and Woodford (2003). Examples in dynamic games include Tsutsui and Mino (1990), Karp (1996a) and Driskill (1997). To the extent that positive externalities are important, we should expect more dynamic models with multiple stable equilibria along the lines of Eaton and Eckstein (1997), Driskill and McCafferty (2001) and macroeconomic models such as Benhabib and Rustichini (1994), Benhabib and Farmer (1994), Farmer and Guo (1994), Xie, 1994, Bond et al. (1996), Benhabib and Nishimura (1998), and Benhabib and Farmer (1999). 6 See McCallum (1999, pp. 625–626) for a discussion of this issue.

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In a series of papers, McCallum (1983a,b, 1990, 1999, 2002) developed and applied a more complete criterion that is consistent with and also subsumes Wallace’s criterion. McCallum’s criterion selects the solution consistent with a forecasting function, i.e., an expectations-formation function, which has as its arguments the minimal set of state variables found in the structural model and which has parameters that are themselves continuous functions around key values of structural parameters. Such a ‘‘minimal state variable’’ or ‘‘MSV’’ solution eliminates ‘‘bootstrap’’ equilibria, i.e., equilibria that occur only because they are expected to occur. A key feature of this criterion is that by design it selects always a unique equilibrium. More recently, a criterion has been proposed that selects an equilibrium that is ‘‘expectationally stable’’ (e-stable) or, equivalently, ‘‘learnable’’ (Evans, 1986, 1989; Evans and Honkapohja, 1999, 2001; Honkapohja and Mitra, 2001; Lettau and Van Zandt, 2001). The basic idea behind this criterion is that economic agents begin with an exogenous ‘‘guess’’ of the value of a parameter in a forecasting function, and then use ‘‘reasonable’’ updating rules to generate future guesses. If these guesses converge to the parameter values of one of the multiple rational expectations forecasting functions, then the equilibrium implied by this forecasting function is the one selected.7 The MSV criterion and the e-stability criterion select the same equilibrium in some models, but select different equilibria in others. A goal of this paper is to provide a framework for understanding under what conditions one might expect these two criterion to agree on an equilibrium selection. One way in which we attempt to reach this goal is by providing an alternative selection criterion to those mentioned above. This alternative is motivated by a view that multiple equilibria are endemic to these dynamic models because of the conjunction of the rational expectations assumption with an infinite time horizon. The infinite time horizon does not necessarily apply to each individual, but may only apply for the model economy as a whole, as in the case of overlapping-generations models. The criterion we suggest is to select the equilibrium that is the limit of the equilibrium of an analogous finite-horizon model. For expository purposes, we will call this criterion either the ‘‘finite-horizon’’ or the ‘‘backward-induction’’ criterion. Before explaining how this criterion helps understand connections between the MSV and e-stability criterion and why this finite-horizon solution deserves consideration as a selection criterion, let us first consider two interrelated heuristic

7 There is a vast literature on this criterion, much of which is referenced in Evans and Honkapohja (2001). In this paper we focus on e-stability in part because Evans and Honkapohja (2001) have demonstrated that most stability criteria for learning are identical to those for e-stability. Although many of the applications of learning are not focused on the problems of multiple equilibria, this problem remains central in this literature. As Evans and Honkapohja (2001, p. 14) write in their introduction: ‘‘Throughout the book the multiplicity issue will recur frequently, and we will pay full attention to this role of adaptive learning as a selection criterion.’’

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explanations of why we view multiple equilibria in infinite-horizon dynamic rational expectations models as endemic rather than as just a frequent occurrence.8 First, dynamic rational expectations models have a structural equation or structural equations that link at any moment in time the value of a variable or variables at that moment to the expectation of the future adjacent-moment’s value of that variable or variables. Hence, such models have the structure of difference or differential equations. Unique solutions to such models require boundary conditions. We argue that models with an infinite horizon and the rational expectations assumption are inherently one (1) boundary condition short of what is needed for the determination of a unique solution. Second, such models can also be thought of as not having enough equations to allow the possibility of a unique solution for the values of the endogenous variables. To see this, consider the following heuristic way of thinking about the idea of rational expectations. Imagine a model in which the equilibrium value of some endogenous variable, perhaps price, is determined at some time t by the equilibrium condition that supply equals demand, and in which demand and/or supply depends on next-period’s expected price. The market-clearing condition would thus be expressed as Pt ¼ fðPex tþ1 Þ. ex is the expected price. where Pt is price, Ptþ1 The rational expectations assumption is that agents ‘‘know the model.’’ Hence, a natural way to model how agents form their expectations of next-period’s price is to assume they know that next-period’s price will be that price that equilibrates nextperiod demand and supply: ex ex Ptþ1 ¼ fðPtþ2 Þ.

Of course, next-period’s demand in turn depends on the subsequent-period price, which depends on the price after that, and so on for as far into the future as the model permits. In an infinite-horizon model, this means that there is always one more price to be determined – the next-period price – than there are equilibrium conditions. That is, for any arbitrary number of periods n, there are n equations – the equilibrium conditions that demand equals supply – but n þ 1 endogenous variables – the n prices plus the ðn þ 1Þth expected price. This problem is reminiscent of the parable recounted by Shell (1971) of the hotelier who managed a hotel with an infinite number of rooms. In the tale, a traveler approached the hotelier and asked for a room. The hotelier responded that he was all booked up, but could still make room for him. He would simply move the guest in room #1 to room #2, the guest in room #2 to room #3, and so forth. The equilibrium price at some initial time is analogous to the traveler. In an infinite-horizon model, room can be made for a variety of possible equilibrium 8 While there seems to be widespread agreement that many rational expectations models have multiple equilibria, and the explanation espoused here seems to have been known as a sort of ‘‘folk theorem’’ to some people, no one seems to have stated that it is a universal feature of such models.

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values of this first price by changing all subsequent prices in an appropriate fashion. What is an ‘‘appropriate fashion’’ depends on the details of the model at hand. Seen in this light, multiple equilibria in dynamic rational expectations models are a fundamental feature of infinite-horizon models. Hence, the finite-horizon criterion, which addresses this cause of multiplicity directly, seems useful. Elements of this idea have been noted before. Game theorists, for example, have long known that restricting the strategy space in dynamic games by assuming a finite rather than an infinite horizon reduces the number of equilibria. Along these lines, Chari and Kehoe (1990) and Stokey (1991) noted that a finite horizon restricted the number of equilibria in games between governments and atomistic private agents. Along a different dimension, Blanchard and Fischer (1989), in their discussion of multiple equilibria, note that prices for finite-maturity bonds cannot have multiple rational expectation solutions. What is new in this paper is the use of this idea to encompass a number of approaches to understanding the multiple equilibria phenomenon in a variety of applications. It allows us to see that in some cases, both nonlinear and linear models, some of which have been thought to have multiple equilibria for different reasons, in fact have multiple equilibria for this same basic reason.9 It also sheds light on how the e-stability and MSV selection criteria are related to each other and to the finite-horizon criterion. In particular, it suggests that in many models the finite-horizon criterion and the MSV criterion are intimately related and select the same equilibrium. The MSV solution is designed to eliminate ‘‘bootstrap’’ equilibria. The finite-horizon solution also eliminates bootstrap equilibria because such equilibria can only occur in an infinite-horizon model. In contrast, it also suggests that, in many models, whether the e-stability criterion selects the same equilibrium as the MSV and finite-horizon criterion depends on crucial details of the model and on the exact specification of the features of the updating rules of the e-stability approach. In the remainder of the paper, we show how the four aforementioned selection criteria work when applied to some well-known models. The examples consist of a linear model of McCallum (1983a, 1999) that builds on the model of Cagan (1956), the nonlinear classic Muth (1961) model of inventory speculation as amended by McCafferty and Driskill (1980), and overlapping generations models, one in which the only asset is land, and one in which the only asset is fiat money.

2. McCallum’s example This model consist of a money demand function, a money supply function, and an equilibrium condition. The linearity and small number of parameters in this model 9 Turnovsky (1995, p. 135), for example, writes that the sources of multiple equilibria in the nonlinear model of McCafferty and Driskill (1980) ‘‘are associated...with solving nonlinear systems’’ and ‘‘are very different from the sources of nonuniqueness discussed...in conjunction with the Taylor model.’’ McCafferty and Driskill used the modifier ‘‘non-linear’’ in their title in part because they also believed the multiplicity arose from solving a non-linear system. The analysis in this paper suggests this is not so.

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allows a particularly clear presentation of the basic ideas. One interesting result is that the MSV, finite-horizon, and e-stability criteria all select the same equilibrium for a set of parameter values for which the model is what McCallum (2002) called ‘‘well-formulated.’’ A well-formulated model excludes parameter values over which there exists an infinite discontinuity in the solution of the model. In contrast, with parameter values for which the model is not well-formulated, this model has an MSV solution that is not e-stable and is not the limit of the finitehorizon solution. The lack of agreement between the MSV and e-stability criterion in this type of case has been emphasized by Evans and Honkapohja (1999) as a possibility in ‘‘sensible’’ models. In contrast, McCallum (1999) seems to have been motivated to develop the idea of a well-formulated model as an explanation of why such models are not economically well-motivated. The simplicity of this model allows us to pinpoint fundamental similarities among the MSV, e-stable, and finite-horizon solutions that help explain these results and help explain in a related sense why the not-well-formulated case might be thought of as ‘‘not sensible.’’ It also suggests under what more general conditions one might expect the e-stability criterion to not agree with the MSV and finite-horizon criteria. 2.1. An infinite-horizon model 2.1.1. Structure Consider the following example from McCallum (1983a).10 Money demand is given as mdt ¼ g þ pt þ aðpex tþ1  pt Þ;

ap0; t ¼ 1; 2; . . . ,

(2.1)

where mt denotes the log of the nominal money stock at time t, pt denotes the log of the price level at time t, pex tþ1 denotes the expectation, based on information available at time t, of what will be the log of the price level at time t þ 1, a and g are parameters. Money supply follows the rule mst ¼ m0 þ mpt1 ; t ¼ 1; 2; . . . ,

(2.2)

where m0 and m are parameters and p0 is given. For expository ease, we will assume m0 ¼ 0. This money supply function is basically a device that allows illustration of concepts, i.e., is a deus ex machina, and is not derived from analysis of deeper motivations of the monetary authority. The money market is assumed to always be in equilibrium: mst ¼ mdt ;

t ¼ 1; 2; 3; . . . .

Substituting the behavioral relationships (2.1) and (2.2) into this equilibrium condition for time t, we have mpt1 ¼ g þ pt þ aðpex tþ1  pt Þ; 10

t ¼ 1; 2; 3; . . . .

(2.3)

We omit random variables from the model for expository ease; inclusion of them would not affect the analytic points made here.

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For expositional purposes, we will refer to this equation as either the equilibrium condition or as the structural model. 2.1.2. A heuristic solution and the fundamental indeterminacy To provide a heuristic understanding of the fundamental indeterminacy of this model, assume perfect foresight: pex tþ1 ¼ ptþ1 . Making this assumption and rearranging (2.3) yields the following second-order difference equation:   m g 1a ptþ1 þ pt  pt1 ¼ ; t ¼ 1; 2; . . . . (2.4) a a a The solution for this difference equation is given by pt ¼ Alt1 þ Blt2 þ p; t ¼ 0; 1; 2; . . . ,

(2.5) g 1m,

where A and B are arbitrary constants, p ¼ and 8 9 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     1< 1a 1  a 2 4m=  , l1  þ  2: a a a; 8 9   sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 < 1 1a 1a 4m= l2 ¼  , þ þ 2: a a a; which are solutions to the characteristic equation   1a m l2 þ l  ¼ 0. a a

(2.6.i)

(2.6.ii)

(2.7)

The only ‘‘natural’’ boundary condition is that p0 is given. To be able to solve for the two unknowns A and B we need another boundary condition. Any criterion that selects out a unique equilibrium must provide this additional boundary condition. We now move on to applying the four selection criteria to this model. Following the literature, we restrict our analysis to parameter values for which the roots of (2.7) 4m 2 are real, i.e., to parameter values such that ð1a a Þ þ a X0. Note that for real roots, 11 jl1 jpjl2 j. 2.1.3. The stability selection criterion Now, for pairs ða; mÞ of the parameter space such that ao0 and ð2a  1Þomp1, jl1 jo1 and jl2 jX1. In this case, the stability criterion, which requires that pt cannot grow without bound, can be invoked to supply the boundary condition of B ¼ 0. 11

From Eq. (2.7), we can write l1 ¼ x  y,

l2 ¼ x þ y, ð1aÞ2 4m 1=2 where x   ð1aÞ X0 for non-imaginary roots. Hence, jl2 jXjl1 j. 2a 40 because ao0 and y  ½ a2 þ a 

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The solution parameter A is then determined by the initial condition that p0 (which is given) equals A þ p. In all other regions of the ða; mÞ parameter space for which the characteristics roots 2

are real, though, the stability criterion has no bite. When ao  1 and m ¼ ð1aÞ 4a , 0ol1 pl2 o1 and all paths of pt converge to p, regardless of the values of A and B, as long as A and B are finite. When ao0 and mp2a  1, or when 1oao0 2

and m ¼ ð1aÞ 4a , jl2 jXjl1 jX1 and the stability criterion is also of no help: given that p0 is given, imposition of the boundary condition that limt!1 pt is bounded (which implies A ¼ B ¼ 0) leads to the inconsistency (for arbitrary p0 ) that pt ¼ p0 and pt ¼ p. The stability criterion, then, selects a unique equilibrium only when the model’s parameter values fall within some of the permissible regions of the parameter space. As noted, some researchers view the existence of parameter regions for which there are multiple stable equilibria as interesting precisely because of this possibility, while others view this as giving rise the most problematic type of multiple equilibria.12 And regardless of one’s position on this point, one might agree with McCallum (1999, p. 625) that there are still interesting questions to ask of the model even when parameter values are such that the model is unstable. We now turn to these alternative selection criteria that can apply over more regions of the permissible parameter space. 2.1.4. The minimal state variable solution McCallum’s approach to dealing with the fundamental indeterminacy of infinitehorizon rational expectations models is to look for solutions that rule out ‘‘bubble’’ or ‘‘bootstrap’’ effects. Operationally, his approach looks for solutions for which the associated forecasting function, i.e., reduced-form solution, depends only on the minimal set of state variables. When this criterion alone does not produce a unique solution, the subsidiary criterion that ‘‘solution formulae be valid for all admissible parameter values’’ (McCallum, 1983a, p. 140) is invoked. State variables in the model are identified from (2.3), and are pt1 and the constant g.13 McCallum’s procedure requires us to look for a reduced-form solution of the form pt ¼ lpt1 þ gy,

(2.8)

where l and y are functions of the structural parameters m and a: l ¼ lðm; aÞ,

(2.9.i)

y ¼ yðm; aÞ.

(2.9.ii)

12

Blanchard and Fischer (1989), for example, write: The second type of multiplicity, the multiplicity of stable equilibria...is more unsettling. It has led to different reactions. One has been nihilistic, arguing that there is little that can be said by economists about economic dynamics under such conditions (p. 260).

13

The constant g is a parameter, but can also be treated as ‘‘state’’ variable because, in contrast to a and m, it enters the structural model linearly.

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Furthermore, McCallum’s subsidiary principle imposes the restriction lð0; aÞ ¼ 0.

(2.10)

That is, when m ¼ 0, l ¼ 0. The logic here is that if m ¼ 0, the structural model (as seen from (2.3)) no longer depends on pt1 . Hence, if m ¼ 0, a reduced-form function of the state variables would not have pt1 in it, and our function l ¼ lðm; aÞ should reflect this. Using the method of undetermined coefficients (see, among many, Blanchard and Fischer, 1989), McCallum found two solutions: pt ¼ li pt1 þ yi g;

i ¼ 1; 2,

(2.11)

1 where li is given by (2.6) and yi ¼ 1þal . i To eliminate one of these possible solutions, McCallum’s subsidiary criterion requires that the solution be valid for m ¼ 0 and consistent with a minimal state representation. If m ¼ 0, then the minimal state is just a constant. Hence, only (2.6.i), the smaller characteristic root, yields the minimal state solution when m ¼ 0, and the selected solution (even when ma0) is pt ¼ l1 pt1 þ y1 g. As McCallum (1999) notes, the MSV solution does not require stability, and exists for all parameter values such that the roots (2.6.i) and (2.6.ii) are real. In terms of imposition of a second boundary condition, the MSV solution always eliminates the effect of l2 . Hence, in the perfect-foresight (2.5), the MSV criterion imposes the boundary condition that B ¼ 0 for all permissible pairs ða; mÞ. The MSV solution by design eliminates all but one solution for parameter values that give rise to real roots as solutions to the characteristic equation. We now ask two questions about this solution. First, can this selection criterion be motivated by an appeal to alternative behavioral assumptions? Second, under what conditions does this criterion lead to the same equilibrium choice as the e-stability, i.e., adaptive learning, criterion? To answer these questions, we need to first introduce another selection criterion.

2.2. The finite-horizon model and backward induction solution Now consider a finite-horizon version of the above model, where t ¼ 0; 1; 2; . . . ; T. Let pet denote the equilibrium price at any time t for this finite-horizon model, and let pt denote the equilibrium price in the MSV solution. The only difference in the structural model associated with this finite-horizon assumption is that the only reason people would hold money in period T is for transactions purposes. Hence, a natural boundary condition to impose is14 mdT ¼ g þ peT . 14

(2.12)

We should note that the specification of a demand for money function in a final period that is derived from explicit foundations, e.g., money in the utility function or cash-in-advance constraints, could be more problematic. Approaches such as found in Bacchetta and Van Wincoop (2000), in which a government end-of-period tax to be paid in money makes final-period holdings of money valuable, would generate an equation like (2.12). As pointed out by Bewley (1980), this device to make fiat money valuable is no more arbitrary (but perhaps less elegant) than the assumption of an infinite time horizon.

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Equilibrium at T is thus given by mdT

msT

zfflfflffl}|fflfflffl{ zfflffl}|fflffl{ g þ peT ¼ me pT1 .

(2.13)

The finite horizon does two things for us. First, it eliminates the infinite regress problem inherent in dynamic rational expectations models, i.e., there are now as many equilibrium conditions as there are endogenous prices. Second, it establishes by backward induction the form of the equilibrium price process, i.e., it establishes that it is linear in the state variables, namely the previous-period price, a constant term g, and the stochastic disturbance e. To emphasize this, we rewrite (2.13) as peT ¼ lT1 peT1 þ gyT ,

(2.14)

where lT1  m and yT ¼ 1. We can now construct the backward-induction rational-expectation solution. Using (2.14), the assumption of rational expectations, and the structural (2.3), we can write the following recursive relationship for any two adjacent values peTk and peTðkþ1Þ ; k ¼ 0; 1; 2; . . . ; T  1: peTk ¼ lTðkþ1Þ peTðkþ1Þ þ gyTk .

(2.15)

Substituting this into the equilibrium condition at any time T  k and rearranging yields g  gayTðk1Þ m þ . (2.16) pe peTk ¼ 1  a þ alTk Tðkþ1Þ 1  a þ alT1 Hence, the coefficients obey the following recursive relationships: m lTðkþ1Þ ¼ ; k ¼ 1; 2; 3; . . . ; T  1, 1  a þ alTk lT1 ¼ m, yTk ¼

1  ayTðk1Þ , 1  a þ alTk

yT ¼ 1.

(2.17.i) (2.17.ii) (2.18.i) (2.18.ii)

The solution given by Eqs. (2.16)–(2.18) is defined over all permissible pairs ða; mÞ, including those that lead to complex roots to the characteristic (2.7). Of most interest to us, though, is the limiting behavior of the finite-horizon solution as ðT  kÞ ! 1. Understanding this behavior allows us to make comparisons among the various selection criteria. To this end, we now restrict the parameter space ða; mÞ to only include those pairs for which l1 and l2 are real. We take up this task by first analyzing the behavior of the sequences of reduced-form parameters lTk ; and yTk . 2.2.1. Behavior of the sequence flTk g; k ¼ 1; 2; . . . ; T We consider first the recursive relationship for the lTk ’s. Note that the stationary values of this difference equation are the same as the two roots l1 and l2 . The question of interest is whether the sequence flTk g; k ¼ 1; 2; . . . ; T  k, converges to

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either l1 or l2 as ðT  kÞ ! 1. What we will first show is that the sequence defined by (2.17.i, ii) converges to l1 . That is, we will prove the following proposition: Proposition 1. Let the values of the terms of the sequence flTk g; k ¼ 1; 2; . . . ; T, be generated by the iterative process m lTðkþ1Þ ¼ . 1  a þ alTk Then lim

ðTkÞ!1

lTk ¼ l1 .

Proof. See appendix.

&

2.2.2. Behavior of yTk; ; k ¼ 0; 1; 2; . . . ; T Now consider analysis of the sequence fyTk g; k ¼ 0; 1; 2; . . . ; T. Because the evolution of lTk is independent of yTk , (2.17.i) is a linear difference equation with non-constant coefficients with solution ! ! ðk1Þ ðk2Þ Y Y X ðk1Þ yTk ¼ bTl yT þ bTl dTm þ dTðk1Þ , m¼0

l¼0

l¼mþ1

k ¼ 0; 1; . . . ; T,

ð2:19Þ

where bTk 

a 1  a þ alTk

(2.20)

dTk 

1 . 1  a þ alTk

(2.21)

and

One question is whether the sequence fyTk g converges as ðT  kÞ ! 1. For this to happen, it must be that limðTkÞ!1 bTk o1. Now, because limðTkÞ!1 lTk ¼ l1 , this is equivalent to a o1. (2.22) 1  a þ al1 This inequality is satisfied if and only if mo1: Proposition 2. If mo1, then Proof. See appendix.

a 1aþal1

2

o1. If 1omo ð1aÞ 4a , then

a 1aþal1

41.

&

2.2.3. A turnpike property This leads us to the following turnpike property: Proposition 3. Assume mo1 and that p0 ¼ pe0 . For any sequence fet g, and for any t40 and any arbitrarily small e40, there exists a horizon-length T4t sufficiently large such that j pet  pt j oe.

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Proof. At t, pt ¼ l1 pt1 þ y1 g, and pet ¼ lTðtþ2Þ pet1 þ yTðtþ1Þ g. As T  k ! 1; lTk ! l1 . Using the well-known properties of limits and infinite sums, solution a (2.19) goes to y as T  k ! 1 if and only if 1aþal o1, which is true if and only if 1 mo1. Hence, limT!1 pt  pet ¼ 0. & The connection between the MSV solution and the limit of the finite-horizon solution as the horizon goes to infinity is exact as long as mo1. This is in part because the backward induction solution to the finite-horizon model implies a minimum-state-variable reduced form. Note also the right-hand side of (2.17.i) is zero (0) when m ¼ 0. This means that iterations of lTk are continuous around m ¼ 0, and consequently the limiting value of lTk as T  k goes to infinity must also be continuous around m ¼ 0. Because convergence is always to l1 and not to l2 , this convergence is consistent with McCallum’s ‘‘subsidiary principle.’’ Consequently, we can think of the limit of the finite-horizon model as providing a mechanism that generates the MSV solution: backward induction generates both a minimal-state reduced-form solution and a solution that obeys McCallum’s ‘‘subsidiary principle.’’ 2.2.4. Non-agreement of the MSV and finite-horizon criteria Only when m41 does the finite-horizon solution not converge to the MSV solution. As noted earlier, McCallum (2002) pointed out a characteristic of this structural model when m41 that he denoted as a model not being not ‘‘wellformulated.’’ By well-formulated, McCallum means that the parameter space for a and m should not include values that lead to an infinite discontinuity in the unconditional mean of pt , if this mean exists, and that the parameter space should include the value of zero for m. The logic here suggests that this exact connection between the MSV criterion and the finite-horizon criterion will hold for any linear model that is well-formulated. We should note that even in models not well-formulated, a stable MSV solution exists, as does a finite-horizon solution. The limit of the finite-horizon solution, though, does not converge, and thus the two solutions differ. This observation will help us understand why the same non-agreement between the MSV criterion and the e-stability criterion occurs in models that are not well-formulated. 2.2.5. The solution selected on the basis of e-stability Evans (1986) suggested the concept of e-stability as a selection criterion. In terms of this model, we imagine that agents assume that expectations are governed by a rule of the same form as is implied by (2.11), but that they do not know the rationalexpectations values of l and y. That is, they believe pep tþ1 is related linearly to pt as follows: pep tþ1 ¼ fN pt þ gyN ,

(2.23)

where fN ali and yN ayi ; i ¼ 1; 2. By substituting this ‘‘perceived law of motion’’ (PLM) into the structural model, the ‘‘actual law of motion’’ (ALM) can

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be found:  pt ¼

   m ayN  1 p þg . 1  a þ afN t1 1  a þ afN

(2.24)

Because fN ali , Evans (1986, p. 149) argues ‘‘it would thus be natural for the forecast to be revised to’’ pep tþ1 ¼ fNþ1 pt þ gyNþ1 þ pNþ1 et ,

(2.25)

where fNþ1 ¼

m , 1  a þ afN

(2.26)

yNþ1 ¼

ayN  1 . 1  a þ afN

(2.27)

Evans (1986, p. 149) defined e-stability in terms of whether a system returns to a rational expectations equilibrium reduced-form function after such a function suffered a small disequilibrium disturbance. Operationally, in terms of Eqs. (2.25)–(2.27), e-stability is satisfied when these three sequences converge to the values of the chosen rational expectations equilibrium reduced-form function.15 Note that the stationary values of (2.25) are the same as the two roots l1 and l2 , and, if the stationary value of f is l1 , the stationary value of (2.26) is y1 . The question of interest is whether the sequence ffN g; N ¼ 0; 1; 2; . . ., converges to either l1 or l2 , and to what if any finite values the sequence fyN g converge. Of course, the answers to these questions is immediate, because (2.26) and (2.27) are isomorphic to (2.17.i) and (2.18.i), respectively. Furthermore, if the sequences generated by (2.17.i) and (2.18.i) converged, they did so for any arbitrary initial value. Hence we have the following proposition: Proposition 4. If mo1, then the MSV selection criterion and the finite-horizon selection criterion and the e-stability criterion all select the same equilibrium. When m41, the parameter recursions generated by the mappings from perceived to actual laws of motion share the same feature as the parameter recursions generated by the finite-horizon solution: the sequence fyN g does not converge. Thus, in this case, there exists an MSV solution that is not selected either on the basis of estability or on the basis of thew finite-horizon solution. The lack of agreement between the e-stability and MSV criterion appears to have motivated McCallum (2002) to develop the concept of well-formulated models. The analysis here indicates that the problem is that the limit of the finite-horizon value of yTk is not a finite number. Interestingly, the lack of convergence is not with respect to f, which does converge to l1 . 15

Evans actually looked at local stability around a linearized rational expectations function.

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3. The Muth model We now take up a nonlinear model, the classic Muth (1961) model of inventory speculation as amended by McCafferty and Driskill (1980). We choose this model for several reasons. First, when viewed in comparison with the linear model of the first section, it demonstrates that the essential reason for multiple equilibria in this variant of the Muth model is not nonlinearities, as sometimes previously thought, but the assumption of an infinite horizon. Second, because of the work of Evans (1989), it permits an explicit comparison (in models with multiple stable equilibria) of the results of the MSV, finite-horizon, and e-stability selection criteria, and illustrates why in this model the MSV and finite-horizon select the same equilibrium and why the e-stability criterion may or may not select the same equilibrium as these other two. The key feature of the Muth model which breaks this agreement between the e-stability criterion and the MSV and finite-horizon criteria is the presence of what Evans and Honkapohja (2001, p. 205) denote as ‘‘mixed dating of expectations.’’ This model also provides an example in which there are two stable roots but in which, because of the non-linearity, the model is well-formulated in the sense of McCallum (2002). That is, like in the preceding McCallum example with m41, this model can generate two equilibria, both of which have a root to the characteristic equation that is less than one (1) in absolute value. But unlike the McCallum example, the region of parameter values for which this occurs do not include values at which infinite discontinuities occur. Closely connected with this, we think, is that in this model the limit of the finite-horizon solution is not unbounded. Also in contrast to the McCallum example, this possibility of two stable roots arises here more ‘‘naturally.’’ By more naturally, we mean that values of m41 in the McCallum model seem unlikely choices by any sensible monetary authority. In the Muth model, though, two characteristic roots both of which are less than one (1) in absolute value arise from plausible parameter values. 3.1. The infinite-horizon model and the MSV solution Consider Muth’s classic rational expectations model of inventory speculation (Muth, 1961). The model consists of flow demand and supply functions, a speculative demand function for stocks, a market-clearing equation, the rational expectations assumption, and an initial condition: C t ¼ c0  bpt ;

b40

Pt ¼ gE t1 pt þ ut ;

(Flow demand),

g40 (Flow supply),

(3.1) (3.2)

I t ¼ a½E t ptþ1  pt 

ðInventory speculation),

(3.3)

I t  I t1 ¼ Pt  C t

(Market equilibrium),

(3.4)

I 1 given

(Initial condition),

(3.5)

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where t ¼ 0; 1; 2; . . . : Pt represents production in the ith period, C t represents consumption in the ith period, pt represents market price, E tj is the conditional mathematical expectation operator, and ut is a zero-mean, serially uncorrelated random variable with finite variance s2u . Muth derives (3.3) from maximization of expected utility of profits, and shows that a¼

b K , s2t;1

(3.6)

b is a non-negative parameter that is a measure of risk aversion and s2 is the where K t;1 conditional variance of next period’s price forecast. As shown in McCafferty and Driskill (1980), the equilibrium price path for all t is described by the following firstorder stochastic difference equation, which we denote as the equilibrium price function: ut pt ¼ lpt1  þ rc0 ; t ¼ 0; 1; 2; . . . , (3.7) b þ að1  lÞ where l and a satisfy the following two equations: b K ½b þ gl2 ¼ ðg þ bÞl, s2u a¼

ðg þ bÞl ð1  lÞ2

(3.8)

(3.9)

1 and r ¼ bþgþað1lÞ . For this model, there exist two real solutions to (3.8) if and only if

bþg 44K; bg

K

b K . s2u

(3.10)

Throughout the remainder of this paper, we assume this restriction is satisfied. Denote the smaller root of (3.8) as l1 ðb; g; KÞ and the larger root as l2 ðb; g; KÞ. As shown in McCafferty and Driskill (1980), 0ol1 ol2 .

(3.11)

Denote by pi;t the equilibrium price associated with the equilibrium price function (3.8) when l ¼ li ; i ¼ 1; 2; . . . : That is, ut þ ri c0 ; t ¼ 0; 1; 2; . . . , pi;t ¼ ðli Þðpi;t1 Þ  (3.12) b þ ai ð1  li Þ i where ai ¼ ðgþbÞl and ri ¼ bþgþa1i ð1li Þ. ð1l Þ2 i

When 0ol1 o1 and l2 41, the equilibrium price function associated with l2 is non-stationary. Existence of such explosive solutions is a well-known feature of many rational expectations models. As noted in the introduction, many researchers have simply ignored this possible solution as uninteresting because of the implied non-boundedness of the price process, while others have viewed this as a possible

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explanation of the real-world phenomena of price bubbles More problematic, though, has been the case emphasized by McCafferty and Driskill in which 0ol1 ol2 o1.16 In such a case, the stability selection criteria does not eliminate all but one of the equilibria.17 The MSV solution, though, does select a unique equilibrium in this situation. To see this, note that when K ¼ 0, Eqs. (3.1)–(3.5) imply a MSV solution without any lagged values of p. McCallum’s subsidiary principle then chooses between l1 and l2 based on which is zero when K ¼ 0. One can readily verify that limK!0 l2 ¼ þ1 while limK!0 l1 ¼ 0. Hence, the MSV solution is described by the equilibrium price function (3.8) with i ¼ 1. 3.2. The finite-horizon model Consider now a finite-horizon version of this model, where t ¼ 0; 1; 2; . . . ; T. The only difference in the structural model associated with this finite-horizon assumption is that utility-maximization by inventory speculators implies that I T ¼ 0. This condition serves as a second boundary condition, and permits a solution by backwards induction much as in the analysis of the McCallum examples. Let pet denote the equilibrium price for this finite-horizon model. The changes in the solution to the infinite-horizon model generated by imposition of this new boundary condition are characterized by the following proposition: Proposition 5. The equilibrium price path for the above finite-horizon Muth inventory speculation model is described by the following first-order stochastic difference equation: peTk ¼ lTðkþ1Þ peTðkþ1Þ þ pTk uTk þ rTk c0 ; k ¼ 0; 1; 2; . . . ; T,

(3.13)

where lTk , pTk , rTk and aTk obey the following recursive relationships: lTðkþ1Þ ¼

pTk ¼

aTðkþ1Þ , b þ g þ aTk ½1  lTk  þ aTðkþ1Þ

1 , b þ aTk ½1  lTk 

aTk ¼ ðKÞfb þ aTðk1Þ ½1  lTðk1Þ g2 ; 16

(3.14)

(3.15)

K

b K , s2u

(3.16)

An example given by Evans (1989) of parameter values for which this occurs is: b ¼ 0:1; g ¼ 0:4; K ¼ 2:5. In this case, l1 ¼ 0:1; l2 ¼ 0:66. 17 We might note that other selection criteria have been proposed, such as a minimum variance criterion in Taylor (1977). With two stable roots, McCafferty and Driskill (1980) showed this criterion was ambiguous for this model. Recently, Roberts (1998) has shown that risk-aversion rules out the possibility of rational probabilistic explosive bubbles such as studied in Blanchard (1979) because in this case the rational expectations solution becomes process-inconsistent in the sense of Flood and Garber (1980). His argument cannot apply in the case of two stable roots, though.

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rTk ¼

1  ðaTðK1Þ þ gÞrTkÞ þ aTk rTðk1Þ , b þ aTk ð1  lTk Þ

(3.17)

where k ¼ 1; 2; . . . ; T, and where the following boundary conditions apply: aT1 lT1 ¼ , b þ g þ aT1 aT1 ¼ Kb2 ,

(3.18.i) (3.18.ii)

pT ¼

1 , b

(3.18.iii)

rT ¼

c0 . b þ g þ aT1

(3.18.iv)

Proof. See appendix.

&

We can now state the following turnpike proposition: Proposition 6. Assume parameter values are such that two real roots exist as solutions to Eq. (3.8). For any t40 and any arbitrarily small e40, there exists a horizon-length T4t sufficiently large such that for a given sequence fut g, j pet  p1;t j oe. Proof. See appendix.

&

Thus, the finite-horizon and MSV criteria both select the same unique equilibrium. As in the linear model of the previous section, the backward-induction solution establishes that the equilibrium price function depends only on the minimal set of state variables. Furthermore, the recursion formula’s for the equilibrium price function parameter lTk are such that as K ! 0, lTk ! 0. Again, this feature might be thought of as a counterpart to McCallum’s ‘‘subsidiary principle.’’ 3.3. The e-stability approach In another explicit analysis of this version of the Muth model, Evans (1989) classified the two solutions to the infinite-horizon Muth model according to whether or not they were e-stable. As noted in the discussion of the McCallum examples, his concept of e-stability asks whether or not agents who start with incorrect expectations (of a particular sort) but update expectations (perhaps in meta-time) according to a learning-like rule eventually converge to the rational expectations solution. This approach, like the finite-horizon model here, gives rise to recursive relationships between the parameters of the equilibrium reduced-form price function, albeit perhaps in meta-time. Evans (1989, p. 299) distinguished between two types of incorrect expectations. One type, which was used to determine what Evans called ‘‘weak’’ e-stability, closely resembles the type used in the e-stability analysis of the McCallum example. In

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particular, agents were assumed to believe the reduced-form solution to the above Muth model was ptþ1 ¼ fN pt þ yN utþ1 þ cN c0 ,

(3.19)

where fN ; cN and yN were initially assumed to be constant disequilibrium values. This implies pex tþ1 ¼ fN pt þ cN c0 ,

(3.20.i)

pex t ¼ fN pt1 þ cN c0 .

(3.20.ii)

Substituting this perceived law of motion into the structural model yields the following actual law of motion: ! ! Kð1  fN Þ  gfN ðyN Þ2 ðyN Þ2 pt ¼ pt1 þ ut , bðyN Þ2 þ Kð1  fN Þ bðyN Þ2 þ Kð1  fN Þ c0 . ð3:21Þ þ 2 bðyN Þ þ Kð1  fN Þ Using the same updating scheme used with the Cagan model, this yields the following recursions: fNþ1 ¼

Kð1  fN Þ  gfN ðyN Þ2 , bðyN Þ2 þ Kð1  fN Þ

(3.22.i)

yNþ1 ¼

ðyN Þ2 , bðyN Þ2 þ Kð1  fN Þ

(3.22.ii)

cNþ1 ¼

1  gcN . bðyN Þ þ Kð1  fN Þ

(3.22.iii)

2

What Evans (1989, p. 304) found was that for some parameter values both solutions were expectationally stable.18 To see why the backward-induction criterion and the e-stability criterion may diverge in this case while they necessarily converged in the well-formulated McCallum model, first remember that there was only one future expected variable in the McCallum structural model: pex tþ1 . In the Muth model, though, two ex expectations variables show up in the structural model: pex tþ1 and pt . The e-stability ex ex assumption used by Evans implied that both ptþ1 and pt depended on the same value of f, namely fN . Thus, the recursion formula for fNþm differs from the 18

Evans actually analyzed a differential equation formed from these recursions, but the stability results are unaffected when using the above recursions. He also went on to develop a stronger concept in which the perceived law of motion depended on additional lags. For this case, called ‘‘strong e-stability,’’ he found only the solution for l1 was expectationally stable. Our purpose here is to show why we would not expect the same congruence between expectational stability and backward induction that we found in the Cagan model analysis, and the comparison with weak e-stability suffices.

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recursion formula for lTðkþ1Þ . Hence, there is no deep reason to suspect the implied recursion from the time-varying perceived law of motion and the recursion from Evans’ time-invariant perceived law of motion will generate sequences of coefficients that converge to the same fixed point. To further illustrate this point, note that if we were to assume that agents believed the perceived law of motion to have time-varying parameters, that is, believed the perceived the law of motion to be ptþ1 ¼ b lt pt þ b ytþ1 utþ1 , then the actual law of motion would be 0 1 0 1 gb lt1 þ ð K2 Þð1  b lt1 Þ B C B C byt 1 Cpt1 þ B C ut pt ¼ B @ A @ A K K lt Þ lt Þ b þ ð 2 Þð1  b b þ ð 2 Þð1  b bytþ1 bytþ1

(3.23)

(3.24)

and the implied recursion would generate values of b lt1 and b yt from b lt and b ytþ1 . For b example, lt1 would equal the coefficient on pt1 in (3.24). Furthermore, with some algebra one can show that these recursion formulas for b lt1 and b ytþ1 would be identical to those for lTðkþ1Þ and pTk in the finite-horizon case, which obviously means they are different from those of (3.22). All that differs between the finitehorizon analysis and the time-varying e-stability analysis is the unique terminal condition imposed in the backward-induction case. Note that the recursions move ‘‘backwards’’ from those of Evan’s, e.g., b lt1 is generated from b lt . All of this means that the standard ‘‘forward’’ e-stability criterion will not in general select the same equilibrium as does the backward-induction criterion whenever the structural model has expectational variables that appear in different time periods. A ‘‘backward’’ e-stability criterion that assumes a perceived law of motion that has time-varying parameters will, though, continue to select the same equilibrium as the backward-induction criterion.

4. Overlapping generations models We now analyze two overlapping-generations models, one with and one without money. These models emphasize that the key assumption that gives rise to multiple equilibria is the infinite horizon, not an infinite lifespan for optimizing agents. These models have also so frequently served as examples of multiple equilibria that they deserve some attention here. The model without money is based on Calvo (1978), one of the first papers to highlight the possibility of indeterminacy. What is surprising here is that the unique finite-horizon solution is periodic, wish the price of land fluctuating between two distinct values, while the MSV and e-stability solutions both select a solution in which the price of land is a constant function. This leads us to speculate that we might expect the finite-horizon and MSV criteria to select different equilibria when finite-horizon solutions are cyclical.

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The model with money has mixed datings of expectations as does the Muth model. In contrast to the Muth results, though, in this model the MSV, finite horizon, and e-stability criteria all select the same unique equilibrium. The analysis shows that this occurs because the parameter recursion formulas for both the finite horizon and e-stability solutions, while different, share the property that they are consistent with McCallum’s ‘‘subsidiary principle.’’ We argue that this congruence is to be expected for the finite-horizon solution but is not a generic feature of the e-stability solution. 4.1. The Calvo model 4.1.1. Infinite-horizon model Calvo (1978) analyzed an infinite-horizon model in which each individual lives for two periods. All individuals born at the beginning of period t are known as members of generation t. All individuals have identical preferences, and every generation is the same size, normalized to one (1). Homogenous, non-depreciating, and nonrenewable land, fixed in quantity at one unit, is the only store of value. Labor is inelastically supplied by the young and is normalized at one (1) unit. Output is produced via a constant returns to scale technology. The marginal product of the one unit of land is the fixed amount R units of output, and the marginal product of the one unit of labor is the fixed amount w. Perfect competition prevails, so workers earn w and landowners receive rent R. Land is owned by the old. That is, at the beginning of period t, members of generation (t  1Þ own the land and collect the rent R. After collecting the rent, the members of generation (t  1Þ sell the land to the members of generation t for the real price qt . People are assumed to have perfect foresight. Calvo assumed relatively general preferences. The key points, though, can be made by assuming preferences are Leontief. For members of generation t, denote their consumption at t, i.e., when they are young, as ct and their consumption at t þ 1, i.e., when they are old, as xtþ1 . Preferences are thus specified as

ct xtþ1 ; U ¼ min . (4.1) ac ax At time t, generation t receives real income w and can purchase either goods or land. Thus this generation’s period-t and period-t þ 1 budget constraints are w  c t ¼ qt ,

(4.2.i)

R þ qex tþ1 ¼ xtþ1

(4.2.ii)

and their demand and savings function at time t are aw cdt ¼ , a þ pt xdtþ1 ¼

w , a þ pt

(4.3.i)

(4.3.ii)

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S t ¼ w  ct ¼

wpt , a þ pt

(4.3.iii)

where pt  Rþqqtex . With the above specification of tastes, resources, and technology, tþ1

and with the assumption of an equilibrium condition at time t that the savings demand of generation t equals the value of the available land, the structural model is described by the following difference equation: 1 w  aR ; qt a0, qex tþ1 ¼  qt þ a a

w  aR ex ; w  aR for qt ¼ 0. qtþ1 2 R

(4.4.i)

(4.4.ii)

With perfect foresight, qex tþ1 ¼ qtþ1 , and (4.4) describes a difference equation. Indeterminacy arises when a41 and ðw  aRÞ40. As Calvo noted, in this case there are an infinite number of values for q0 for which the sequence fq0 ; q1 ; . . .g, qt X0, both solves (4.4) and converges to the steady-state value q ¼ waR 1a . That is, there are multiple equilibria. We depict these possibilities in a phase diagram in Fig. 1, with the thick line depicting Eq. (4.4) and with two possible equilibrium paths depicted. In anticipation of analysis of the finite-horizon analogue of this model, both possibilities are depicted for initial value q0 ¼ 0.19 One depicted path has q1 coming from the open interval ðwaR a ; w  aRÞ. For this path, the phase diagram shows that the sequence fq2 ; q3 ; . . .g converges to q ¼ waR 1a . A second path is depicted with q1 ¼ ðw  aR_Þ. For t ¼ 2; 3; . . ., this path continually cycles, with pt taking the alternate values of zero ð0Þ and w  aR. Let us consider which of all these equilibria would be selected by the MSV criterion. For this restricted model, because there are no state variables, the MSV solution would be a constant, and would simply be the steady-state value: w  aR . (4.5) qMSV ¼ 1a Now let us consider which would be selected on the basis of e-stability. Assume agents believe the perceived law of motion to be given as qtþ1 ¼ fN qt þ cN . (4.6) Substituting this into the structural model and updating yields fN ¼ 0; cNþ1 ¼ acN þ w  aR.

(4.7)

This implies convergence of cNþk to waR 1a as k ! 1. That is, the e-stability criteria selects the same solution as the MSV criterion. Again, this congruence of the MSV and e-stability criterion is in part due to the lack of mixed dating of expectations. We now take up the finite-horizon version of this model. 19

It should be clear from the phase diagram that an infinite number of values of q0 2 ð0; w  aRÞ would also lead to sequences converging to the steady state.

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1.5 1.375 1.25 1.125 1

q(t+1)

0.875 0.75 0.625 0.5 0.375 0.25 0.125 0 0

0.125 0.25 0.375 0.5 0.625 0.75 0.875

1

1.125 1.25 1.375 1.5

q(t) Fig. 1. Calvo phase diagram.

4.1.2. Finite-horizon model As Calvo noted (1978, pp. 332–333) with a finite horizon, ‘‘uniqueness is, in principle, recovered...’’20 The terminal condition implied by a finite horizon is that qT ¼ 0, where T is the time period after which no generations are born. That is, the young at T will not want to save. If qT ¼ 0, then generation T consumes w, leaving R for consumption by the old of generation T  1. Demand by the old of generation T  1 is given by (4.3.ii). Setting this equal to R lets us solve for the equilibrating value of pT1 : pbT1 ¼

w  aR , R

(4.8)

qT1 where the ‘‘hat’’ signifies an equilibrium value. Because qT ¼ 0 and pT1 ¼ Rþq , T then

qbT1 ¼ w  aR.

(4.9)

Now, when qbT1 ¼ w  aR, the equilibrium condition for T  2 implies that qbT2 ¼ 0. Consequently, qbT3 ¼ w  aR, qbT4 ¼ 0, and so on. That is, we have the 20

We might ask why Calvo did not pursue the finite-horizon solution. It appears he thought other problems arose for the solution of a finite-horizon model. He went on to write (p. 333) that ‘‘a new, puzzling difficulty might pop up with finite horizon when conditions for the type of indeterminancy of this paper prevail...equilibrium q0 will be very sensitive to changes in the terminal conditions... .’’

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following proposition: Proposition 7. The equilibrium of the infinite-horizon model that is the backwardinduction solution to the finite-horizon model is the one in which there are regular oscillations between the two periodic points ð0Þ and ðw  aRÞ. This cycle is an example of what Grandmount (1985, p. 1005) called backwards perfect foresight (b.p.f.) dynamics. He viewed these as ‘‘fictitious’’ and introduced them in part because of technical considerations due to dealing with backwardbending offer curves. Our view is that there is nothing inherently unnatural in computing an equilibrium backwards in time from a terminal condition. Backward e-stability: Now, imagine agents assume the values of q are generated by the following time-varying process: qtþ1 ¼ ktþ1 ,

(4.10)

where k is a scalar. Substituting this perceived law of motion into the actual law of motion (4.4.i), we get qt ¼ aktþ1 þ w  aR

(4.11)

which suggests the natural recursion kt ¼ aktþ1 þ w  aR.

(4.12)

Again, notice how the assumption of a time-varying parameter forces us to work backwards in time, from ktþ1 to kt . Now, for any initial value qT ¼ kT , recursion (4.12) implies that eventually, i.e., for some n sufficiently large, kTn ¼ 0. Now, once kTn ¼ 0, the sequence fkTðnþmÞ g; m ¼ 1; 2; . . ., will perpetually oscillate between ðw  aRÞ and zero ð0Þ. That is, the backward-e-stability solution converges to a solution that oscillates like the backward-induction solution, regardless of the initial value kT . 4.1.3. Why the MSV and finite-horizon criterion diverge The contrast here is to the earlier examples, where (with the exception of models not well-formulated) the MSV and the limit of the finite-horizon solution coincided. In this manifestation of the Calvo model, this cannot happen because there is no unique limiting value of qTn as ðT  nÞ ! 1: the equilibrium value of the price of land at any t ¼ T  n depends on whether n is an odd or even number. Note that if we assumed ao1, it is straightforward to show that the stability, the MSV, the e-stability and the backward-induction criterion all select the same equilibrium: q ¼ waR 1a . Of course, if ao1, the issue that attracted Calvo in the first place, namely indeterminacy of stable equilibria, is no longer a problem. 4.2. A monetary example We now analyze an OLG model with fiat money used to finance a government deficit. This model, unlike the Calvo model, generates both multiple steady states

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and indeterminacy around one of the steady states. It also exhibits mixed dating of expectations. We find that the MSV, e-stability, and finite-horizon criterion all select the same equilibrium, for reasons that highlight under what conditions we might expect this type of result even with mixed dating of expectations. We contend this result aids understanding of analysis of a closely related model that has been exhaustively analyzed in terms of adaptive learning selection criterion (Lettau and Van Zandt, 2001). What they showed is that which equilibrium is selected on the basis of stability under adaptive learning depends critically on informational assumptions embodied in the learning rules. Our example illustrates what sorts of informational assumptions can lead to recursive relationships that mimic those generated by a finite-horizon model. Finally, Marimon and Sunder (1993) provide some experimental evidence on this type of model. This evidence is far from conclusive, and was not gathered to evaluate the selection criteria reviewed here. Nonetheless, it is suggestive about approaches to empirical investigation of the relevance of these selection criterion. 4.2.1. The infinite-horizon model Consider an example similar to the one above and also much like that in Lettau and Van Zandt (2001), in which fiat money instead of productive land is the only store of value. Again, in each period t, generation t of size one ð1Þ is born, and each such cohort lives for two periods. In contrast to the above OLG model, all generations in this model have preferences represented by the following CES utility function in which the elasticity of substitution is ð 12Þ: U ¼ ½yðxtþ1 Þ1=2 þ ð1  yÞðct Þ1=2 2 ; y 2 ð0; 1Þ,

(4.13)

where again xtþ1 is consumption by generation t in its old age and ct is consumption by generation t in its youth.21 Young generations receive an endowment w that they can consume that period or use to purchase nominal money, mt , at price p1 , the t inverse of the price level: mt (4.14) w ¼ ct þ . pt The old receive no endowment, and must live off of their savings: mt ¼ xtþ1 . ptþ1

(4.15)

Hence, generation t’s lifetime budget constraint is w ¼ ct þ pex tþ1 xtþ1 , where

pex tþ1

cdt ¼ 21

pex ¼ tþ1 pt . Consequently, wapex tþ1 , 1 þ apex tþ1

(4.16) demand and saving functions for generation t are (4.17.i)

This functional form is chosen to allow a closed-form solution that also permits multiple stable equilibria.

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xdtþ1 ¼ St ¼

pex tþ1

w , 2 þ aðpex tþ1 Þ

w , 1 þ apex tþ1

(4.17.ii)

(4.17.iii)

where a ¼ 1y y (note this is a different meaning for a than in either Section 2 or in the preceding section). The government runs a constant real deficit of magnitude d that they finance via money creation. That is, mt ¼ mt1 þ dpt .

(4.18)

At t, the nominal value of savings must equal the quantity of nominal money, which implies pt ðSt  dÞ ¼ St1 . ex This implies the following relationship among pt ; pex t ; and ptþ1 :    1 þ apex w tþ1 pt ¼ . 1 þ apex w  d  apex t tþ1

(4.19)

(4.20)

The two important features of (4.20) are that there is mixed dating of expectations and that there are no state variables. Assuming rational expectations, (4.20) implies that pt evolves through time according to the following non-linear difference equation: ptþ1 ¼ fðpt ; dÞ,

(4.21)

where fðpt ; dÞ ¼

w þ dpt ð1 þ apt Þ þ wpt ð1 þ apt Þ . a½w þ dpt ð1 þ apt Þ

(4.22)

The properties of f that are of most interest to us are, for pt X0: fð0; dÞo0,

(4.23.i)

f0 40,

(4.23.ii)

f00 o0.

(4.23.iii)

The values of p in the steady state where fðpÞ ¼ p are thus 2 ffi3   sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   14 w  d wd 2 wad 5 4 , p1 ¼  2 ad ad ðadÞ2 2 ffi3   sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   14 w  d wd 2 wad 5 . 4 þ p2 ¼ 2 ad ad ðadÞ2

(4.24.i)

(4.24.ii)

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What interests us is that there are parameter values for which two positive real values of p satisfy this quadratic equation. That is, there are two (2) such steady states when w  d40,

(4.25.i)

ðw  dÞ2 44wad.

(4.25.ii)

5 16 ;

For example, if a ¼ d ¼ 1; and w ¼ 4, the two roots are: p1 ¼ 1:6; p2 ¼ 8. The phase diagram for this example is displayed in Fig. 2. In this diagram, the sequence of ptþk ’s are found by climbing up the ‘‘staircase’’ that starts from an initial value pt ¼ 4. Because of the concavity of fðpt Þ, the sequence will converge to p2 . Two features are of interest here. First, as is clear from the phase diagram, p2 is the ‘‘stable’’ solution in the sense that, for any arbitrary p0 4p1 , limt!1 pt ¼ p2 . Second, there are an infinite number of these ‘‘stable’’ solutions, each indexed by a different value of p0 4p1 . That is, there is the same type of indeterminacy as in the Calvo model. 4.2.2. The MSV solution With no state variables, candidates for the MSV solution are a constant function that satisfies (4.21), i.e., either p1 ðdÞ or p2 ðdÞ. We write these two constant functions as functions of d in anticipation of applying the subsidiary principle to eliminate one of these two candidate solutions. In particular, the MSV solution should be a

15

12.5

pi(t+1)

10

7.5

5

2.5

0 0 -2.5

2.5

5

7.5 pi(t)

Fig. 2. ptþ1 ¼ fðpt Þ.

10

12.5

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continuous finite function of d around d ¼ 0. This implies that the MSV solution is p1 : pMSV ¼ p1

(4.26)

4.2.3. The finite horizon model To create a finite-horizon analogue, a decision must be made about how to close the model at time T. With a finite horizon and nominal money that yields no return, we must specify some device that makes people demand money in the final period. To this end, we assume the government finances its real deficit of d by collecting lump-sum taxes (equal to d) that must be paid in fiat currency. This means that generation T has the following demand for money: mdT ¼ dpT .

(4.27)

Because the government doesnot finance the period T deficit with fiat money, the supply of fiat money in period T is just mT1 . Hence, mT1 ¼ dpt .

(4.28)

Now, the nominal value of savings by generation T  1 just equals mT1 : pT1 S T1 ðpT Þ ¼ mT1 .

(4.29)

This implies the following boundary condition: dpT ¼ S T1 ðpT Þ.

(4.30)

bT , that Given our functional form for S, this means there is a value of p, call it p solves w dpT ¼ ; pT X0. (4.31) 1 þ apT Given this value, we now have the backward-induction solution for the model. At any time T  ðk þ 1Þ; k ¼ 0; 1; 2; . . . ; T  1, the equilibrium (4.20) is

w  d  ade pTk w e pTðkþ1Þ , (4.32) ¼ 1 þ ae pTðkþ1Þ 1 þ ae pTk where e p denotes the equilibrium inflation factor for this finite-horizon model. This leads to the following recursive relationship: e pTk ¼

w  ðw  dÞ½e pTðkþ1Þ ð1 þ ae pTðkþ1Þ Þ . a½w þ de pTðkþ1Þ ð1 þ ae pTðkþ1Þ Þ

(4.33)

This, of course, has the same functional form as the recursive relationship (4.21): e pTk fðe pTðkþ1Þ Þ.

(4.34)

pTk . That is, the But the recursion here generates values e pTðkþ1Þ as a function of e more insightful way of writing (4.34) is as e pTðkþ1Þ ¼ f1 ðe pTk Þ.

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For the finite-horizon case, we can thus think of the recursive relationship (4.21) as generating values of pt as a function of ptþ1 , and we can think of the phase diagram depicted in Fig. 2 as simply ‘‘running backwards.’’ That is, the phase diagram for the finite-horizon model just re-labels the x-axis as e pTðkþ1Þ and the y-axis as e pTk . This is displayed in Fig. 3. In this diagram, the sequence e pTðkþnÞ is traced out by descending down the staircase from an initial value of e pTk . Because of the concavity of f, e pTðkþ1Þ converges to p1 . The starting point for this recursion is b pT . Hence, if b pT op2 , the sequence converges to p1 . Hence, to prove that the backward-induction criterion selects p1 , we need to prove that b pT op2 . Lemma 8. If the parameter restrictions implied by (4.25) are satisfied, then b pT op2 . Proof. See appendix.

&

Thus we have the following proposition: Proposition 9. Assume the parameter restrictions implied by (4.25) are satisfied. Then limTðkþ1Þ!1 e pTðkþ1Þ ¼ p1 .

15

12.5

10

pi(T-k)

7.5

5

2.5

0 0

2.5

5

7.5

10

pi(T-(k+1)) -2.5 Fig. 3. pTk ¼ fðpTðkþ1Þ Þ.

12.5

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Proof. Given the concavity of f and the above lemma, convergence to l1 follows immediately. & Note also from the phase diagram that this convergence occurs for any arbitrary terminal value pT op2 , not necessarily the value consistent with our terminal (4.31). That is, we can also think of ‘‘backward e-stability’’ as selecting p1 . The e-stability selection criterion: We now ask whether the usual e-stability criterion selects the same equilibrium as that selected by the finite-horizon criterion. Again, assume agents believe the perceived law of motion is ex pex tþ1 ¼ pt ¼ pN .

(4.35)

Substituting this into the actual law of motion (4.21) and rearranging yields pt ¼

w . w  d  adpN

(4.36)

Hence, the natural updating rule would be pNþ1 ¼ f ðpN ; dÞ, f ðpN Þ 

w . w  d  adpN

(4.37.i) (4.37.ii)

Analysis of this difference equation is almost identical to that of the first example based on McCallum (1983a). In particular, this Riccati equation will insure convergence of pN to p1 for pN o wd ad . In this case, it is clear that p1 is the expectationally stable solution. The interesting contrast here is between this model and the example based on Muth (1961) and McCafferty and Driskill (1980). There, the presence in the structural model of expectations taken at different dates led to a structuralparameter dependent e-stability criterion: for some parameter values, the criterion chose the same equilibrium as did the finite-horizon criterion, while for other values, it chose a different equilibrium. Here, the e-stability criterion always chooses the same equilibrium as does the finite-horizon criterion. This difference between the two examples, both of which have mixed datings of expectations, emphasizes that what governs agreement between the e-stability criterion and the finite-horizon criterion is the continuity of the e-stability recursion formula around a key parameter value. In this model, when d ¼ 0, we have that the right-hand side of (4.37.i), the recursion formula f ðpN ; dÞ, equals one (1). The continuity of this function with respect to d around d ¼ 0 thus insures that convergence will always be to l1 . This also emphasizes that agreement between e-stability and finite-horizon criteria will always depend on the exact details of the assumption about what is a reasonable updating scheme. This point is reinforced by the analysis of Lettau and Van Zandt (2001) of a similar model. They found that the selection of the low- or high-inflation equilibrium by various adaptive learning schemes was crucially dependent on whether these schemes assumed updating based on current or lagged information.

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Finally, the experimental evidence of Marimon and Sunder (1993) sheds some light on these results. They conducted experiments built around a model similar to the one here, and found strong evidence that the data clustered around the lowinflation equilibrium. They did not test for whether the experimental data were consistent with the limit of the finite-horizon equilibrium as the horizon goes to infinity or, equivalently, with the MSV solution. While they did conclude that the data were consistent with adaptive learning as a behavioral assumption, a look at their Fig. 3 (p. 1087) suggests that the experimental inflation rates clustered immediately around the low-inflation equilibrium value, which is the MSV solution as well as the limit of the finite-horizon solution as the horizon goes to infinity. This provides at least suggestive evidence that the MSV and finite-horizon selection criterion may predict real behavior.

5. Summary and conclusions This review has argued that multiple equilibria are an inherent feature of infinitehorizon dynamic rational expectations models, but not of finite-horizon dynamic rational expectations models. Hence, any selection criterion that chooses a unique member of the set of infinite-horizon-model equilibria must in essence provide an additional boundary condition that is not a natural feature of the infinite-horizon structural model. The requirement that a solution be stable imposes this additional boundary condition, but fails in general to restrict the set of equilibria to a unique solution. While some researchers view this as a desirable feature in that it provides potential explanations of economic phenomena, it has also led to a search for other criterion that choose a unique equilibrium. The MSV selection criterion, by design, eliminates all ‘‘bootstrap’’ or ‘‘bubble’’ equilibria by requiring solutions to be functions of the minimal set of state variables found in the structural model. The analysis in this paper shows that ‘‘bootstrap’’ or ‘‘bubble’’ equilibria occur only in an infinite-horizon model. Hence, using the limit of the equilibrium of a finite-horizon analogue of the infinite-horizon model as a selection criterion will, in a wide class of models, select the same unique equilibrium as does the MSV criterion. The one example in which the MSV criterion selected an equilibrium different from the one selected by the finite-horizon criterion was an OLG model in which land was the only store of value and in which preferences between young and old consumption exhibited no substitutability. In this model, the finite-horizon solution, no matter the length of the horizon, always retained time as a state variable: the relevant environment of each generation depended on whether there were an even or odd number of periods from when that generation was born until terminal time T. That is, there was no unique limit of the value of land as the horizon went to infinity. In contrast, in all other examples, while the (unique) finite-horizon solution did have time as a state variable, it disappeared in the limit as the horizon approached infinity, i.e., the limiting solution was autonomous.

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The conjecture would be, then, that for any model in which the solution of a finitehorizon analogue has a unique limit, the finite-horizon and MSV criterion are isomorphic. The frequency with which economically interesting models without such a limit arise is an issue on which researchers can disagree. Clearly, we have one example in this paper of such a model, but it’s restrictive two-generational discretetime structure suggests more flexible versions would not share this feature. Subject to this proviso, it seems that the MSV and finite-horizon criteria are operationally identical. The finite-horizon criterion, though, has several additional attributes that augment interest in it, and, by implication, in the MSV criterion. First, many finite-horizon models have behavioral relationships grounded in economic theory that generate a natural terminal condition. Equilibria that are the limit of solutions to such models share this feature. Thus, this selection criterion brings to bear the economic analysis that generates this terminal condition, and is in that sense less ‘‘arbitrary’’ that other criteria.22 Second, because finite-horizon models provide people (and economists) with much of their intuition about the workings of dynamic models, solutions to such models seem a natural ‘‘focal point’’ about which agents can be assumed to coordinate their ‘‘belief functions’’.23 Examples of how economists use finite-horizon analysis to motivate infinite-horizon assumptions abound. In their classic application of game theory to growth and exploitation of a common natural resource, Levhari and Mirman (1980) solved a finite-horizon problem via backward induction before moving to the infinite-horizon solution. Before introducing infinite-horizon models that are the centerpiece of their analysis, Obstfeld and Rogoff (1996) start with a two-period model. Many expositions of such things as no-Ponzi-game restrictions on infinite-horizon budget constraints usually use finite-horizon ideas to provide intuition.24 Note also how many explanations of the time-inconsistency problem use a two-period example similar to that of Fischer (1980) as the key aid to understanding.25 The backward-induction solution also sheds light on the conditions under which we might expect the e-stability criterion as developed by Evans (1986, 1989) to select (or not select) the same equilibrium as does the MSV and finite-horizon criterion. 22 The use of quotation marks around arbitrary is meant to convey the idea that one person’s concept of arbitrary is another person’s concept of reasonable. We argue that the arbitrariness of the criterion espoused in this paper is of a lesser degree in that more economic analysis is brought to bear on the issue. 23 Schelling (1960) argued that players in a game with multiple Nash equilibria might coordinate on a particular equilibrium by using ‘‘focal points,’’ i.e., information from culture and experience not specified in the strategic form of the game. Karp (1996b) made a similar argument to the one used here about the desireability of using Markov strategies in a dynamic game specification. See Farmer (1993) and Matheny (1999) for a discussion and definition of belief functions. Also see Benhabib and Rustichini (1994, p. 2) for an argument about the possibility of ‘‘variables extrinsic to economics’’ acting as ‘‘coordination or selection devices among equilibria.’’ 24 A no-ponzi-game restriction does not allow an infinitely-lived agent to borrow more that the present value of his or her lifetime resources, i.e., does not allow chain-letter schemes. See Shell (1971) for a discussion of the arbitrariness of such a restriction in an infinite-horizon model. 25 See, for example, Turnovsky (1995, Chapter 11, Section 10) and Blanchard and Fischer (1989, Chapter 11, Section 4).

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Basically, both the finite-horizon solution and the e-stability solution involve recursions of either solution parameters or solution values. In the case of mixed dating of expectations, though, these recursions are not likely to be isomorphic. But if the e-stability recursions are continuous around the same structural parameter values as are the finite-horizon recursions, then the two criterion lead to the same choice. There appears to be no deep reason, though, for such a property to be found for any general class of models. The ‘‘learning’’ approach, as an extension of the estability criterion, thus should not be thought of as in general likely to select the finite-horizon equilibrium. It is useful to further compare and contrast the finite-horizon selection criterion with learning approaches. Learning approaches provide the needed additional boundary condition by positing an initial disequilibrium value for some parameter in the model and then positing an updating rule. One similarity is that they both have, in the language of game theory, an element of ‘‘perfectness’’ as providing a ‘‘refinement’’. The specifics, though, are different. In the learning approach, this ‘‘perfectness’’ is associated with ‘‘trembles’’ by atomistic agents about what are the parameters or the values of the rational expectations solution.26 In the finite-horizon approach, the ‘‘perfectness’’ arises directly because the finite horizon leads to construction of a backward-induction solution. The key difference between the two approaches is that the finite-horizon criterion is built upon components more familiar to economists than ideas about how people learn. The learning approach, though, surely has great intuitive appeal to many economists as an accurate description of how the world works. Perhaps future work extending such studies as Flood and Garber (1980) in the empirical tradition and Marimon and Sunder (1993) in the experimental tradition can sort out which approach is most relevant.

Acknowledgements I would like to thank Roger Farmer, Zvi Eckstein, Ben Eden, Robert Flood, Larry Karp, Bennett McCallum, and members of the Georgetown University Macroeconomics Seminar for useful discussions and comments.

Appendix A Proof of Proposition 1. Eq. (2.17.i) is a Riccati equation with known solution (see, for example, Mickens, 1990, p. 213) lTk ¼

26

½1 þ Dð1  ð1  a=aÞð1=l2 ÞÞðl1 =l2 ÞTk  1 l1

þ Dðl1 =l2 ÞTðkþ1Þ

See Guesnerie (2002) for a discussion of this.

,

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where D is a constant whose value is determined by the initial condition on lT1 . Because j ll1 jo1, limðTkÞ!1 lTk ¼ l1 . & 2

m a 1 Proof of Proposition 2. Note that l1 ¼ 1aþal , which implies that 1aþal ¼ al m . 1 1 al1 We seek to prove, then, that m o1. First consider the case of mp0. This implies that l1 o0. Now, l1 satisfies the equation   1a u 2 ðl1 Þ ¼  l1 þ . a a dg u Let ðl1 Þ2  f ðl1 Þ and ð1a a Þl1 þ a  gðl1 Þ. Now, dl1 41 (because ao0). The m intercept of g is a. Hence, gðl1 Þ ¼ 0 at some e l1 4  1, and the intersection of f and g must occur at l1 4  1. & m m Now consider the case of 0omo1. Clearly, if f ðm a Þogð a Þ, then l1 o  a. Now, m2 m m m m 2 ¼ a2 , and gð a Þ ¼ a2 . Hence, for 0om, f ð a Þogð a Þ if and only if m om, or, equivalently, mo1. 2 m m 0 m 0 m Finally, consider the case in which 1omo ð1aÞ 4a . If f ð a Þ4gð a Þ and f ð a Þog ð a Þ, 2 m m m m m m m then l1 4 a . Now, f ð a Þ ¼ a2 , and gð a Þ ¼ a2 . Hence, f ð a Þ4gð a Þ if and only if m 0 m 1a m2 4m, or equivalently, m41. Now, f 0 ðm a Þ ¼ 2 a and g ð a Þ ¼  a . Hence, we need m 1a 1a to show that 2 a o  a or equivalently, that mo 2 . By assumption, 2 1a 1a mo ð1aÞ ¼ ð1a 2 Þð2aÞ. Now, ð2aÞo18ao  1, i.e., is a fraction, so, by assumption, 4a 1a m is less than a fraction of ð 2 Þ. Hence, m must be less than ð1a 2 Þ.

f ðm a Þ

Proof of Proposition 4. At t, pt ¼ l1 pt1 þ y1 g, and pet ¼ lTðtþ2Þ pet1 þ yTðtþ1Þ g. As T  k ! 1; lTk ! l1 . Using the well-known properties of limits and infinite sums, a the (2.19) goes to y as T  k ! 1 if and only if 1aþal o1, which is true if and only 1 if mo1. Hence, limT!1 pt  pet ¼ 0. & Proof of Proposition 5. First consider the equilibrium (3.4) at t ¼ T: pT ¼ aT1 ½E T1 peT  peT1  þ c0 . gE T1 peT þ uT þ be

(P.3.1)

Hence, the period-(T  1) expectation of period-T price can be found as aT1 c0 . E T1 peT ¼ peT1 þ b þ g þ aT1 b þ g þ aT1

(P.3.2)

So, the equilibrium price function for peT is peT ¼ lT1 peT1 þ pT uT ; lT1 

aT1 1 . ; pT  b b þ g þ aT1

(P.3.3)

Knowing this price function, the period-T forecast variance can be computed as s2T;1 ¼

s2u . b2

(P.3.4)

Now, using this in the definition of aT1 yields aT1 ¼ Kb2 ¼

K . ðpT Þ2

(P.3.2) and (P.3.5) establish the boundary (3.17) and (3.18).

(P.3.5) &

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Knowing the equilibrium price function peT , the equilibrium price function for peT1 can be constructed by the same steps as used in construction of the equilibrium price function peT . Backward induction then shows that for any k ¼ 2; 3; . . . ; T, the equilibrium price function will have the form peTk ¼

aTkþ1 ½1  lTkþ1   glTðkþ1Þ peTðkþ1Þ b þ aTk ð1  lTk Þ

uTk b þ aTk ð1  lTk Þ

1  ðaTðK1Þ þ gÞrTkÞ þ aTk rTðk1Þ þc0 , b þ aTk ð1  lTk Þ 

(P.3.6) (P.3.7)

(P.3.8)

where lTðkþ1Þ ¼

aTðkþ1Þ ½1  lTðkþ1Þ   glTðkþ1Þ , b þ aTk ½1  lTk 

(P.3.9.i)

pTk ¼

1 , b þ aTk ½1  lTk 

(P.3.9.ii)

rTk ¼

1  ðaTðK1Þ þ gÞrTkÞ þ aTk rTðk1Þ . b þ aTk ð1  lTk Þ

(P.3.9.iii)

To establish the third recursive relationship (3.16), note that the one-period-ahead forecast variance at T  k is E Tk ½ðe pTðk1Þ  E Tk peTðk1Þ Þ2 .

(P.3.10)

Substituting (3.13) into (P.3.10), we have ðe pTðk1Þ  E Tk peTðk1Þ Þ2 ¼ ½pTðk1Þ uTðk1Þ 2 .

(P.3.11)

Hence, s2Tðk1Þ;1 ¼ s2u ½pTðk1Þ 2 .

(P.3.12)

Substituting this into (3.8) yields aTk ¼

b K s2u ½pTðk1Þ 2

.

(P.3.13)

Substitution of (3.15) for pTðk1Þ into this yields (3.16). Proof of Proposition 6. First we need to establish is that limðTkÞ!1 lTk ¼ l1 . To this end, note that (3.14) and (3.16) form a coupled difference-equation system in {lTk ; aTk g. Let T  k ¼ k; T  ðk þ 1Þ ¼ k þ 1, and so on, for notational ease. Furthermore, define xk ¼ ak ð1  lk Þ. Using this definition, rearrange (3.14) as xkþ1 lkþ1 ¼ . (P.3.2.1) b þ g þ xk

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Hence, 1  lkþ1 ¼

b þ g þ xk  xkþ1 . b þ g þ xk

(P.3.2.2)

Eq. (3.16) can be written as akþ1 ¼ K½b þ xk 2 . By a rearrangement of the definition of x, we have xkþ1 . akþ1 ¼ 1  lkþ1

(P.3.2.3)

(P.3.2.4)

So, xkþ1 ¼ K½b þ xk 2 . 1  lkþ1

(P.3.2.5)

Combine (P.3.2.2) and (P.3.2.5) to yield xkþ1 ¼

K½b þ xk 2 ðb þ g þ xk Þ  fðxk Þ. K½b þ xk 2 þ b þ g þ xk

(P.3.2.6)

For xk 4  b, the properties of f are fð0Þ ¼

f0 ¼

Kb2 ðb þ gÞ 40, Kb2 þ b þ g

K 2 ðb þ xÞ4 þ 2ðb þ xÞðb þ g þ xÞ2 40, fKðb þ xÞ2 þ b þ g þ xg2

lim f0 ¼ þ1,

x!1

(P.3.2.7)

(P.3.2.8) (P.3.2.9)

where the subscript on x is suppressed. Given the definition of x, if two real roots l1 and l2 exist, then there must be two real critical values that solve (P.3.2.6). That is, fðxk Þ intersects the locus xkþ1 ¼ xk at two points, say x1 and x2 ; 0ox1 ox2 . Because fð0Þ40 and f is monotonically increasing in x, fðxk Þ4xk for xk ox1 , and f0 o1 at xk ¼ x1 . Now, the value of xT1 is computed from the boundary conditions on lT1 and aT1 , and is xT1 ¼

Kb2 ðb þ gÞ Kb2 þ b þ g

(P.3.2.10)

which is equal to fð0Þ. Now, fðxÞ is upward-sloping, so it must be that fð0Þox1 , because fðxÞ ¼ x at x1 . Hence, xt1 ox1 . Because fðxÞ is monotonically increasing, and because xT2 ¼ fðxT1 Þ, it must be that xT1 oxT2 ox1 . Now xT3 ¼ fðxT2 Þox1 . Again, by monotonicity of f, xT3 oxT4 ox1 . Because f is continuous and monotonic, it must be that xTk oxTðkþ1Þ ox1 ; k ¼ 1; 2; 3; . . . ; T, and limk!1 j x1  xTk j¼ 0. Hence, limk!1 xk ¼ x1 . Because the steady-state value of x and steady-state value of l are monotonically related by the following

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relationship: l¯ ¼

x¯ b þ g þ x¯

it must be that limðTkÞ!1 lTk ¼ l1 .

(P.3.2.11) &

Now, limðTkÞ!1 pTk ¼ bþa 1 follows immediately. 1 ð1l1 0Þ (P.3.9.iii) as aTk rTk ¼ b þ g þ aTðk1Þ þ aTk ð1  lTk Þ 1 . þ b þ g þ aTðk1Þ þ aTk ð1  lTk Þ

Finally,

rewrite

ðP:3:2:12Þ

This linear first-order difference equation with non-constant coefficients has a solution that approaches r1 as (T  kÞ ! 1. Proof of Lemma 8. As a preliminary, we first prove that 1op1 o2op2 : Let w HðpÞ  p2  ðwd ad Þp þ ðadÞ. The zeroes of HðpÞ are then p1 and p2 . H is convex, with w Hð0Þ ¼ ad which is greater than zero, and Hð1Þ ¼ 1 þ 1a which is greater than zero. Let b p solve H 0 ðpÞ ¼ 0, i.e., b p ¼ 12 ðwd pop2 : Because both zeroes ad Þ. From (4.24), p1 ob are real, it must be that Hðb pÞo0. Hence, 1op1 ob p. Now, H 0 ð2Þ ¼ 4adðwdÞ . From (4.25.ii), 4adwoðw  dÞ2 ¼ wðw  2dÞ þ d2 . Hence, ad 4adow  d  wd ðw  dÞ. Because ðw  dÞ40, 4adow  d and H 0 ð2Þo0. Now, H 0 ðp2 Þ40. Because H 0 ðpÞ is monotonically increasing, it must be that p2 42. Hence, 1op1 o2op2 . Eq. (4.31) is a quadratic function in p that we will denote as d w GðpÞ : GðpÞ  ðpÞ2 þ ad p  ad . GðpÞ is convex, with Gð0Þo0. Hence there is one negative and one positive zero for GðpÞ, and the positive zero is b pT . Now, w e e G  H ¼ ad p  2w . Let p solve G  H ¼ 0; i.e., p ¼ 2. Consequently, for any p42, ad G4H. Now, at p2 ; Hðp2 Þ ¼ 0, so Gðp2 Þ40. Also, from above, Hð2Þo0, so Gð2Þo0: Therefore, GðpÞ ¼ 0 for some p 2 ð2; p2 Þ. & References Bacchetta, P., Van Wincoop, E., 2000. Does exchange-rate stability increase trade and welfare? American Economic Review 90 (5), 1093–1109. Benhabib, J., Farmer, R.E.A., 1994. Indeterminacy and increasing returns. Journal of Economic Theory 63, 19–41. Benhabib, J., Farmer, R.E.A., 1999. Indeterminacy and sunspots in macroeconomics. In: Taylor, J.B., Woodford, M. (Eds.), Handbook of Macroeconomics, vol. IA, Elsevier Science B.V., Amsterdam, pp. 387–448 (Chapter 6). Benhabib, J., Nishimura, K., 1998. Indeterminacy and sunspots with constant returns. Journal of Economic Theory 81, 58–96. Benhabib, J., Rustichini, A., 1994. Introduction to the symposium on growth, fluctuations, and sunspots: Confronting the data. Journal of Economic Theory 63, 1–16. Bewley, T., 1980. The optimum quantity of money. In: Kareken, J.H., Wallace, N. (Eds.), Models of Monetary Economics. Federal Reserve Bank of Minneapolis, Minneapolis, MN.

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