Eductive stability in real business cycle models

Eductive stability in real business cycle models George W. Evans, Roger Guesnerie, Bruce Mcgough To cite this version: George W. Evans, Roger Guesnerie, Bruce Mcgough. Eductive stability in real business cycle models. 2010. ฀halshs-00565011฀ HAL Id: halshs-00565011 https://halshs.archives-ouvertes.fr/halshs-00565011 Preprint submitted on 10 Feb 2011 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. WORKING PAPER N° 2010 - 47 Eductive stability in real business cycle models George W. Evans Roger Guesnerie Bruce McGough JEL Codes: E1, E30, E32, E6 Keywords: PARIS-JOURDAN SCIENCES ECONOMIQUES 48, BD JOURDAN – E.N.S. – 75014 PARIS TÉL. : 33(0) 1 43 13 63 00 – FAX : 33 (0) 1 43 13 63 10 www.pse.ens.fr CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE – ECOLE DES HAUTES ETUDES EN SCIENCES SOCIALES ÉCOLE DES PONTS PARISTECH – ECOLE NORMALE SUPÉRIEURE – INSTITUT NATIONAL DE LA RECHERCHE AGRONOMIQUE Eductive Stability in Real Business Cycle Models George W. Evans University of Oregon University of St. Andrews Roger Guesnerie Paris School of Economics Collège de France Bruce McGough Oregon State University November 13, 2010 Abstract We are indebted for helpful comments from participants in seminars at Chicago, Columbia, the Delhi School of Economics, New York University, Oregon, and St. Andrews, and in particular from D. Pearce, H. Uhlig, T. Sadzik and M. Woodford. Financial support from National Science Foundation Grant no. SES-1025011 is gratefully acknowledged. 1 Abstract : We re-examine issues of coordination in the standard RBC model. Can the unique the unique rational expectations equilibrium be "educed" by rational agents who contemplate the possibility of small deviations from equilibrium? Surprisingly, we …nd that coordination along this line cannot be expected. Rational agents anticipating small but possibly persistent deviations have to face the existence of retroactions that necessarily invalidate any initial tentative "common knowledge" of the future. This “impossibility” theorem for eductive learning is not fully overcome when adaptive learning is incorporated into the framework. Résumé : Nous réexaminons la question de la coordination des anticipations dans le modèle standard de cycles réels. L’équilibre à anticipations rationnelles unique peut-il être déduit de raisonnements d’agents rationnels confrontés à la possibilité de petites déviations de l’équilibre ? De façon quelque peu inattendue, nous donnons une réponse négative à cette question. Des agents rationnels faisant face à des possibilités de petites déviations persistantes ne peuvent écarter des rétroactions qui invalident nécessairement toute hypothétique « Connaissance Commune » initiale de l’avenir. Ce théorème d’impossibilité de l’apprentissage « divinatoire » a des contreparties qui mettent en exergue des di¢cultés parallèles de l’apprentissage « adaptatif » dans ce contexte. Codes JEL : E1, E30, E32, E6 2 1 Introduction This paper examines the question of expectational coordination in a simple Real Business Cycle model. The long run focal point for expectational coordination, as usual in economic modelling, is the rational expectations (in this simple model, perfect foresight) equilibrium. Our analysis puts emphasis on the expectational robustness of the equilibrium, using what may be called the “eductive” viewpoint, (see Evans and Guesnerie (1993, 2005) and Guesnerie (2002) for an introductory conceptual assessment and Guesnerie (2005) for a collection of studies along these lines). “Eductive learning” can be contrasted with the more standard “evolutive” or “adaptive” learning viewpoint,1 which is also introduced at a later stage in the paper. The logical framework and the central results for eductive learning, as well as the connections with evolutive learning, are well understood in many contexts, and in particular within simple models of the overlapping generations type in which agents are short lived (see Gauthier and Guesnerie (2004) for an assessment that puts emphasis on the consistency of di¤erent viewpoints on expectational coordination). In contrast, in an RBC (real business cycle) model, agents are long lived and in fact have in…nite life. The long-life assumption plays an important role in the working of the world under examination. In particular, long-lived agents take into account their permanent income, rather than income over a short horizon, a fact that has a key impact on the understanding and design of macroeconomic policies. The question under scrutiny here is the e¤ect of the introduction of long-lived agents on expectational coordination: does it make expectational coordination more or less robust? The answer, based on our “eductive” assessment of coordination, is that coordination is necessarily weak. There is no collective image of the future, close but not identical to the “true,” self-ful…lling, image, which is able to trigger (a common knowledge of) the self-ful…lling image. Thus every such image is subject, at some stage, to be invalidated by facts: in this simple world, a “crisis,” here an expectational crisis, is in some sense unavoidable. However, the extent of weakness of expectational coordination, and metaphorically the plausibility of the crisis, depends upon a certain number of characteristics of the system that we exhibit. Also, the “real-time” amendment of the collective image of the future must necessarily rely on adaptive learning, the success of which in maintaining the collective image of the future, i.e. in some sense in avoiding the crisis, depends on features of the system that we stress. The paper proceeds as follows. In section 2, we present the model and its equilibria. We then provide a number of preliminary results, together with their intuition, on the connections between long-run and short-run individual expectations and ag1 For the adaptive learning approach see, for example, Marcet and Sargent (1989), Woodford (1990) and Evans and Honkapohja (2001). 3 gregate, long-run or short-run, e¤ects. In section 3, we present from two di¤erent viewpoints the eductive criteria that serve to assess expectational robustness of the equilibrium. Section 4 provides conditions for weak eductive stability, gathered in three propositions. Section 5 shows that strong eductive stability, whatever its exact de…nition, necessarily fails. Section 6 focuses on the possibility of maintaining a plausible image of the future in the presence of real-time adaptive learning. Section 7, which precedes the Conclusion, contrasts our results with those for a model of capital accumulation in which agents have short lives. 2 The model, equilibrium and the in‡uence of beliefs on states 2.1 The model and equilibrium We consider a standard RBC model, except that for simplicity we assume a …xed labor supply and omit exogenous productivity shocks.2 These simplifying assumptions, which amount to a focus on a nonstochastic discrete-time Ramsey model, are not critical to our results and are made in order to clarify the central features of our analysis. Elimination of both random shocks and labor-supply response to disequilibrium expectations can be expected to facilitate coordination on the REE.3 Despite eliminating these in‡uences we establish that a strong form of eductive stability fails. 2.1.1 The household problem There is a continuum of identical in…nitely-lived households, indexed by ! 2 [0; 1]. Households own capital, kt (!), and one unit of labor, each supplied inelastically. At time t = 0, household ! solves max E0 (!) 1 X t U (ct (!)), where 0 < < 1, (1) t=0 subject to kt+1 (!) = (1 + rt )kt (!) + qt ct (!); (2) with initial wealth k0 (!) given. Here qt is the wage rate, and rt is the rental rate for capital, at time t, and the utility function U (c) is increasing, strictly concave and smooth. We further impose a No Ponzi Game (NPG) condition that the present value of their limiting lifetime wealth be nonnegative. If, at this stage, one does not assume 2 The seminal papers developing the RBC model include Kydland and Prescott (1982), Long and Plosser (1983) and Prescott (1986). 3 For example, the weak eductive stability conditions, given below, can be shown to be stricter when labor supply is elastic. 4 that the future is deterministic, E0 (!) captures the expectations of agent ! formed using his subjective distribution. Let us assume at this stage that k0 (!) = k0 for all agents. Iterating forward the household ‡ow budget constraint, and imposing the NPG and transversality conditions, gives the lifetime budget constraint of the household; thus, we rewrite the consumer program as: 1 X t max E0 (!) U (ct (!)) t=0 subject to 1 X Rt ct (!) = t=0 1 X Rt qt + (1 + r0 )k0 (!), where t=0 Rt t Y = (1 + ri ) 1 i=1 and R0 = 1. The …rst-order condition for the household optimization problem is the Euler equation U 0 (ct (!)) = Et (!) ((1 + rt+1 )U 0 (ct+1 (!))) . (3) 2.1.2 Equilibrium Goods are produced by …rms from capital and labor using a constant returns to scale production function f (K; L), satisfying the usual assumptions, under conditions of perfect competition. Thus rt ; qt are given by rt = fK (Kt ; 1) qt = fL (Kt ; 1) R1 where Kt = 0 kt (!)d! and where fK = @f =@K and fL = @f =@L. For convenience, below, we also write f (K) in place of f (K; 1) and use the notation f 0 = fK and f 00 = fKK . In addition we have the aggregate capital accumulation equation Kt+1 = (1 where Ct = R1 0 )Kt + f (Kt ; 1) Ct ; ct (!)d!. We can now de…ne the (unique) perfect foresight steady state. De…nition 1 The perfect foresight steady state Kt = kt (!) = K, Ct = ct (!) = C, 5 rt = r and qt = q is given by 1 r q C = (1 + r) = fK (K; 1) = fL (K; 1) = f (K; 1) ; K In steady state, agents consume what is left after depreciated capital is replaced. Since, f (K; 1) = fK (K; 1)K + q we also know that C = rK + q: If K0 = K then under perfect foresight the economy stays in the steady state for all t, and if K0 6= K, then there is a unique perfect foresight path that converges to the steady state as t ! 1. We now assume that we are initially in steady state, with k0 (!) = K for all !, and we examine the robustness of expectational coordination on this equilibrium. 2.2 2.2.1 Beliefs, actions, plans and realizations Preliminaries Consider an individual agent facing the consumption/savings problem (1)-(2). The behavior of the agent is in part determined by his beliefs about the future values of wages and interest rates. At the most general level, an agent’s beliefs may be stochastic, and so summarized by a sequence of joint density functions fFt (q t ; rt )g where q t and rt are the time t wage and interest rate histories, respectively. Here, we restrict attention to deterministic beliefs, i.e. to point expectations, which is satisfactory in our nonstochastic setting for beliefs that are small deviations from perfect foresight. The beliefs of agent ! may therefore be summarized by real sequences of expected wages and interest rates. We choose to assume that the agent understands the relationship between aggregate capital and input prices, that is, the agent knows rt = f 0 (Kt ) ; and qt = f (Kt ) f 0 (Kt )Kt ; so that, in fact, his beliefs are completely captured by a sequence of real numbers identifying his point expectations of future capital stock: we denote these beliefs by ~ e (!) = fKte (!)gt 0 ; where K0e (!) = K0 = K is known to all agents. A beliefs K ~ e = fK ~ e (!) : ! 2 [0; 1]g. pro…le is the collection of all agents’ beliefs: K 6 The key ingredient of the analysis is the understanding of the e¤ect of change in individual expectations on changes in individual actions or plans (particularly when these changes occur around the equilibrium). Taking as reference point the perfect foresight steady-state path K0 = K; Kt = K, for all t, we analyze the e¤ects on an agent’s plans of small changes, around the steady state-values, in the agent’s individual initial capital k0 (!); in the aggregate initial capital and in the agent’s point expectations Kte (!). The analysis relies on the following lemma. ~ e (!)) be the value function associated with the problem (1)Lemma 1 Let V (k0 (!); K (2) and the NPG condition, given initial stock k0 (!), aggregate capital K0 and beliefs ~ e (!). The derivatives of the value function, at the steady-state path k0 (!) = K; K K0 = K; Kte (!) = K for all t 1; are as follows: 1. @V @k0 (!) 2. @V @K0 3. @V @Kte (!) = 1 U 0 (C) =0 = 0: The proof of the Lemma is provided in the Appendix. The …rst result of the Lemma is fairly intuitive from examination of the intertemporal budget constraint. The third result relies on the fact that, given the individual budget constraint of period t as written above, kt+1 (!) = (1 + rt )kt (!) + qt ct (!); the equilibrium plan Kt = kt (!) = K, Ct = ct (!) = C; is still feasible, to …rst-order approximation, when rt changes from r to r + fKK (K; 1)dKte (!) and qt changes from q to q + fKL (K; 1)dKte (!), since KfKK (K; 1) + fKL (K; 1) = 0: (The second result uses the same argument at period 0). In other words, the expected price changes induced by the change in expectations on the capital stock have no income e¤ect, to the …rst order approximation. For that reason, it has no welfare e¤ect, as stressed in the second and third parts of the Lemma. 2.2.2 Further insights Lemma 1 provides a useful computational conclusion, which is most easily exploited by allowing for a modi…ed notation that measures quantities as deviations from steady state: set dkt (!) = kt (!) K and dKte (!) = Kte (!) K, and similarly for any individual or aggregate quantity or point expectation thereof. Using the Lemma we can now explicitly compute, to …rst-order, the consumption function. Because we are restricting attention to beliefs time paths within a small neighborhood of the steady state, we will often identify a particular variable’s time path with its …rst-order approximation; and we will use our deviation notation to capture this ~ e (!), initial wealth identi…cation. Suppose, for example, that agent ! has beliefs K 7 k0 (!), and that ct (!) is his optimal consumption decision at time t. Then we will also ~ e (!), initial wealth dk0 (!) and that dct (!) is his optimal say that he has beliefs dK consumption decision at time t.4 With this notation, an intuitively straightforward consequence of the previous lemma is: Corollary 1 Consider any time path of beliefs dKte (!), and any initial capital stock dk0 (!) and dK0 . Let dct (!) be the associated sequence of consumption decisions. Then X 1 t dk0 (!) = dct (!): (4) t 0 Corollary 1 follows from Lemma 1 and implies in particular thatPgiven beliefs dKte (!), t if k0 (!) = K then the optimal consumption path satis…es dct (!) = 0: The natural welfare interpretation of the second part of the Lemma can again be stressed: a change in the expected path of aggregate capital, and the corresponding expected price changes that it triggers, have no …rst-order impact on welfare. For this reason, we call this the welfare corollary. The next Lemma exploits the welfare corollary as well as the above individual Euler equation. Lemma 2 Consider any time path of beliefs dKte (!) and assume dk0 (!) = 0. Let dct (!) be the associated sequence of consumption decisions. Then ! t X dct (!) dc0 (!) = + drse (!) ; C C s=1 where = CU 00 (C)=U 0 (C); and dc0 (!) = 1 X C s drse (!) s=1 ! ; where drte (!) = fK (K; 1)dKte (!): The above Lemma (which is proved in the Appendix) brings two facts into sharp relief. The …rst, expressed in the second formula, is the sensitivity of current consumption, dc0 (!); to expectations of the distant future. The second, captured in the 4 We may compute the agent’s optimal consumption as dct (!) = X @ct (!) @ct (!) dk0 (!) + dK e (!); @k0 (!) @Kse (!) s s 1 where the partial derivatives are evaluated at steady-state beliefs. 8 …rst formula, is still more striking since it shows that dct (!); the plans for even distant future levels of consumption, can be extremely sensitive to a change in expectations. As will be seen later, this is a key issue for the assessment of expectational stability as we approach it here. 3 The robustness of expectational coordination: “eductive stability” criteria We shall …rst provide a de…nition of “eductive” stability based on rather abstract game-theoretical considerations. This will be the “high-tech” view on expectational coordination. We shall however see later how the sophisticated “high-tech” viewpoint meets simpler considerations, that may be termed “low-tech”. This …rst analysis will voluntarily leave in the shadow the time dimension of the problem. We shall reintroduce time in order to see how to adapt the general ideas to our in…nite horizon setting. 3.1 Local eductive stability: the high-tech view We are in a world populated of rational economic agents (in all the following, we shall assume that these agents are in…nitesimal and modelled as a continuum). The agents know the logic of the collective economic interactions (the underlying model). Both the rationality of the agents and the model are Common Knowledge (CK). The state of the system is denoted E and belongs to some subset E of some vector space. Note that E can be a number (the value of an equilibrium price or a growth rate), a vector (of equilibrium prices,...), a function (an equilibrium demand function), an in…nite trajectory of states, as will be the case in this paper, or a probability distribution. Emphasizing the expectational aspects of the problem, we view an equilibrium of the system as a state E such that if everybody believes that it prevails, it does prevail.5 Under eductive learning, as described below, each agent contemplates the possible states of the economy implied by the beliefs and associated actions of the economy’s agents. Coordination on a particular equilibrium outcome obtains when this contemplation, together with the knowledge that all agents are engaged in the same contemplation, rules out all potential economic outcomes except the equilibrium. If coordination on an equilibrium is implied by the eductive learning process, then we 5 Note that E is such that the assertion “it is CK that E = E ” is meaningful. 9 say that the equilibrium is “eductively stable”or “strongly rational.”6 The argument can be either global or local. We now introduce the local version. Formally, we say that E is locally eductively stable (or locally strongly rational) if and only if one can …nd some non-trivial “small” neighbourhood of E , V (E ), such that Assertion A implies Assertion B: Assertion A : It is “hypothetically” CK that E 2 V (E ). Assertion B : It is CK that E = E . Assertion A is at this stage hypothetical.7 The mental process that leads from Assertion A to Assertion B is the following: 1. Because everybody knows that E 2 V (E ), everybody knows that everybody limits their responses to actions that are best responses to some probability distributions over V (E ). It follows that everybody knows that the state of the system will be in a set E(1) E. 2. If E(1) is a proper subset of V (E ), the mental process goes on as in step 1, but based now on E(1) instead of V (E ). In this case it follows that everybody knows that the state of the system will be in a set E(2) E. 3. The process continues inductively provided that at each stage, E(n) is a proper subset of E(n 1). In the stable case, we then have a decreasing sequence E(n) E(n 1) :::: E(1) V (E ).8 When the sequence converges to E , the equilibrium is “locally strongly rational” or “locally eductively” stable. Here “locally” refers to the fact that the initial neighbourhood is small.9 Intuitively, an equilibrium is eductively stable if whenever agents believe the economy is within a neighborhood of the equilibrium, and these beliefs are common knowledge to all agents, then the agents can conclude that the aggregate behavior implied 6 We remain rather vague on the full game theoretical background of our investigation (for a deeper discussion of some of the issues, the reader may refer to Guesnerie and Jara-Moroni (2011)). A study within a “normal form” framework echoing the preoccupations of the present paper can be found in Matsui and Oyama (2006). 7 Although it might be sustained by some policy commitment. 8 A given model or economic environment may be naturally allied with several distinct common knowledge assumptions: these common knowledge assumptions will all impose the recursive reasoning process described above, but will di¤er in the initial restrictions on agents’ beliefs; and because of the central role the initial restriction plays in the eductive learning process, di¤erent common knowledge assumptions may produce di¤erent stability results. 9 If the initial neighbourhood were equal to the whole space E, then the word global would replace the word local. 10 by these beliefs results in an economy that is within a strictly smaller neighborhood of the equilibrium.10 3.2 Local eductive stability: the low-tech view The de…nition, i.e. the description of the successful deletion of non-best responses, starting under the assumption that the state of the system is close to the equilibrium state, re‡ects the “local” version of a “hyper-rationality” viewpoint. Another plausible intuitive de…nition of local expectational stability would be the following: there exists a non-trivial neighbourhood of the equilibrium such that, if everybody believes that the state of the system is in this neighbourhood, it is necessarily the case, whatever the speci…c form taken by everybody’s belief, that the state is in the given neighbourhood. With the above formal apparatus, the de…nition would be: one can …nd some nontrivial “small” neighbourhood of E , V (E ), such that if everybody believes that E 2 V (E ), then the state of the system will be in E(1); a proper subset of V (E ). The absence of such a neighbourhood V (E ) does signal risks of instability: there can be facts falsifying any universally shared conjecture on the set of possible states, whatever the proximity of the conjectured set to the equilibrium, unless V (E ) is reduced to the equilibrium E itself. Again, with the initial belief assumption, individual actions are best responses to some probability distributions over V (E ), and the stability property puts emphasis on the non-falsi…ability of the initial belief. The criterion is low-tech, in the sense that it refers to the rationality of agents, but not to CK of rationality or of the model.11 In contrast, the low-tech criterion appeals only to the results of the …rst step of the high-tech criterion: in the low-tech setting, agents’ actions depend only on their beliefs about the state of the system, and not on their beliefs about other agents’ beliefs. However, the high-tech and low-tech criteria turn out to be equivalent in this abstract setting as been previously stressed in the literature, and producing a formal statement is left to the reader.12 Some words are …nally in order concerning the connections between the “eductive” viewpoint and the “evolutive learning” viewpoint. Let us only say that the failure to …nd a set V (E ), for which the the equilibrium is locally strongly rational, signals a tendency for any near-equilibrium states of beliefs, a priori reachable through some 10 When this …rst step conclusion obtains, and the neighbourhood is small, then the next step follows. Because of common knowledge, agents can also conclude that all agents will reach the same conclusion, and therefore all agents can conclude that all agents will modify their beliefs appropriately; the mental process repeats ad-in…nitum, with a (in…nitely) decreased support. 11 It does not even require full knowledge of the model. 12 The equivalence can for example be deduced from assumptions analogous to the weak assumptions of Guesnerie and Jara-Moroni (2009). Their results also allows one to show that the analysis may concentrate on heterogenous point-expectations. 11 “reasonable” evolutive updating process, to be triggered away in some cases, a fact that threatens the convergence of the corresponding learning rule.13 3.3 Eductive stability: the time dimension The time dimension of our problem, and in particular the in…nite horizon, as well as the fact that agents are in…nitely-lived, creates a certain number of problems with our general framework. The equilibrium E under consideration is given by Kt = K for all t, and the …rst issue is concerned with the notion of a neighbourhood V (E ), which is less straightforward here than in timeless or short-horizon contexts. We begin with a simple, natural restriction on the initial time paths of beliefs, that they lie within an "-neighborhood of the steady state, which might be called the “cylinder” assumption. In later sections we consider alternative initial assumptions. The cylinder assumption is: B1: For some " > 0 su¢ciently small, Kt 2 D(") [K "; K + "] for all t.14 We will refer to the assumption that B1 is common knowledge as CK1: CK1: B1 is common knowledge. Given our emphasis on point expectations, which is justi…ed15 for a small neigh~ e satis…es B1 provided that bourhood of beliefs, we say that a beliefs pro…le K e Kt (!) 2 D(") for all ! and t. The neighborhood V (E ) is the collection of all beliefs satisfying B1. The low-tech eductive approach simply assumes that agents’ beliefs satisfy B1, that is, lie in the neighborhood V (E ), and for eductive stability, we require that the implied trajectory of capital is consistent with B1. Under the high-tech eductive approach we require further that B1 is common knowledge: agent ! 0 knows that Kte (!) 2 D("), and that agent ! 00 knows that agent ! 0 knows that Kte (!) 2 D("), etc. If the equilibrium is eductively stable under the high-tech approach, then agents are able to deduce at time zero that Kt = K for all t. We will henceforth say that the steady state is strongly eductively stable if it is eductively stable according to the high-tech de…nition. We use the word “strong” to indicate the simultaneous coordination at time zero of the entire time path of aggregate capital, using the high-tech approach. Later we will distinguish this notion of stability from less demanding concepts discussed below. 13 And certainly forbids a strong form of “monotonic” convergence. More on this subject can be found in Guesnerie (2002), Guesnerie and Woodford (1991) and Gauthier and Guesnerie (2004). 14 We will later introduce alternative assumptions B1’ and B2. 15 since the expectation of a random variable remains in the support of the random variable. 12 Our hope would be to demonstrate strong eductive stability as follows. The criterion leads one to concentrate, at the …rst stage of the mental process, on the in…nite-horizon aggregate plans of the agents. In the same way, second-order beliefs are concerned with aggregate plans implied by …rst order beliefs, so on. At the end of the mental process, assuming strong eductive stability, agents correctly predict the next period’s state, and all future states resulting from the guessed actions and plans. Strong eductive stability implies a low-tech version of strong eductive stability, which requires that the plans of the agents, as deduced at the …rst step of the iteration process by everybody from the B1 assumption, be compatible with B1. In a sense, as noted above, this condition should be almost su¢cient for (high-tech) strong eductive stability. This line of argument is developed in Section 5. As we will see in Section 5, even in its low-tech version, strong eductive stability is impossible to achieve. Because of this, we consider less demanding concepts of eductive stability. The …rst concept, which can regarded as an alternative form of strong eductive stability, is inherently low-tech. This option consists of looking at the real-time consequences of the beliefs: we ask whether beliefs satisfying B1 trigger actions that generate an actual path of the system compatible with B1. Under the alternative approach just described, we allow individual agents to hold …xed or varying beliefs over time, even though these beliefs may have been (perhaps repeatedly) contradicted, provided they stay close to the equilibrium in the sense of B1, and we require that this implies that aggregate capital continues to satisfy B1. This approach is considered in Section 5.3. Another real-time option is to mix adaptive learning considerations with the eductive dimension stressed here: as is standard under adaptive learning, agents are assumed to modify their beliefs over time in response to the realizations of the capital stock according to a natural class of adaptive rules; and, in line with the eductive approach, for stability we require that both beliefs and realizations satisfy B1. This approach is studied in Section 6. Whatever the approach to the coordination is, a necessary condition for eductive stability is that the initial beliefs necessarily trigger a …rst period realization of aggregate capital that is compatible with the belief restriction B1. We call this “weak eductive stability,” and now turn to this concept. 4 4.1 Weak Eductive Stability Further on the eductive view: weak eductive stability as a necessary condition As just argued, the “strong” stability question is whether CK1 will allow agents to coordinate on the unique perfect foresight equilibrium Kt = K for all t 1. 13 Answering this question will be the focus of Section 5. In the current Section, we determine when a necessary condition for this is met: under B1 will the optimal plans of agents necessarily lead to consumption and saving decisions c0 (!); k1 (!) that satisfy K1 2 [K "; K + "]? If an equilibrium meets this necessary condition, we say that it satis…es weak eductive stability. De…nition 2 The steady state K is weakly eductively stable if, given the initial condition k0 (!) = K for all agents !; and that for all agents beliefs satisfy B1, the aggregate capital stock in period t = 1 implied by the agents’ optimal plans satis…es K1 2 D("). 4.2 Weak eductive stability: a …rst result We establish Lemma 6 in the Appendix which identi…es the worst-case expectations of the agents in the sense of inducing a maximum threat to the validity of the initial conjecture. In words the lemma says that the worst-case expectations are on the boundary, i.e. they arise when agents believe that the capital stock will remain at one of the boundaries of the cylinder. Using this lemma, we may now establish our …rst stability result.16 Theorem 1 Under B1, the steady state is weakly eductively stable if and only if 2 Cf 00 <1 (1 ) (5) Proof: Let " > 0 be small, so that linear approximations to optimal behavior are valid. Weak eductive stability is equivalent to showing that for any beliefs pro…le satisfying B1 then dK1 2 I(") ( "; "): By Lemma 6 it su¢ces to …nd necessary and su¢cient conditions guaranteeing that dK1 2 I(") provided that dKte (!) = e for all ! and t, and for e = ". Using dk1 (!) = dc0 (!), recalling from Lemma 2 that ! 1 X C s dc0 (!) = drse (!) ; s=1 and noting drse (!) = jf 00 j "; we get that 2 dk1 (!) = Cf 00 (1 ) 16 ": In the following theorem we by assumption exclude the case in which Throughout the paper we exclude analogous nongeneric cases. 14 2 Cf 00 = ( (1 )) = 1. From Kt = R1 0 kt (!)d! we thus have 2 Cf 00 (1 ) dK1 = ": Since weak eductive stability requires jdK1 j < ", the result follows. Note that the formula (5) makes intuitive sense: high and low f 00 promote eductive stability. As an example, suppose that production is Cobb-Douglas, so that f (K) = K , for 0 < < 1, and that utility takes the constant elasticity form (1 ) U (c) = (c 1)=(1 ), for > 0. Then 1=(1 K= r+ ) and f 00 = ( 1)K 2 . Using these and the steady-state equations for and C it can be computed that if, say, = 1=3, r = 0:05 and = 0:10, then we have weak eductive stability if and only if > 2=3. This condition is perhaps plausibly satis…ed, but we have obtained the somewhat surprising result that even weak eductive stability of the RBC model cannot be taken for granted. 4.3 Weak eductive stability: further results A natural question is whether weak eductive stability can be obtained if assumption B1 is strengthened. We consider the following alternative assumption in which the possible deviations of Kt from K shrink to 0 over time at a geometric rate. B1’: For some 0 < a < 1 and " > 0 su¢ciently small, Kt 2 [K t. at "; K + at "] for all Theorem 1’: For a > 0 su¢ciently small, the steady state is weakly eductively stable under B1 if and only if f 00 C 2 < 1: Proof: Suppose again that all agents have homogeneous expectations given by one of the extremes Kte (!) = Kte = K + at 1 e for all t = 1; 2; 3; : : :, where e = ": Then dCt = dC0 + = dC0 + 15 C C f 00 Pt f 00 e s=1 1 1 dKse at : a Since also P1 t=0 t dCt = 0, we obtain : 1 X t=0 t dC0 + 1 X C t t=0 which yields f 00 e 1 1 at a = 0; 2 Cf 00 e : (1 a) dC0 = Considering a ! 0 establishes the result. Unsurprisingly, by imposing stronger belief assumptions, the condition we obtain is weaker than the previous one. However, it puts emphasis on similar features of the system, i.e. 2 ; C; f 00 ; ; with similar intuitive e¤ects. We reserve consideration of a more general class of assumptions to the case of strong eductive stability, to which we now turn. 5 5.1 Strong eductive stability: impossibility theorems The impossibility of strong eductive stability under CK1 The question of local “strong eductive stability” was discussed in Section 3.3 as a suitable formulation of the high-tech approach that takes into account the time dimension. For our economic model we can state this explicitly as follows: De…nition 3 The steady state is strongly eductively stable if CK1 implies that it is common knowledge that the equilibrium path, Kt = K for all t, will take place. In Theorem 1 we gave a condition for a minimal consistency requirement: given beliefs B1, the …rst period plans of agents will necessarily be consistent with CK1 in the …rst period t = 1, if and only if the stated condition (5) is satis…ed. However, this is only a weak necessary condition for consistency in the stronger sense just de…ned. In t = 0, given their expectations of the aggregate capital stock fKte (!)g1 t=1 each 1 agent formulates an optimal dynamic consumption plan fc (!)g . This implies a n ot 1 t=0 R1 trajectory for the aggregate consumption Ct = 0 ct (!)d! and hence, using t=0 Kt+1 = (1 )Kt + f (Kt ; 1) Ct ; (6) a trajectory for the aggregate capital stock fKt g1 t=1 . Recall that D(") = [K "; K +"]. For local strong eductive stability of the steady state, it is necessary – indeed, for the reasons sketched in Section 3.1, almost su¢cient – that the following condition be met: 16 for every " > 0 su¢ciently small, the implied trajectory of aggregate capital fKt g1 t=1 lies in a strictly smaller cylinder D1 ("0 ) = D("0 ) D("0 ) , i.e. Kt 2 D("0 ) for all t = 1; 2; 3; : : :, for some 0 < "0 < ". This implies that for strong eductive stability, for all expectations satisfying CK1, we must have Kt 2 D(") for all t = 1; 2; 3; : : :. We now show that, in fact, for all parameter values, the plans made by agents at time t = 0, based on CK1, can fail to be compatible with beliefs. That is, Theorem 2 The steady state is never strongly eductively stable under CK1. Proof: Suppose that all agents have homogeneous expectations Kte (!) = Kte = K + e for all t, where e = ": Their consumption plans thus satisfy dCt dC0 = + C C We have already shown that dC0 = dCt = f 00 e C tdr = ( (1 dC0 + C )) 1 tf 00 e: 2 00 f eC; which implies t , for t = 1; 2; 3; : : : 1 Linearizing (6) around the steady state we have dKt+1 = (1 + f 0 dKt+1 = 1 dKt )dKt dCt , or dCt : For t > =(1 ) it can be seen that dCt is bounded away from 0; with the opposite sign to e; and grows linearly with t. Hence for t su¢ciently large jdCt j > 2 1 ", which implies jdKt+1 j > " if jdKt j < ". It follows that there must be a time t at which jdKt j > ". This establishes the result. Theorem 2 shows that strong eductive stability never holds under CK1. We remark that the proof in fact shows that the low-tech version of strong eductive stability also necessarily fails. The Theorem 2 result should be contrasted with the fact that, with short-lived agents, eductive stability holds under assumptions that are reminiscent of (though often more stringent than) standard conditions (such as determinacy, saddle-path con…guration). The long-run horizon dramatically a¤ects expectational coordination, as evaluated from the strong eductive viewpoint. In the next section we show that the instability result does not depend on our speci…c choice of the neighbourhood. 5.2 The general impossibility of strong eductive stability We now consider a more general class of beliefs, which nests B1 and B1’: 17 B2: There exists a speci…ed deterministic sequence f"t g1 t=1 with 0 < "t < " and " su¢ciently small, such that Kt 2 [K "t ; K + "t ] for all t. CK2: B2 is common knowledge. When the Common Knowledge assumption is CK2, De…nition 3 of strong eductive stability is modi…ed in the obvious way. Theorem 3 The steady state is never strongly eductively stable for any common knowledge assumption CK2. Proof: Fix a CK assumption with f"t g1 t=1 that satis…es CK2. Suppose that all agents have homogeneous expectations Kte (!) = Kte = K + "t for all t. Their consumption plans satisfy ! t dCt dC0 f 00 X = + "s : (7) C C s=1 From Lemma 2 we have f 00 dC0 = C 1 X s "s s=1 ! ; (8) which implies dCt = f 00 C t X "s s=1 1 X s "s s=1 ! , for t = 1; 2; 3; : : : : (9) From the linearization of (6) around the steady, as in the proof of Theorem 2, we have dKt+1 = 1 dKt dCt : (10) P1 If ( s=1 "s ) = +1 then (7)-(8) imply that jdCt j ! 1 as t ! 1. It follows from (10) that for t su¢ciently large, jdKt j < "t < " implies jdKt+1 j > ", so that strong eductive stability fails. P Suppose instead that ( P1 s=1 "s ) is …nite. Then P1 "t s! 0 as t ! 1. Furthermore, "s for some a > 0, and hence " = (1 + 2a) since 0 < < 1, we have 1 s=1 s=1 s there exists T > 0 such that t X "s (1 + a) s=1 1 X s=1 18 s "s for all t T . Then t T and (9) imply dCt = jf 00 j C 1 X s s=1 "s t X "s s=1 ! Q; P 00 s where Q = jf j Ca 1 "s > 0. Choose T1 > T su¢ciently large so that "t < Q=2 s=1 for all t > T1 . By (10) it follows that jdKt j < "t and t T1 imply jdKt+1 j Q=2 > "t+1 and again strong eductive stability fails. We remark that if the common knowledge assumption for a sequence f"t g1 t=1 satis…es CK2 except that "t = 0 for some proper subset of times t, the contradiction obtains more directly by focusing attention on such times. We also note that, again, our proof in fact demonstrates the failure of even low-tech strong eductive stability. The negative result of Theorem 3 means that the hyper-rationalistic viewpoint of strong eductive stability is never conclusive. Our hyper-sophisticated agents cannot convince themselves that the equilibrium will prevail. In a sense, here in the RBC model, expectational coordination must appeal to bounded rationality considerations. 5.3 A “real-time” alternative view of stability: another impossibility theorem Another way to read the above statement is that there are always initial plans in line with initial beliefs of the type B2, which turn out to be inconsistent with such beliefs. It follows that CK2 cannot trigger common knowledge of the equilibrium. In this section we consider an alternative, real-time perspective of strong eductive stability, following the low-tech approach. B2 continues to describe a (common) set of possible beliefs: the beliefs of individual agents are taken from this set, and now may or may not change through time. At time t the earlier speci…c beliefs of agents will generally have been falsi…ed by events. However we suppose that their beliefs always obey the initial restrictions B2 and that the actual path up to t also satis…es Ks 2 [K "s ; K + "s ] for s t. Will some of the possible paths of the system generated from such beliefs falsify the assumed initial restrictions B2? The connection of this question with the notion of strong stability indicated in De…nition 3 is not immediate.17 Let us go to the formal analysis. Our alternative low-tech de…nition of strong eductive stability, for collective beliefs of type B2, is the following. 17 We know from the previous analysis that initial plans turn out to be incompatible with B2; however, the actual path of aggregate capital will clearly di¤er from the path determined by these initial plans, and so violation of B2 by the actual path of aggregate capital is not obvious. 19 De…nition 4 We say that K is strongly eductively stable in the alternative sense if, for all t, Kt 2 [K "t ; K + "t ] implies Kt+1 2 [K "t+1 ; K + "t+1 ] when each agent ! chooses ct (!) optimally given expectations fKse (!)g1 s=t+1 consistent with B2. This de…nition says that if up to period t, the actual path of capital has remained compatible with the initially assumed restriction on beliefs, (making them close to the equilibrium beliefs, in the sense of assumption B2), then it will remain compatible at period t + 1,... and hence for ever. For this de…nition we still have: Theorem 4 Under the beliefs restriction B2, the steady state is never strongly eductively stable in the sense of the alternative de…nition. In fact, stability in the sense of the alternative de…nition implies here to strong eductive stability in the sense of the …rst de…nition. To understand this, we compare the planned and real aggregate trajectories of capital associated with beliefs consistent with B2: the planned trajectory conditions the agents’ time t choices on the expected aggregate capital at time t + s for s 0, whereas the real trajectory conditions their time t decisions on the realized aggregate capital at time t and the expected aggregate capital at time t + s for s > 0. Note that in this real-time approach, beliefs at time 0 over the whole trajectory, to in…nity, determine aggregate K1 ; then the beliefs from time 1, up to in…nity, determine K2 , and so on. We claim that, when beliefs from time 1 up to in…nity are supposed identical to what they were at time 0 (a case permitted under our de…nition), the real-time trajectory is close (in fact at a second-order distance) to the planned trajectory. This comes from the fact that although the planned aggregate capital and the realized aggregate capital, along the real time path, di¤er, this has only a second order e¤ect on the change of consumption of individual agents, as shown in Corollary 1, (which stresses that changes in aggregate K0 do not matter). Hence, to …rst-order approximation, the planned and real aggregate trajectories of capital, when beliefs are maintained as explained, are the same. Thus Theorem 4 follows form Theorem 3. A formal proof requires only notational care, and is omitted. This analysis con…rms the pessimism of the …rst analysis. Trembling beliefs, of the type described here, are subject to real-time falsi…cation, either in the long run or in the short run (leading then to what may be called a crisis?). This brings us to the next viewpoint which must necessarily mix bounded rationality with eductive considerations. Bounded rationality will lead us to introduce learning of the evolutive type. 20 6 6.1 Combining eductive and evolutive learning The framework We have found that the steady state K is not strongly eductively stable according to the various de…nitions given above. At the same time it is known that it is locally asymptotically stable under certain statistical learning rules.. At …rst sight, this suggests a signi…cant contrast between stable adaptive and unstable eductive approaches. A better way to consider the connection is to combine these approaches. We will again endow agents with expectations about the future path of the aggregate capital stock. These expectations are restricted to belong a set, which for convenience we will take to be described by B1. The set B1 can here be viewed as describing a collective belief which provides bounds to individual beliefs. Starting from k0 (!) = K these assumptions generate, in accordance with the analysis of Section 4, a range of possible values for K1 . In line with the eductive approach, agents’ decisions today are based on an assessment of the whole future.18 Now, however, the “expected” trajectory at time t is supposed not only to re‡ect initial beliefs but to respond to observed actual capital, and in the spirit of evolutive approach, we specify a set of adaptive learning rules that determine the way initial expectations change along the real time trajectory of aggregate capital. Then, coming back to the collective belief interpretation of B1, reminiscent of the “eductive” approach, we then ask if the implied paths fKt g1 t=1 will necessarily satisfy B1, i.e. if the collective belief which serves as a frame for the individual beliefs is subject to falsi…cation. If for some nonempty subset of adaptive learning rules, falsi…cation is impossible, then we say that the steady state is weakly eductively stable under evolutive learning, and if this occurs for all adaptive learning rules within the set of adaptive learning rules under consideration, we say it is strongly eductively stable under evolutive learning. Let us …rst make more precise what the evolutive learning process is about, and introduce a simple evolutive learning scheme in the nonstochastic RBC model. We assume, in line with standard adaptive learning studies, that all the agents use the same adaptive learning rule.19 We also assume that what is learned is not the whole future trajectory, but some individually relevant summary statistics of the future. Let 18 In the adaptive learning literature, within in…nite-horizon models, this approach has been followed, for example, in Sargent (1993, pp. 122-125), Preston (2006), Eusepi and Preston (2008) and Evans, Honkapoha and Mitra (2009). An alternative approach in the adaptive learning literature is based on one-step-ahead “Euler equation” learning. See, e.g. Evans and Honkapohja (2001), Ch. 10, and Evans and McGough (2009). 19 However, our strong instability results under adaptive learning extend to the case in which heterogeneous adaptive learning rules are permitted, since small perturbations of each agent’s learning rule leads to a small perturbation of the aggregate path for Kt . 21 us be more precise. 6.2 The real-time system In the real-time system we assume that at each time t each agent ! re-solves their dynamic optimization problem. That is, at each t agent ! chooses ct (!), given kt (!), to solve their consumer program, given their time t expectations about future wages and interest rates, where these are a function of their time t expectations of future 1 e aggregate capital, Kt;t+j (!) j=1 . From Corollary 1, for any speci…ed expectations, P 1 s the optimal path of consumption satis…es dkt (!) = 1 dct+s (!), and from s=0 P s C e Lemma 2 we have dct+s (!) = dct (!) + j=1 drt;t+j (!): Combining these equations e 00 e and using drt;t+j (!) = f dKt;t+j (!) we obtain dct (!) = (1 ) C dkt (!) f 00 1 X s e dKt;t+j (!); (11) j=1 which gives, in deviation from steady-state form, the consumption at t of agent ! as a function of its current wealth kt (!) and the time path of expected future aggregate 1 e capital dKt;t+j (!) j=1 . It can be seen that the agent’s decision ct (!) depends on a e su¢cient statistic for dKt;t+j (!) 1 j=1 ^ e (!) = dK t 1 , given by (1 ) 1 X s e dKt;t+j (!): j=1 ^ te (!) is the time t discounted sum of the expected future aggregate capital Thus dK ^ te (!) is a su¢cient statistic for dct (!). stock. Together with dkt (!) the quantity dK 1 e The proportionality factor (1 ) ensures that if dKt;t+j (!) = e for all j = e 20 ^ (!) = e. Furthermore, it is easily seen that under B1, dK ^ e (!) 1; 2; 3; : : :, then dK t t must lie in the interval [ "; "]. From (11) we have dct (!) = (1 ) 2 dkt (!) 20 (1 C ) ^ te (!); f 00 dK Adaptive learning in nonstochastic models with in…nite horizons often assumes “steady state learning” in which forecasts are the same at all horizons. See, for example, Evans, Honkapohja and e Mitra (2009). In the current context this would mean dKt;t+j = e at t for all j, with the value of e ^ e updated over time. Our formulation in terms of dKt allows for greater generality, while retaining a single su¢cient statistic that is updated over time. In stochastic models, the time pattern of interest rates can be estimated and updated using recursive least squares. For technical reasons this procedure cannot be used in nonstochastic systems. Intuitively, in a nonstochastic equilibrium the asymptotic lack of temporal variation makes impossible consistent estimation of the time-series parameters. See Evans and Honkapohja (2001, pp. 152-154). 22 and from the linearized household accumulation equation (as in the proof of Theorem 3) we have dkt+1 (!) = 1 dkt (!) dct (!): ^ e (!) over time: Finally we specify a simple adaptive scheme for the revisions of dK t ^ e (!) = (1 dK t ^ e (!) + dKt ; )dK t 1 where 0 < 1 parameterizes how expectations adapt to current information about the actual capital stock. We are now in a position to describe the “real-time” evolution of the system. For the sake of simplicity, we start from an initial situation, in which the time zero ^ 0e (!) = dK ^ 0 ; so that the system has homogeneous belief is the same for everybody dK expectations for all t; together with initial actual K0 at or near K (so that dK0 is near 0): The homogeneity assumption allows us to calculate the resulting time-path, but is also illuminating in the sense that we would hope the system to be stable under learning if we start near the steady state and with a small commonly-held expected deviation of expected future capital from the steady state. By combining the above expressions for dct (!) and dkt+1 (!) we obtain dkt+1 (!) = ^ te (!). The system can thus be written as dkt (!) dK dKt+1 = dKt ^e dK t+1 = (1 where ^ te dK ^ e + dKt+1 ; )dK t (12) (13) 2 = Cf 00 (1 ) denotes parameter that determines weak eductive stability in the sense of Section 4. We can now return to the previously suggested de…nitions of eductive stability under evolutive (adaptive) learning and give formal de…nitions. De…nition 5 The equilibrium is weakly eductively stable under adaptive learning if there exists 0 < 1 such that the trajectory fKt g1 t=1 ; triggered by (12)-(13), remains in the cylinder D("); de…ned in B1, for all K0 near K (dK0 near 0) and all e jdK0;j j ", j = 1; 2; 3; : : :. De…nition 6 The equilibrium is strongly eductively stable under adaptive learning if for each 0 < 1 the trajectory triggered by (12)-(13) remains in the cylinder D("), e de…ned in B1, for all K0 near K (dK0 near 0) and all jdK0;j j ", j = 1; 2; 3; : : :. 23 Stability obtains under these de…nitions if, starting near the steady state K0 = K, the real-time evolution of Kt stays for all t in the cylinder D("); provided the initial ^0 ". For convenience, we (homogeneous) expectations satisfy B1, so that dK will henceforth refer to “weakly eductively stable under adaptive learning” as simply “weakly stable under adaptive learning,” and similarly for “strongly eductively stable under adaptive learning.” 6.3 The results The …rst result is again an impossibility result. Theorem 5 The equilibrium cannot be strongly stable under adaptive learning. Proof: The result can be obtained as a consequence of the argument in Section 5.3. The trajectory under adaptive learning is continuous in = 0 (for small > 0; the trajectory is close to the = 0 trajectory under an appropriate metric). Impossibility then follows by continuity as a result of the impossibility of strong eductive stability in the alternative sense. A second proof, obtained by explicit computation of the path under adaptive learning, is given in the Appendix. A striking feature of this result is that the instability arises in the “small gain” limit of small , which in the adaptive and least-squares learning literature is usually viewed as a stabilizing case. In our approach, the problem is that in this case the initial collective belief will be falsi…ed, which we view as a fragility of expectational coordination. Our …nding of strong instability under adaptive learning is even stronger than it may appear. Theorem 5 implies that at some time t0 we must have jdKt0 j > ", but is it nonetheless possible that the discounted sum of future aggregate remains in Pcapital s ^ t j " for all t where dK ^ t = 1 (1 D(") for all t, i.e. that jdK ) 1 dK t+j ? The j=1 answer is again no: from the proof of Theorem 5 it can be seen that for su¢ciently ^ t1 j > ": small there exists a time t1 such that jdKt1 j > " (1 ) 1 and hence jdK Hence the collective belief associated with B1 can be destroyed in two senses: actual capital goes away from the cylinder, and the summary statistics of future capital also goes away from the cylinder. We next turn to weak stability under adaptive learning. We show …rst that it is necessary for our de…nition that the system (12)-(13) be asymptotically stable.21 Lemma 3 The equilibrium is weakly stable under adaptive learning only if the system (12)-(13) is asymptotically stable for some 0 < 1. 21 Recall that througout the paper we rule out nongeneric cases. Thus we rule out eigenvalues of (12)-(13) that lie on the unit circle. 24 Proof: This follows from standard results for discrete-time linear systems. If the equilibrium is weakly stable under adaptive learning then the system (12)-(13) must be stable for all initial conditions. Because the system is linear this implies that its eigenvalues lie inside the unit circle, which in turn implies asymptotic stability. Lemma 4 The evolutive system (12)-(13) is asymptotically stable if and only if 1 <4 2: Proof: The system can be written as dKt+1 e ^ t+1 dK = 1 1 (1 + ) dKt ^ te dK : (14) Let A denote the 2 2 matrix that governs the dynamics. For asymptotic stability we need both eigenvalues within the unit circle. Equivalently we require jdet(A)j < 1 and jtr(A)j < j1 + det(A)j. Since det(A) = 1 the …rst condition is satis…ed for all 0 < < 1. Using tr(A) = 2 (1 + ) leads to the stated condition. Lemma 4 implies asymptotic stability for all 0 < < 1 if < 2 and for some ^ e j " for t su¢ciently 0 < < 1 if 2. Asymptotic stability implies that jdKt j; jdK t large. However this is only a necessary condition for weak stability under adaptive learning. Based on Section 4 we already know that < 1 is a requirement for weak stability under adaptive learning, since this gives the condition for K1 to satisfy B1, for K0 = K and all possible beliefs that satisfy B1, as ! 1; and in fact for < 1 weak stability under adaptive learning holds for < 1 su¢ciently close to one.22 We can summarize the results in the following Theorem. Theorem 6 Under adaptive learning the weak and asymptotic stability properties of (12)-(13) are as follows: 1. For 0 < < 1; the steady state is weakly stable under adaptive learning for all 2 ( ( ); 1]; where ( ) is an increasing function from (0; 1) onto (0; 1): 2. For 1 < ; the steady state is not weakly stable under adaptive learning, although: 3. For 1 < 2; the learning process is asymptotically stable whatever . 22 Based on the proof of Theorem 5 given in the Appendix, it can be shown that < 1 implies weak stability under adaptive learning. For < 1 near one, the eigenvalues 1 ; 2 of A are real and ^ e = e we have dK1 = negative and tend to f0; g as ! 1. For dK0 = 0 and dK e and the path 0 of Kt is given by t 1 dKt = ( 1 e( t1 2) 2) from which it can be seen that jdKt j reaches a maximum at t = 1, which implies the stated result. 25 4. For 2 ; the learning process is asymptotically stable for < 4 2+ . This theorem emphasizes the relevance of what we earlier called weak eductive stability for the understanding of real-time learning. Indeed, the coe¢cient ; stressed in Theorem 1, plays a key role either in the understanding of stability under adaptive learning or in the analysis of the plausibility of the asymptotic stability of the adaptive learning process. Whatever the viewpoint taken, a higher signals higher expectational fragility. If < 1; if the learning rule reacts quickly enough to information on the path, i.e. > ( ); and the more quickly the higher , then the trajectory necessarily remains in the cylinder de…ning the collective belief. If > 1;then the path may eventually go away and falsify the initial belief, whatever the speci…c (asymptotically stable or not) adaptive earning rule used by the agents Finally, we remark that while for given the asymptotic stability condition < 4 1 2 is easier to satisfy when > 0 is small, it is small values of that generate the failure of strong stability under adaptive learning. Small > 0 under adaptive learning leads to a cumulative movement of aggregate capital away from the steady state value, which over …nite time periods, as ! 0, track the possible Kt paths deduced by agents in our eductive setting.23 7 A …nite-horizon model with capital In the introduction we stated that the strong instability results of this paper result from the long planning horizon of agents in the RBC model. This is re‡ected in our proofs, which show that the violation of the CK assumption arises not because dK1 is too large, but because, eventually, dKt is too large. However, one might still ask whether strong instability would arise in a model with capital in which agents are short-lived but the economy is in…nitely-lived. To answer this question, we contrast our eductive stability results with those for an overlapping generations model with capital. This model can also be interpreted as a variant of our model in which agents are myopic (i.e. they only consider next period as the relevant horizon) instead of being far-sighted and envisaging the whole future. The previous analysis stresses that farsighted agents …nd it very di¢cult to coordinate expectations. The next analysis shows that indeed myopia, makes expectational coordination easier. 23 An interesting feature of the instability under adaptive learning, which can be seen in the proof of Theorem 5 given in the Appendix, is that the instability is associated with long cyclical movements in Kt . 26 7.1 The OLG Model We stick here to the standard overlapping generations (OLG) terminology, although we keep in mind the just evoked dimension of myopia versus farsightedness of the problem. Consider a two-period OLG model with capital, along the lines of Diamond (1965). Population is constant and normalized to one, and all markets are competitive. Let ! t be an agent born at time t. He is endowed with one unit of labor, which he supplies inelastically for real wage qt . He then allocates his income between savings s(! t ) = k(! t ) and consumption c1 (! t ). In period t + 1, this agent is now old: he rents his savings for net real return rt+1 , consumes the gross return plus pro…ts and dies. Thus agent ! t solves the following problem max E(! t ) u (c1 (! t ); c2 (! t )) s.t. c1 (! t ) + s(! t ) = qt e c2 (! t ) = (1 + rt+1 (! t ))s(! t ) + (! t ) (15) (16) Notice that when agent ! t makes his savings decision, he does not know the value of rt+1 . Below we assume constant returns to scale production so that = 0. The agent ! t ’s …rst-order condition is given by e uc1 (c1 (! t ); c2 (! t )) = (1 + rt+1 (! t ))uc2 (c1 (! t ); c2 (! t )) : (17) Equations (15)–(17) may be used to compute the savings decision of agent ! t based on current and expected future factor prices: e s(! t ) = s(qt ; rt+1 (! t )): Firms hire workers and rent capital in competitive factor markets, and employ constant returns to scale technology to manufacture goods: Y = f (K; L); thus pro…ts are zero and factors prices are given by the respective marginal products. Capital is inelastically supplied “in the morning” by the old and depreciation is zero: the capital accumulation equation is given accordingly by Z Z e Kt+1 = s(! t )d! t = s(qt ; rt+1 (! t ))d! t : Assuming agents know the relationship between real interest rates and marginal products, and so form expectations of aggregate capital instead of real interest rates, we have Z e Kt+1 = s fL (Kt ; 1); fK (Kt+1 (! t ); 1) d! t ; (18) e where Kt+1 (! t ) is agent ! t ’s forecast of aggregate capital tomorrow. Equation (18) captures the dynamics of the economy: given aggregate capital today and forecasts 27 of aggregate capital tomorrow, the actual value of aggregate capital tomorrow can be determined. It also highlights a key di¤erence between the OLG model and the RBC model: in the OLG model aggregate capital depends only on one period ahead forecasts; in the RBC model, aggregate capital depends on forecasts at all horizons. 7.2 Common Knowledge and Eductive Stability To motivate the de…nition of eductive stability, we consider the following thought experiment at time t = 0: Let K be a steady state of (18): K = s fL (K; 1); fK (K; 1) : Assume that at time t = 0 every old household has capital k0 (!) = K. This determines the wage, and therefore the income, of the young, as given by q. Each young agent ! t forecasts capital stock tomorrow, K e (! t ), and determines his savings decision s(q; fK (K e (! t ); 1)). He then contemplates the savings decisions of other agents. We again make the common knowledge CK1: CK1: It is common knowledge that for some " > 0 su¢ciently small, Kt 2 D(") [K "; K + "] for all t: These CK beliefs are assumed held by all agents at all times, i.e. for all ! t for all t: The de…nitions of weak and strong eductive stability under CK1 are identical to the de…nitions given in Sections 4 and 5. We have the following results: Lemma 5 The steady state K is weakly eductively stable if and only if < 1: (19) Proof: To see this, note that (19) holds if and only if there is for small " > 0, whenever jK e (! t ) Kj " it follows that 2 (0; 1) such that @s=@r fL (K; 1); fK (K; 1) fKK K; 1 s fL (K; 1); fK (K e (! t ); 1) K < ": CK1 implies in particular that K1 2 D("): If (19) holds this implies that s(^ !0) 2 e D( ") for all ! ^ 0 . Because each agent ! 0 knows that K (^ ! 0 ) 2 D("), he concludes that s(^ ! 0 ) 2 D( ") for all ! ^ 0 and hence that K1 2 D( "). Thus it isTcommon knowledge n that K1 2 D( "). Iterating the argument it follows that K1 2 1 n=1 D( ") = fKg: In contrast if (19) fails then for some beliefs compatible with CK1, the aggregate capital stock at t = 1 implied by the agents’ optimal plans fails to satisfy K1 2 D("): Theorem 7 The steady state K is strongly eductively stable if and only if condition (19) holds. 28 Proof: From the Lemma it follows that it is common knowledge that K1 = K if and only if (19) holds. For agents at time t = 1 the situation is identical to the situation at time t = 0. Thus at t = 1 agents will conclude that K2 = K and this implies that it will be the case that K2 = K and this will also be common knowledge for agents at t = 0. By induction it follows that the fact that the equilibrium path will be Kt = K, all t, is common knowledge. Because in general OLG models can have multiple steady states, further analytic results at this level of generality are not available.24 Therefore, to apply our results, we specify particular functional forms and conduct numerical analysis. Assume utility is time separable and takes the constant relative risk-aversion form u(c1 ; c2 ) = 1 1 c11 + c21 2 ; for 0 < < 1; and assume that production is Cobb-Douglas, f (K; L) = K L1 . In this case there is a unique positive steady-state level of capital, and parameter values for and completely characterize the model. For all parameters examined – 2 (0; 1) and 2 (0; 100) – the steady state is strongly eductively stable. This example provides a striking contrast to the coordination problems we have demonstrated for the RBC model with in…nitely-lived agents. 8 Conclusions The di¢culties of expectational coordination can be ascertained from two sides, the “eductive” one and the “evolutive” one. In both cases, farsighted agents are sensitive to the whole path of expectations and long-run expectations signi…cantly matter. Long-run concerns do unsurprisingly in‡uence present decisions and future plans. But the sensitivity to expectations of long-run plans envisaged today is extreme. And this is at the heart of the impossibility of strong “eductive” stability. Indeed, in the eductive framework there is negative short-run feedback, which may be destabilizing, and positive long-run feedback, which will be destabilizing. That is, an expectation that aggregate K will persistently exceed K will lead agents to reduce their capital in the coming period, which, if the e¤ect is strong enough, can be destabilizing. But, in addition, the optimal dynamic plans of agents call for them to eventually accumulate capital in excess of the conjectured level of aggregate K. Hypothetical Common Knowledge of the equilibrium cannot trigger Common Knowledge of the equilibrium, whatever the speci…c characteristics of the economy, an extreme form of expectational 24 We remark that, when there are multiple steady states, eductive stability provides a natural selection criterion. 29 instability which has no counterpart in previously studied models25 . If evolutive learning is incorporated into the model, so that expectations evolve adaptively over time, these two sources of instability remain pivotal. If the adaptation parameter is large then unstable overshooting can arise in the short-run, while if the adaptation rate is small then low-frequency swings over the medium run will necessarily generate instability. 25 We do not claim however that the di¢culty occurs in every model with in…nitely lived agents. For example, it would not occur in the world of Lucas (1978), where the assets returns do not depend upon “extrinsic” uncertainty, but only upon intrinsic uncertainty. 30 Appendix Proof of Lemma 1: These results follow from the envelope theorem. The Lagrangian is given by !! X X t e e L= U (ct ) + t ; Rst qt+s ct+s + (1 + rte ) kt (!) t 0 s 0 where s Y Rst = (1 + rt+i ) 1 ; i=1 Qs e e ~ e (!), and similarly and Rst = i=1 (1 + rt+i ) 1 is the point expectation of Rst given K e for qt+s and rte . The …rst result follows easily from @L = @k0 (!) 0 (1 + r0 ) : since K0 = K it follows that (1 + r0 ) = 1; and since utility of wealth, 0 = U 0 (c0 ). 0 is the time zero marginal To obtain the second result, notice that because production has constant returns @q0 @r0 to scale, it follows that @K + @K K0 = 0: Now simply compute 0 0 @L = @K0 0 @q0 @r0 + K0 : @K0 @K0 The …nal result obtains as follows: T 1 X @L = @KTe (!) t=0 We may compute Also, for t T t t X @ Re q e @KTe (!) s 0 st t+s ct+s ! + T T @ (q e + (1 + rTe )kT (!)) = 0: @KTe (!) T 1, we have @ Re = @KTe (!) st @ qe = @KTe (!) s+t s+1 00 f 0 t+s else Kf 00 t + s = T 0 else 31 T (qTe + (1 + rTe )kT (!)) : Thus X @ e e Rst qt+s e @KT (!) s 0 T t = ct+s Kf 00 X c) f 00 (q s s T t T t = Kf 00 T t + r X s s 0 T t = Kf 00 r 1 1 ! = 0: Proof of Lemma 2: The individual Euler equation (3) implies t= ct (!) = c0 (!) Taking logs gives Qt s=1 (1 log(ct (!)) = log(c0 (!)) + (t= ) log + rse (!))1= + (1= ) t X : log(1 + rse (!)): s=1 Taking small changes, as argued above, we have t X dc0 (!) dct (!) = + C C drse (!) s=1 ! Using the welfare corollary we obtain 1 X 0= C t 0 t dct (!) = 1 1 where we have used the identity The result follows. dc0 (!) +( (1 C P1 t=1 t Pt s=1 ) ) drse (!) = (1 +1 X s ! drse (!) ; s=1 ) 1 P1 s=1 s drse (!). Discussion and statement of Lemma 6 ~ e ) be the deviation of aggregate capital in period one given an arLet dK1 (dK ~ e . Weak eductive stability implies that if dK ~ e satis…es B1 bitrary beliefs pro…le dK e ~ ) 2 ( "; ") I("). To establish conditions su¢cient for weak eductive then dK1 (dK ~ e that satisfy B1 and that stability, it is useful to identify the beliefs pro…le(s) dK ~ e ). Some additional notion is helpful: denote by maximize the magnitude of dK1 (dK e dS~ ( ) an agent speci…c constant beliefs time path: dKte (!) = for all times t. Abusing notation slightly, we will also use dS~e ( ) to indicate a constant and homogeneous beliefs pro…le: dKte (!) = for all agents ! and times t. 32 ~ e be any beliefs pro…le consistent with B1. Then Lemma 6 (A.1) Let dK dK1 dS~e (") dK1 dS~e ( ") : ~e dK1 dK The proof follows fairly straightforwardly from the previous Lemma. Constructive proof of Theorem 5: We consider dK0 = 0 and dK0e (!) = e, with e = ", for all !. This implies dK1 = e. The dynamics of the system are then given by (14), which can equivalently be written as dKt+2 = (2 (1 + ))dKt+1 (1 )dKt with dK0 = 0 and dK1 = e. The eigenvalues of the dynamics are complex if 2 (1 + ) < 4 , and therefore for > 0 su¢ciently small and are given by ; = or 1n 2 2 p p i 4 (1 + ) (1 + )2 o ; = r(cos( ) i sin( )), where 2 r = 1 and p p 1 r sin = 4 (1 + )2 : 2 When the roots are complex the solution meeting the initial conditions is given by dKt = At t = T = 2 e t r sin( t). r sin we have dKT = p p 4 s 4 = sin 1 Taking the limit as 2 e (1 + 2 (1 4(1 )2 (1 ) =(4 ) , where + )2 : ) > 0 tends to zero it can be veri…ed that lim dKT ( !0 ( )) = 1 where the sign is opposite to the sign of e. It follows that for > 0 su¢ciently small we have jdKt j > " for values of t near T ( ( )), and hence the equilibrium is not strongly stable under adaptive learning. 33 References [1] Diamond, P.A. (1965), National debt in a neoclassical growth model, American Economic Review 55, 1126-1150. [2] Eusepi, S. and B. Preston (2008), Expectations, learning and business cycle ‡uctuations, NBER Working Paper 14181. [3] Evans, G.W. and R. 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