Living in an imaginary world that looks real

Journal of Economic Dynamics & Control 41 (2014) 209–223

Contents lists available at ScienceDirect

Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

Living in an imaginary world that looks real Maciej K. Dudek Vistula University and INE PAN, Warsaw, Poland

a r t i c l e i n f o

abstract

Article history: Received 12 January 2014 Accepted 22 January 2014 Available online 10 February 2014

In the paper we show – using standard approaches, general equilibrium modeling and the assumption of complete rationality – that the macroeconomic environment is endogenous and is indeterminate. Specifically, it is argued – without resorting to sunspot type arguments – that microeconomic fundamentals do not suffice to characterize the economy at the macro level. In particular, we show how perceptions of rational agents of the workings of the economy (a) shape the environment, (b) affect the environment sufficiently to ensure that rational economic agents find the observed environment consistent with their beliefs even though it is not. As a by-product, we illustrate that endogenous macro uncertainty can arise as an outcome if rational economic agents whose expectations are anchored on endogenous variables expect them to arise. Finally, we show that systematic errors can persist indefinitely under rationality. & 2014 Elsevier B.V. All rights reserved.

JEL classification: D83 D84 D91 E32 Keywords: Endogenous environment Imaginary parameters Real parameters Beliefs consistency Macro-uncertainty Permanent errors

1. Introduction We construct an economy occupied by fully rational economic agents who base their actions on beliefs that stem from an underlying theory, which includes a complete description of the economy at the micro-level. Individual actions based on private beliefs lead to real outcomes and determine the actual allocation. Rational economic agents observe the ensuing allocation and confront their beliefs with the observed outcomes and find their beliefs consistent with the observed environment despite the fact that their beliefs do not reflect reality. Naturally, we consider the former condition to be in fact a prerequisite for logical macroeconomic modeling as it has long been recognized that macroeconomic systems are selfreferential. However, at the same time it may appear that the assumption of complete rationality implies that the former and the latter condition are mutually exclusive as systematic errors of perception cannot perpetuate indefinitely. In the paper we argue the opposite and show in a general equilibrium framework that rational agents can consider themselves to be correct all the time even though they constantly err. More precisely, we show that economic agents can become convinced that they possess a complete description of the economy despite the fact that their data-verified beliefs do not correspond to the true description of reality. We derive our results in a number of steps. First we make a noncontroversial observation and note that beliefs held by economic agents influence their decisions and consequently shape market equilibria and in turn determine aggregate

E-mail address: [email protected] http://dx.doi.org/10.1016/j.jedc.2014.01.021 0165-1889 & 2014 Elsevier B.V. All rights reserved.

210

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

outcomes. Naturally, we require that the beliefs themselves be in equilibrium, i.e., that the beliefs held by economic agents correspond to the observed equilibrium outcomes. In other words, we do impose on the beliefs the standard notion of consistency and require that they be at a fixed point. In that sense, our contribution fits fully into the standard framework. However, we show that the standard requirement that the beliefs be consistent with observables does not suffice to identify a model. Specifically, we show in a general equilibrium model based on Matsuyama (1999) that beliefs held by economic agents can sufficiently affect the equilibrium dynamics to ensure that the equilibrium dynamics are consistent with the beliefs despite the fact that in reality the dynamics are generated by a different, but still endogenous, process. In other words, we show that it can be the case that agents rationally consider themselves to be correct all the time despite the fact that they happen to be constantly wrong. Formally, we construct a model in which perceptions held by economic agents affect the equilibrium and the observed dynamics are consistent with the underlying perceptions. However, the observed dynamics are generated by a process distinct, but still endogenous, from the one deemed correct by economic agents. Despite the fact that our agents err in equilibrium, we never depart from the assumption of complete rationality. In fact, all our agents are fully rational all the time. Specifically, our rational agents postulate a theory that is to describe the workings of the economy, and, in particular, the theory comprises a complete description of the micro-structure of the economy. Then given the theory, agents behave rationally and, in particular, derive the correct macro-level relationships stemming from the underlying micro-level description. Having derived the macro-level relationships rational economic agents test – using the observables (macro-level data generated by an endogenous process) – the theory and find the theory consistent with the data. In that sense, our agents are both rational and correct since their perceptions of reality are confirmed by the data. However, we show that at the same time our agents are wrong since the micro-structure they postulate to occur is in fact nonexistent. There are numerous contributions that we consider related to our paper. Specifically, in spirit we perceive our paper to be most closely related to the sunspot idea of Cass and Shell (1983) who point out that forward looking equations can admit more than one solution. In our paper, however, the results are derived without appealing to the presence of exogenous coordination devices and without shocks to expectations. Moreover, in the case of our paper agents form expectations with regard to endogenous equilibrium variables with expectations being always rational and given by a time invariant rule. Our approach can be considered to be complementary to that presented in Hommes et al. (2013) who show that it can be the case that expectations based on a simple linear forecasting rule can be in fact consistent with the underlying process even if the underlying process is nonlinear. Naturally, we share main premise of Hommes et al., however, pursue a dual approach as we try to reconcile the concept of Consistent Expectations Equilibrium with perfect rationality of economic agents. Hommes et al., on the other hand, extend the concept of CEE in the opposite direction and study the existence and stability of Stochastic Consistent Expectation Equilibria while adhering to the notions of bounded rationality. The paper shares a major theme with a recent contribution by Eusepi and Preston (2011) who studied a feedback mechanism between private decisions and perceived aggregate equations. However, in our context the link between micro and macro relationships is fully identified and known by economic agents. Moreover, agents in our model are fully rational whereas agents described by Eusepi and Preston must rely on constant gain learning and only have a limited picture of the economy. Finally, in our context the uncertainty is endogenous and constitutes an outcome rather than an assumption. From the conceptual point of view our contribution can be viewed as a constructive response to the challenge posed by Grandmont (1998) who introduced the notion of a self-fulfilling mistake. Specifically, we provide an explicit example of an economy where rational economic agents err, but never learn that they do as the underlying observables make not only the identification of errors impossible, but, in fact, justify also the original misperceptions and, thus, make the mistakes selffulfilling. We consider our paper to be related to the work of Sorger (1998) who in his contribution presents an example of an economy where economic agents make a self-fulfilling mistake. Specifically, Sorger shows that it can be the case that economic agents who believe that the interest rate follows a random process decide to accumulate physical capital at the rate, which is consistent with private beliefs, and at the same time results in the path of the interest that looks as if it were random validating the beliefs. In the paper, we share the basic premise expressed by Sorger; however, our contribution makes an extra step – we constructively bring the concept of CEE proposed by Hommes (1998) to the standard of the REE of Lucas (1972) – as it never departs from the assumption of complete rationality. Agents in our model are always fully rational; they incorporate and understand the micro-structure of the economy. In particular, they are aware of the true relationship that describes the actual values of the interest rate, whereas in Sorger's case agents are boundedly rational and the actual relationship defining the interest rate escapes their attention. Many contributions, e.g., Brock et al. (2006), argue, in particular, that there could be numerous descriptions of macroeconomic data. Specifically, Brock at el. show that a rational external observer could, in principle, mistakenly accept a model as valid even though the reality is described with an unrelated model. In the paper we make a similar point, but we differ substantively as mistakes in our framework are made within the model by an internal actor whose actions shape the reality, which, thus, is endogenous and affected by the mistakes. From the technical point of view we build on and extend the noise traders literature, which has gained some popularity without earning a universal appeal since it incorporates irrational behavior, originated by Grossman and Stiglitz (1976). Specifically, in this paper, we show that we effectively can, without ever departing from the notion of complete rationality, obtain the effects that appear when irrational noise traders are present. This occurs, in our framework, as the imputed behavior of nonexistent irrational agents who are presupposed to exist affects the beliefs of fully rational agents – all

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

211

agents – and in turn their behavior. As a consequence, it affects the ensuing dynamics, which always remains consistent, as demanded by Grandmont (1998), with the beliefs of rational agents. We want to emphasize that our agents always remain rational, but they presuppose – incorrectly – that some anonymous agents need not be. Formally, in our model rationality prevails at all times, but it is not common knowledge that it does. Our approach falls broadly into the category where simple rules can lead to a complex outcome popularized by Wolfram (2002) in other sciences. However, we believe that we manage to strengthen the argument even further as in our case the rule is not imposed on the system, but is determined within the system and is consistent with the system. In summary, we perceive the paper as being complementary to the existing strands of literature. We consider the key insights of our paper to be novel and of interest. Specifically, we constructively offer a dual perspective to the mainstream of the literature on macro dynamics. In particular, we argue that the existing macroeconomic environment is endogenous and constitutes a reflection of human minds. In other words, we reverse the standard causality and show that not so much the minds of rational agents must adjust to the existing exogenously given macro-environment, but rather that the perceptions held by economic agents can affect the environment sufficiently to make the environment endogenous and consistent with the underlying – possibly misguided – perceptions. The paper comprises five sections. In Section 2 we outline the key ingredients of the model. The equilibrium is described in Section 3. The consistency of beliefs is shown in Section 4. We placed conclusions in Section 5. 2. Model In the paper we rely on a variant of the Diamond (1965) OLG model. At any point in time a cohort of continuum of measure one of the agents is born. Specifically, agent i A ½0; 1 born at time t lives for two periods and her preferences are represented with Uðci1t ; ci2t þ 1 Þ ¼ ðci1t  βit Þci2t þ 1 ;

ð1Þ

i βt

where denotes a mean β person specific preference shock independent across agents and time drawn from distribution F β ð Þ. The assumption of independence allows us, in particular, to state that we must always have Z b 1 8 tj βj dj ¼ β: ð2Þ ba a t It is apparent at this stage that we make a specific assumption with regard to the functional form of the utility function. The assumption, technical in nature, is primarily made to facilitate the smoothness of algebraic manipulations and to ensure tractability. Naturally, we want the reader to be aware that our choice of the utility function does in fact make our substantive problem harder as the number of degrees of freedom is restricted. The supply side differs marginally from the one in the standard Diamond model. In particular, we assume that agents in the second period of their lives, in addition to their rental income (if any), receive additional income. Our assumptions imply that the problem of agent i born at time t can be summarized as max E½Uðci1t ; ci2t þ 1 ÞjΩit  ¼ E½ðci1t βit Þci2t þ 1 jΩit  fsit g

ð3Þ

subject to ci1t þ sit ¼ yi1t

ð4Þ

ci2t þ 1 ¼ ð1  δþ r t þ 1 Þsit þyi2t þ 1 ;

ð5Þ

and

i Ωt

denotes the information set of agent i at time t, r t þ 1 denotes the rental price of capital in period t þ1, δ denotes the where i rate of depreciation, and yi1t and yi2t þ 1 denote incomes earned in periods t and tþ1, respectively. Naturally, st denotes the amount saved at time t by agent i. Moreover, we assume that physical capital is the only saving instrument. Consequently, in equilibrium, we have K it þ 1 ¼ sit :

ð6Þ

Note that we do assume that agent i is an expected utility maximizer. Agent i maximizes her expected utility given her i information set Ωt. Naturally, we assume that 8 t; iA ½0; 1jβit A Ωit . The solution is straightforward and can be written as ! E½yi2t þ 1 jΩit  1 i i i y1t  βt  Kt þ 1 ¼ ; ð7Þ 2 E½Rt þ 1 jΩit  where Rt þ 1 ¼ 1 δ þ r t þ 1 denotes the gross real period tþ1 rental price of capital. Observe that to decide on the amount saved, agent i must assess at time t the values of two period tþ1 variables. Specifically, expectations must be formed at time t with regard to the future real value of income yi2t þ 1 and the future value of the gross real

212

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

rental price of capital Rt þ 1 . In our model, based on Matsuyama (1999), in equilibrium we have (see Dudek, 2010 for a detailed exposition of the specifics) 8 tjRt þ 1 ¼ Λ;

ð8Þ

where Λ is a constant, i.e., the real value of the rental price of capital is time invariant. Given that Rt þ 1 is a constant we simply assume that economic agents in our model have sufficient intellectual capacity to notice that this is indeed the case. Formally, we assume that 8 t; ijfRt þ 1 ¼ Λg A Ωit :

ð9Þ

Accordingly, we can write the following: 8 t; ijE½Rt þ 1 jΩit  ¼ E½ΛjΩit  ¼ Λ:

ð10Þ

Moreover, in our model (again see Dudek, 2010 for derivations) we have 8 t; ijyi2t þ 1 ¼ BΛK t þ 1 ;

ð11Þ 1

where B and Λ are constants. Again, we have a dilemma. We must decide whether economic agents have an ability to identify the relationship expressed with Eq. (11). Again, purely subjectively, we decide that this is indeed the case, i.e., that we have 8 t; ijfyi2t þ 1 ¼ BΛK t þ 1 g A Ωit :

ð12Þ

Naturally, now we can write that 8 t; ijE½yi2t þ 1 jΩit  ¼ E½BΛK t þ 1 jΩit  ¼ BΛE½K t þ 1 jΩit :

ð13Þ

Observe that our assumptions expressed in Eqs. (10) and (13) do in fact conform to the standard notion of rationality and simply amount to the informal statement that agents know the model. In particular, agents in our model know how prices and equilibrium quantities are determined. Moreover, they have a complete understanding of the input output flows in the economy. Furthermore, in equilibrium (Dudek, 2010 provides the details), we have the following relationship: 8 t; ijyi1t ¼ AK αt ;

ð14Þ 2

where A is a constant and αA ð0; 1Þ. Now, given Eqs. (10), (13), and (14) we can write the equation that describes the level of investment i at time t, Eq. (7), as h  i 1 α ð15Þ AK t  βit  BE K t þ 1 Ωit : K it þ 1 ¼ 2 It is apparent now that agent i must assess the value of a single future variable to decide rationally on her investment level at time t. Specifically, the agent must form expectations at time t with regard to K t þ 1 . At this stage we encounter a R1 serious modeling issue. Note that in equilibrium we have K t þ 1 ¼ 0 K jt þ 1 dj, i.e., K t þ 1 is in fact determined at time t and is only indexed with a future time index. Should we not then simply assume that K t þ 1 A Ωit ? We subjectively choose not to pursue this path. In other words, we assume that individual investment decisions are private information3. Accordingly, we assume that 8 t; ijK t þ 1 2 = Ωit :

ð16Þ

Observe that our assumption does not preclude economic agents from attempting to assess the value of K t þ 1 rationally given their private information sets. Specifically, we allow our agents to embark on the process of aggregation of Eq. (15) R1 across all agents, which leads to, note that by assumption 0 βjt dj ¼ β, ! Z 1 Z 1 h i 1 AK αt  β  B ð17Þ K jt þ 1 dj ¼ E K t þ 1 jΩjt dj ; Kt þ 1 ¼ 2 0 0 which implicitly defines K t þ 1 , and allows them to solve for the equilibrium value of K t þ 1 . Note that to solve for the fixed point of Eq. (17) a given agent must first assess, given her private information set, the expectations of other agents, then average them out, and then solve an algebraic equation. We assume that agents in our model privately solve problems of this complexity given their private information sets. 1 We want to emphasize that relationships (8) and (11) hold at all times. Furthermore, the two relationships – motivated explicitly in Dudek (2010) following Matsuyama (1999) – describe the within period allocation and are satisfied regardless of the form of expectation formation and even when agents are heterogenous. 2 In fact A ¼ ð1  αÞðα12 θFÞ  α L, BΛ ¼ ð1  γ Þð1  αÞðα12 θFÞ  α Lα , and Λ ¼ γ ð1 αÞðα12 θFÞ  α Lα , where F denotes a fixed cost entailed in the process of production of intermediate goods, L denotes the supply of labor in the economy, α, and γ are parameters of the production functions at different stages of production. Finally, θ is a constant equal to ð1  αÞ1  1=α . 3 For the record, note that it does not suffice to observe the equilibrium values of prices or any other variable to identify the value of K t þ 1 , as in our model economic agents invest the unconsumed part of their purchases.

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

213

Before we proceed further we want to make our essential assumptions transparent. We do not want to imply that our assumptions are unorthodox or special in some sense. Quite the contrary, our driving assumptions are extremely standard and in fact are implicitly taken as given by nearly all modelers. Precisely because of that we want to make them explicit. Lucas (1972) in his contribution assumes – an assumption often criticized – that agents do not observe equilibrium values of aggregate variables and must make their decisions given their rational assessments of those values. Here we make a much weaker assumption. We assume that agents do observe the equilibrium values of aggregate variables. However, at the same time we assume that agents do not observe the private information of other agents. Formally, we make Assumption #1. Information sets of economic agents satisfy, in particular, the following properties: 8 t; ijK it A Ωit

ð18Þ

8 t; ijf…; K t  3 ; K t  2 ; K t  1 ; K t g  Ωit

ð19Þ

= Ωit : 8 t; i; j; ia j : K jt 2

ð20Þ

The first condition simply states that economic agents know the amount they privately invest. The second restriction confirms that agents observe the equilibrium values of aggregate variables.4 Note that the second condition is totally at odds with Lucas's restrictive assumption. Finally, the third condition simply states that economic agents do not know the private decisions of other agents. In summary, Assumption #1 implies that economic agents possess private information and that the equilibrium values of aggregate variables are publicly observable. Agents in our model must form assessments of the future value of the stock of physical capital. We want the assessments to be and ensure that the assessments are in equilibrium rational. However, we want to emphasize that in our model the assessments are privately rational, i.e., they are rational for a given agent given her information set. Specifically, we make the following assumption. Assumption #2. Agents posit theories with regard to the operational structure of the economy and behave rationally given their theories. Theories are confronted with publicly observable data (the equilibrium values of aggregate variables) and are accepted as valid if publicly observed data corroborates the theories. Our second assumption simply states that a given conjecture about the workings of the economy is accepted when the observed aggregate data does not falsify the conjecture. In technical terms, agent i accepts a given theory, Σ, as valid when it i cannot be falsified given her information set, Ωt. Note that theories in our model are privately tested, i.e., they are tested individually by all agents. Consequently, theories can be accepted or rejected only at an individual level. Moreover, our assumptions, thus far, implicitly allow for the possibility that a given theory, Σ, gains uniform private acceptance from all agents, but at the same time the same theory, Σ, would be rejected were the agents allowed to pool their private information. In the remaining sections of the paper we show that indeed it can be the case that more than one theory can be privately accepted as valid. Moreover, we show that it can be the case that a given theory, comprising a complete description of the economy at the micro-level, is privately accepted as valid by all agents despite the fact that it does not reflect the truth (the true workings of the economy).

3. Equilibrium In this section we solve for the equilibrium, given the mind sets of economic agents, adhering to the standard methodological approach. Specifically, at all stages we allow economic agents to derive the relevant equilibrium conditions in a rational and internally coherent manner. 3.1. Beliefs structure The equilibrium value of K t þ 1 is always defined by the following equation, Eq. (17) from the previous section. We should emphasize that Eq. (17) is typical of macro models as normally the optimal amount saved depends on the relative strength of present income, which is a function of the present value of the state variable – AK αt in our case – and the future real income, which is a function of the future state variable – K t þ 1 in our case. In other words, one can expect to obtain an equation of the form of (17) in most dynamic macro models. Despite the fact that Eq. (17) is standard finding its solution need not be simple, as originally noted by Townsend (1983), as it is normally not permissible – different private information sets – to exchange the order of integration. To facilitate a solution for the equilibrium we make the following assumption explicit. 4

t All equilibrium aggregate variables are functions of the state variable, fK τ gττ ¼ ¼  1.

214

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

Assumption #3. All agents at all times are rational. The preceding assumption makes it explicit that we never depart from the notion of rationality. In fact all agents i A ½0; 1 are always rational and able to solve Eq. (17). Nevertheless, before we find the actual solution we would like to start with a formal description of the belief structure – the mind set – of rational agent i A ½0; 1. First of all, we assume that agent i is rational, however, agent i's beliefs do not correspond to the objective truth. Specifically, agent i believes that some agents operating in the economy are in fact naive. In particular, agent i presupposes that 1. agents j A ½0; x [ fig are rational as well and share the view of the world of agent i, 2. agents j A ½x; 1\fig are naive and are not able to solve (17) in a rational manner and form assessments of K t þ 1 using a simplified rule, 3. naive agents, j A ½x; 1\fig, are affected by a common preference shock. More precisely, we assume, the second condition above, that rational agent i believes that naive agents being unable to solve for the equilibrium themselves resort to simple econometric techniques to assess the value of K t þ 1 . Specifically, naive agents are assumed to rely on a simple OLS procedure to estimate, using available data, the coefficients of the following relationship: ^ τ  K Þ þ ητ þ 1 ; K τ þ 1 ¼ K þ ρðK

ð21Þ

where K and ρ^ are de facto the sample mean and the first order autocorrelation, respectively. Furthermore, in our model rational agents believe that naive agents use a well specified model. In particular, rational economic agents believe that naive economic agents find the errors, fητ þ 1 g, in specification (21), to be uncorrelated and mean zero. Naive economic agents are assumed to use specification (21) to assess the value of K t þ 1 . Given that we demand that the error terms in specification (21) be independent and mean zero we can state that the rational assessment of K t þ 1 of naive agents can be expressed as ^ t K Þ þE½ηt þ 1 jΩjt  ¼ K þ ρðK ^ t  K Þ: 8 t; j A ½x; 1\figjE½K t þ 1 jΩjt  ¼ K þ ρðK

ð22Þ

Note that naive agents exist only in the space of beliefs of rational agents. Consequently, formally there is no need to explain the behavior of naive agents. If anything, rational agents must simply take the forecasting technique, irrespective of how absurd it can be, and the ensuing behavior of naive agents as given. In other words, formally parameters ρ^ and K are totally free and, thus, can assume any values. This generates at least two additional degrees of freedom, which by itself makes our problem simpler. Nevertheless, we choose to impose a very serious restriction – the two parameters are the OLS estimates of Eq. (22) with actual data, on the two parameters, ρ^ and K and by doing so, we not only give up two degrees of freedom, but, in addition, we also impose yet another strong consistency requirement. Therefore, we in fact embark on a problem significantly more profound than it is necessary to prove our basic point. Observe that by imposing this additional restriction on the behavior of naive agents, who are just assumed to exist, we make the narrative of our paper not only more palatable to the reader, but also consistent with the paradigm involving a composition of fully rational agents and naive agents – chartists relying on econometric techniques – present in the literature. Naturally, the critical distinction remains; in our case naive agents are just presupposed to exist and in fact they do not, whereas in the literature they are truly present. Recall that by assumption the preference shocks, fβjt g, affecting agents in the model are always independent across agents and across time. However, as stated in condition three above, rational agents in our model believe that this is not the case. In fact, rational agents believe that naive agents are hit5 by a common, mean ε, unobservable,6 and uncorrelated across time, preference shock drawn from distribution7 F ε ðÞ. Formally, we state that 8 t; j A ½x; 1\figjβjt ¼ εt þ 1 ; which, in particular, implies that Z 1 1 βj dj ¼ εt þ 1 : 1x x t

ð23Þ

ð24Þ

Observe that the belief structure of agent i departs from the true description of the economy. In particular, agent i makes several errors. First the agent incorrectly attributes naive behavior to a fraction of agents. Secondly, agent i presupposes that a specific forecasting rule is used by naive agents even though it is not the case. Finally, rational agent i believes that naive agents are affected by a common preference shock, which obviously is not true as assumption (24) contradicts the objective truth captured in assumption (2). Naturally, the belief structure (the mind set) of agent i does not reflect the correct description of the actual environment. 5

We can interpret the disturbance as a sentiment shock affecting naive agents as proposed by Angeletos and La'O (2013). The realization of the shock at time t, εt þ 1 , is indexed with tþ 1 to emphasize that the shock is not observable at time t. Note that there is no need to assume that F ε ðÞ is in anyway related to F β ðÞ as the former distribution exists only in the mind of agent i and the true individual preference shocks are not observable. 6 7

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

215

Despite the fact that agent i at the outset makes several mistakes it is the case that agent i, given her mind set, is fully rational. In other words, agent i, given her belief structure, behaves as a fully rational agent would. Specifically, agent i is assumed to use Bayes's rule, to calculate expectations correctly, and to perform all mathematical operations in accordance with standard algebra. Moreover, we assume that the mind set of agent i is common to all agents. Furthermore, all agents are fully rational, i.e., Assumption #3 holds at all times. Some anonymous agents, however, are only presupposed to be naive even though in fact they are not. Formally speaking our assumptions amount to a simple statement that economic agents in our model are fully rational, but they start their economic activity with mis-specified priors. Naturally, given the assumptions, one is nearly forced to ask whether economic agents are bound to discover the true description of the economy or perhaps more interestingly whether it could be the case that observed data and the ensuing learning process will in fact reinforce the a priori mis-specified priors. In the next section we show that the latter can indeed occur. In other words, we show that over time economic agents become convinced that they have learned and that they understand the economy despite the fact that they have not, i.e., economic agents in our model rationally believe in something that is not true. We derive our basic result by explicitly constructing an equilibrium in which economic agents' beliefs conform to the observed data despite the fact that the beliefs do not reflect reality. 3.2. The evolution of the state variable While solving Eq. (17) for K t þ 1 , given that she is rational and given her mind set, agent i must take into account the presence of naive agents who are presupposed to exist in proportion 1  x. Moreover, agent i believes that the remaining fraction of agents are rational and share her description of the micro-structure of the economy. These assumptions allow agent i to solve Eq. (17) in a rational manner. Rational agents do not observe the behavior of naive agents, but have the necessary mental capacity to redo the optimization problem of naive agents and to estimate their forecasting rule. Knowing the forecasting rule of the naive agents, Eq. (22), agent i can find, using (15), the amount invested by a single naive agent, j, which is given by     1 α K N;j AK t  βjt B ρ^ K t K þ K : ð25Þ tþ1 ¼ 2 The above equation allows agent i to establish the total amount invested by all naive agents given by Z Z 1     1 1 j ð1  xÞ  α AK t B ρ^ K t K þK  KN K N;j β dj; tþ1 ¼ t þ 1 dj ¼ 2 2 x t x

ð26Þ

which, noting assumption (24), becomes KN tþ1 ¼

    1 x ð1 xÞ  α AK t  B ρ^ K t  K þ K  εt þ 1 : 2 2

ð27Þ

Eq. (27) defines the amount that is believed by rational agents to be invested at time t by naive agents. Naturally, the total amount invested at time t is given by K t þ 1 ¼ K Rtþ 1 þ K N t þ 1;

ð28Þ

K Rtþ 1

denotes the amount saved by rational agents. where Eqs. (28) and (27), and the fact that by assumption E½εt þ 1 jΩit  ¼ ε permit agent i to assess rationally the value of K t þ 1 . In particular, several steps of algebra – detailed in Appendix A – allow us to establish that the amount invested at time t by rational agents according to rational agent i is given by ( )   x Bð1  xÞε  2β 2 Bð1  xÞ B2 ð1  xÞ   α R þ AK t þ ρ^ K t  K þK : Kt þ 1 ¼ ð29Þ 2 2 þ Bx 2 þ Bx 2 þ Bx Consequently, the total amount invested by all agents, rational and naive, at time t can be expressed, note Eqs. (27) and (29), as K t þ 1 ¼ K Rtþ 1 þ K N t þ 1 , i.e., Kt þ 1 ¼

  1x x Bð1  xÞε  2β 1 Bð1 xÞ   εt þ 1 : þ AK αt  ρ^ K t  K þ K  2 2 þ Bx 2 þBx 2 þ Bx 2

ð30Þ

Eq. (30) describes the amount believed by rational agent i to be invested by all agents, both rational and naive, at time t. In other words, agent i, being fully rational, believes that the true law of motion in the economy is represented by Eq. (30). Furthermore, all other agents are rational as well – Assumption #3 – and share the same view of the world as agent i does. Thus, we can conclude that all agents believe that the true evolution of the state variable in the economy is described with Eq. (30). We know, however, and agents in the model do not, that Eq. (30) does not represent the actual law of motion. In fact, it only describes the perceived, by all agents, law of motion. Recall, that agents in our model, despite being fully rational, err at the outset as they incorrectly attribute naive behavior to some agents. Specifically, they believe that only a fraction, x, of agents are rational and the remainder, 1  x, are naive, whereas the objective truth is that all agents are rational. Therefore, actual investment is done by rational agents and there is a continuum of measure for one of them. Consequently, the actual

216

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

5

4

K

3

2

1

0 0.4

0.45

0.5

0.55

0.6

0.65

x Fig. 1. The dependence of K t þ 1 as a function of x.

law of motion is given by ( )   1 Bð1 xÞε 2β 2  Bð1  xÞ B2 ð1 xÞ   α þ AK t þ ρ^ K t  K þ K Kt þ 1 ¼ 2 2 þ Bx 2 þ Bx 2 þ Bx

ð31Þ

and Eq. (30) describes only the perceived law of motion. Observe that, Eq. (31), the actual law of motion is very simple. In fact, the state variable follows a purely deterministic process. Nevertheless, the actual evolution, as is commonly known, of the state variable can be quite complex and chaotic, in particular, despite the simplicity of the underlying mathematical description. Fig. 1 illustrates how the asymptotic behavior of the state variable, K t þ 1 , changes with x, which itself is a parameter reflecting human beliefs. Furthermore, as we argue later, human beliefs can be such that the value of x can be chosen accordingly, so that chaotic dynamics actually appears along the equilibrium path. More importantly, observe that several types of the underlying parameters determine the form of the dynamic equation (31). Specifically, parameters: A, B, β and α are real and reflect the values of the fundamentals (preferences, technology, ^ and K appear to be parameters as well. However, given resources, market structure, etc.) in the economy. Furthermore, ε, ρ, our assumption that ρ^ and K are in fact the OLS estimates of Eq. (21) on the actual data, we must conclude that ρ^ and K are in fact endogenous and their values are determined in equilibrium. Similarly, the value of ε is determined by the mean of actual perception errors observed ex post. Furthermore, there exists parameter x, which is imaginary8 and it does not correspond to any tangible economic variable, but simply reflect the beliefs of economic agents. Consequently, we must acknowledge that the actual dynamics as described by Eq. (31) is shaped by the physical description of economy (real parameters β, A, B, and α), the values of three endogenous variables ε, ρ^ and K , and human perceptions (imaginary parameter x). Furthermore, it is the case that the values of the real parameters and endogenous variables must be taken as given, however, the value of the imaginary parameter can be arbitrary as a priori economic agents' beliefs are unconstrained; economic agents can believe what they want. Similarly, we can note that the perceived law of motion, Eq. (30), is also affected by real parameters β, A, B, and α, the values of equilibrium variables ε, ρ, ^ and K , the value of the imaginary parameter, x, and, in addition, by a realization of a random variable, εt þ 1, from an imaginary probability distribution F ε ðÞ. Naturally, the perceived law of motion is slightly more complicated than the actual law of motion as the former involves stochastic disturbances and, yet, in the paper, we show that economic agents rationally consider Eq. (30) to be the correct description of reality despite the fact that it is not as reality which is described with Eq. (31). In other words the perceived law of motion and the actual law of motion are different in form and nature. It is shown in Fig. 2(a) and (b) how a given value of the state variable determines the future value of state variable according to the actual law of motion and the perceived9 law of motion. Clearly, to the first approximation rational agents in our model believe that the economy converges to the steady state and that the actual fluctuations are driven by the preference shocks of naive agents, εt þ 1 . However, the objective truth is different. The actual process is deterministic, but characterized by an unstable steady state with the actual dynamics being 8 9

Of course, the value of x is a real number, we use the term to illustrate that the value is just a manifestation of the beliefs. The perceived law of motion is plotted under the assumption that the realized value of the preference shock is equal to its mean, i.e., εt þ 1 ¼ ε.

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

5

4

4

3

3

K t+1

K t+1

5

217

2

2

1

1

0

0

1

2

3 Kt

4

5

0

0

1

2

3

4

5

Kt

Fig. 2. The evolution of the state variable according to (a) the actual law of motion and (b) the perceived law of motion.

chaotic in nature allowing, thus, rational economic agents to interpret the observables as being realizations of stochastic disturbances. Can it then be the case that the actual dynamics generated by process (31) be in fact indistinguishable from the dynamics described by the perceived law of motion? Before we answer the above question affirmatively we make several additional observations that highlight the difficulty of our problem. Our assumptions, thus far, imply that parameters ρ^ and K are not arbitrary. In fact, by specifying the form – econometric type – of forecasting behavior on the part of the naive agents we have lost two degrees of freedom. Consequently, ρ^ and K must be considered endogenous. Observe that our assumptions imply that ρ^ and K are obtained by analyzing the actual time series data with a simple econometric technique, in fact K is simply the sample mean and ρ^ is the first order autocorrelation coefficient. However, at the same time, the actual law of motion that generates the actual time series data given by (31) does depend on ρ^ and K . Clearly, our assumptions imply that now ρ^ and K are in fact solutions to an intricate fixed point problem and as such assume proper equilibrium values and are endogenous. Recall that we chose not only to make ρ^ and K endogenous, but we also imposed restrictions on the errors, fηt þ 1 g, in Eq. (21), which make our problem even more demanding. Finally, observe that the very ambitious consistency requirements described above that we chose to adhere to are in fact applicable in the subspace of beliefs of agents with regard to the behavior of naive agents who in fact are only presupposed to exist, i.e., they are applicable in the imaginary world. We demand that the attributed behavior of naive agents who do not exist be consistent with the underlying data affected through the beliefs of rational agents by the attributed behavior of naive agents. Naturally, for the entire equilibrium to materialize we must require that the beliefs of rational agents, i.e., all agents, be consistent with regard to the presumed behavior of naive agents, but foremost that they are consistent with the observed macro-level dynamics. Specifically, we must require that the actual dynamics generated with (31) be consistent with the dynamics generated by the perceived law of motion (30). Naturally, we show that this indeed happens to be the case as well. 4. Internal consistency of beliefs Recall that we have already assumed that all agents are fully rational and that a given rational agent i A ½0; 1 believes that only a fraction, x, of agents are rational and the remaining agents are naive. In addition, let us assume that economic agents believe that the actual value of x, which is an imaginary parameter, is equal to 1/2. In addition, let the distribution of perception errors, F ε ðÞ, be such that ε ¼  22:1709467753775. Furthermore, we assume that a given rational agent believes that naive agents observe the state variable and use simple econometric techniques to estimate Eq. (21). Specifically, let us now assume that a given rational agent believes that the actual econometric estimates obtained by naive agents are given by ρ^ ¼ 0:307722381843423

ð32Þ

K ¼ 2:36419926251736:

ð33Þ

and

Furthermore, let us assume that rational agents maintain that the corresponding errors, fητ þ 1 g, are uncorrelated across time. Now, given the above specifications and the assumed state of mind of agents we can determine the actual law of motion, Eq. (31) with the values of ρ^ and K given by Eqs. (32) and (33), and x ¼ 1/2. Furthermore, recall that β, A, B, and α are real and reflect the fundamentals, and their values are fixed and given by β ¼  22:1710014375072, A ¼ 54:0115397471311, B ¼ 35:9658446514713, and α ¼ 1/3. It is commonly known that equations in the form of Eq. (31) can generate data that exhibits very complicated dynamics. Specifically, the values of the state variable described with Eq. (31) can display chaotic behavior, which turns out to be the case here, Fig. 3.

218

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

5

4

4

3

3 K t+1

K t+1

5

2

2

1

1

0

0

1

2

3

4

0

5

0

2000

4000

time

6000

8000

10000

time

Fig. 3. The evolution of Kt over time. (a) The dependence of K t þ 1 on K t and (b) time series data.

3.5

1

3

0.5

2.5 0 2 -0.5

1.5

1

2

2.2

2.4

2.6

2.8

3

-1

0

0.1

0.2

0.3

0.4

0.5

K

^ Fig. 4. The implied values of ρ^^ and K^ when all other values are held constant. (a) K^ held constant and (b) ρ^^ held constant.

We have assumed that economic agents believe that naive agents, who are just presupposed to exist, resort to a simple econometric analysis to estimate the coefficients in their forecasting rule, Eq. (21). Using actual data generated by the actual law of motion (31), economic agents can estimate the coefficients in the forecasting rule of naive economic agents. Specifically, the estimates based on T ¼1 000 000 observations are given by ρ^^ ¼ 0:305930405594571;

ð34Þ

K^ ¼ 2:36053259028082:

ð35Þ

and

Observe that these estimates are endogenous and do depend on the initial specifications in (32) and (33), as the initial specifications do affect the actual law of motion. Therefore, it must be emphasized that the actual estimates (34) and (35) reveal the presence of a fixed point between the actual data and the assumed forecasting rule, which is obtained with the actual data and affects the actual law of motion. Formally we do not have an analytical proof that the fixed point actually exists. Nevertheless, our simulations, see (4), indicate that indeed we deal with a fixed point in the context of our model.10 Furthermore, the autocorrelations of the implied errors, η^ t þ 1 , reflect the assumed value of zero, Fig. 5. Clearly, the supposition that naive agents exist and use a specific forecasting rule obtained with a simple econometric technique applied to the actual data withstands fully the internal consistency test as the actual law of motion that is shaped by the forecasting rule of naive agents generates actual data that validates the original forecasting rule. Nevertheless, to show that rational agents – all agents – are truly in equilibrium, we must not only prove that the beliefs with regard to the presumed actions of naive agents are corroborated by actual data, but more importantly that the observables, actual time series data, are consistent with the perceived law of motion. Recall that the actual law of motion is given by Eq. (31) and the perceived law of motion is given by Eq. (30). Obviously, we must concede that it is impossible to reconcile Eqs. (31) and (30), as the former is purely deterministic and the latter 10 Fig. 4 indicates merely the existence of the first order consistency. Higher order consistency is not typically implied by the first order consistency and needs to be verified separately – Figs. 5 and 6.

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

219

0.04

0.02

0

-0.02

-0.04

0

2

4

6

8

10

12

14

16

lag

Fig. 5. The autocorrelations of η^ t þ 1 .

contains a stochastic disturbance. However, economic agents in the model believe rationally that Eq. (30) holds and, in fact, they are not even aware that Eq. (31) exists. Moreover, agents are assumed to observe only the aggregate values of the state variable, which can look as if they were random. Therefore, it is legitimate to posit that our rational agents could in principle rationally accept the perceived law of motion as the correct description of reality in line with Grandmont (1998) who proposed a basis for a consistency criterion: The ultimate test that this approach will have to pass, however, is that such “learning equilibria” must, to be acceptable, exhibit a reasonable degree of consistency with the agents' beliefs. In this respect, one might envisage situations in which agents think that they are living in a world that is relatively simple, although subject to random (e.g., white noise) shocks, but in which deterministic “learning equilibria” are complex (“chaotic”) enough to make the agents; forecasting mistakes still “selffullfiling” in a well defined sense. Naturally, the phrase “well defined sense” leaves a fair amount of discretion. In this paper, we adopt a metric that builds on Hommes and Sorger (1998) and propose the following operational definition Definition #1. Economic agents accept a given theoretical description of reality as valid if (i) the empirically observed histogram of realizations of a given variable corresponds to the prior distribution, (ii) time series autocorrelations of the realized data reflect those implied by the theory.

Definition #1, despite its simplicity, captures the intuitive notion of empirical testing. Moreover, as noted by Hommes (1998), two distinct processes that satisfy the restrictions imposed on autocorrelations and the histogram can, in fact, be indistinguishable from the statistical perspective. Similarly, Radunskaya (1994) shows that there are deterministic processes that can be indistinguishable from stochastic processes. Therefore, despite the fact that the underlying time series data in our model is deterministic in nature and the perceived law of motion is affected by a stochastic disturbance, Definition #1 can be used as a basis for empirical testing. Recall that we have already noted that economic agents believe – postulate a prior – that the disturbances, εt þ 1 , in Eq. (30) are realizations of a random variable distributed according to F ε ðÞ with mean ε ¼ 22:1709467753775. Moreover, it has been assumed that the disturbance terms, εt þ 1 , are believed to be uncorrelated across time. Naturally, we now have to show that these beliefs are reinforced by empirically observed data. Obviously, the actual description of reality is given by a deterministic equation (31). Therefore, at any point in time the realized values, fK^ τ g, of the state variable must satisfy Eq. (31). However, the agents rationally perceive a different relationship, Eq. (30), as the true description of reality. Therefore, to reconcile their theory with the observables, they fit the observed time series data into the perceived law of motion, (30), which, in particular, implies the following values of the disturbances:

 i α 2 x Bð1 xÞε 2β 1 Bð1 xÞ h  ^ ε^ t þ 1 ¼ þ AK^ t  ρ^ K t K þK  K^ t þ 1 : ð36Þ 1x 2 2 þ Bx 2 þ Bx 2 þ Bx

220

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

0.04

0.02

0

-0.02

-0.04

0

2

4

6

8

10

12

14

16

lag

Fig. 6. The autocorrelations of the realized errors, ε^ t þ 1 .

Now, Eq. (36) can be used to retrieve the values of the disturbances implied by the theory, which allows economic agents to construct the relevant histogram, and in turn calculate the implied mean ε^ ¼ 22:1595234650686, and, finally, the relevant empirical time autocorrelations, given in Fig. 6. Clearly, Fig. 6 confirms that economic agents find the observed values to be consistent with their theories. The distribution of disturbances obtained from the perceived law of motion implies the value of the mean, ε, that corresponds to the assumed one. Similarly, the observed autocorrelations of the imputed disturbances, ε^ t þ 1 , line up with the assumed values. Furthermore, as shown earlier, the actual law of motion, which is endogenous and depends on the forecasting rule of nonexistent naive agents, generates observables that validate the forecasting rule. In other words, observables fully support the underlying theory, which affected the behavior of economic agents and led to equilibrium equations that produced the observables. Naturally, rational economic agents can accept, in light of Definition #1, their theory about the workings of the economy as valid despite the fact that it is not consistent with the true description of reality. It is the case, however, that the actual law of motion could be easily discovered with a simple delay plot, ðK t ; K t þ 1 Þ, as the underlying process is purely deterministic. The possibility of such a simple identification is due to the simplicity of the model and would not be as trivial in higher dimensional models or models affected by noise,11 as argued by Hommes and Sorger (1998). More importantly, as noted by Sorger (1998), the possibility of such a simple identification exists only because the original mistake – consistent with the assumed theory – is made. However, once the mistake is made rational economic agents find the ensuing dynamics to be in fact consistent with the assumed theory. Therefore, formally, they do not have any incentive to experiment with other, possibly simpler, theories. In summary, we claim that we have constructed an equilibrium in which perceptions of rational economic agents shape the environment sufficiently to ensure that the resulting equilibrium dynamics correspond to the one that would be generated in an economy described with the perceptions of economic agents even though the actual dynamics are generated with a different process. Consequently, we have shown that economic agents can be rationally wrong all the time, as their beliefs find constant support in time series data. Moreover, we want to emphasize that we do not just argue that economic agents in our model make the same mistake as an outside observer who fails to specify the model correctly. In fact, our results are much more profound. We construct an equilibrium in the space of beliefs and show that agents' beliefs can affect individual actions, and in turn the macro-environment, sufficiently to ensure that the actual dynamics corresponds to the underlying beliefs even though fundamentally, the actual dynamics are generated by a process distinct, but determined within the system, from the perceived one. Furthermore, in our case, economic agents who are always rational do make mistakes within the system, and the errors are propagated through the system and in turn generate dynamics that are consistent with the dynamics that would have occurred if the errors had been real shocks rather than mistakes in perception. Furthermore, we would like to note that the described and equilibrium consistent behavior of economic agents generates nontrivial economic implications. First of all, we argue that systems that are normally stable can be destabilized by equilibrium consistent beliefs of economic agents. This naturally implies that endogenous fluctuations can occur in an otherwise stable system and can be driven by equilibrium consistent beliefs of economic agents. Secondly, we would like to emphasize that the errors made by agents in our model are not trivial. Agents in our model rationally err. Consequently, they 11 In our case it suffices to add a common preference shock to the model to remove the possibility of a simple identification. We choose not to pursue this option for expositional reasons.

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

221

do not rely on the best available strategies, they only use the best available strategy given their equilibrium verified beliefs. As a result economic agents suffer unnecessary utility losses. Therefore, each agent would change her behavior to increase her utility if the objective truth was revealed to her. In addition, we want to emphasize that our approach does not suffer from the shortcomings identified by Grandmont (1998) and Hommes (1998). Specifically, it is typically the case that for a dynamical system to generate interesting dynamics, the underlying parameters must assume proper values, which raises concerns as the space of admissible values can be small. In our approach, we do not encounter this problem as the actual law of motion is described with an imaginary parameter, x, that can assume any value. Therefore, if the values of the underlying real parameters are not favorable, it is always possible to adjust the value of the imaginary parameter to ensure that the resulting dynamics have the desired properties. Specifically, our example was constructed with B ¼35.9658446514713 and x ¼ 1/2, where B is real and x is imaginary. Obviously, the value of B can appear special, but it is possible to easily recover the same values of the underlying coefficients in Eq. (31) with a proper adjustment of x, should the value of B be slightly different from the one assumed. Consequently, we can state that our results, if not generic, hold for a non-degenerate set of values of the underlying parameters. Our results can be given yet another economic interpretation. Typically it is assumed in macro models that the environment in which agents operate is fixed. Moreover, it is routinely the case that agents are assumed to know the environment. In the previous section, we showed that the standard set of fundamentals, preferences, technologies, and resources combined with assumed universal rationality, do not suffice, even absent the presence of exogenous coordination devices, to define the environment uniquely. In other words, there is no such entity as the environment in which decision making agents are submerged. Quite the contrary, the environment itself is an outcome of the decision making process. Alternatively, we can simply state that the environment in which economic agents do operate is in fact endogenous and, in particular, can be, and in our context is, shaped by rational and data supported beliefs. Furthermore, our model, literally interpreted, in fact implies that aggregate data and aggregate relationships, if any, are simply manifestations of agents' – data verified – beliefs. The results reported in this paper can in fact be interpreted even in a more drastic sense and can be perceived as an extension of observations made by Lucas (1976). In his paper, Lucas argued that aggregate relationships are sensitive to changes in the environment. In this paper, we effectively make a dual observation. We argue that agents' perceptions about the validity of specific aggregate relationships can shape the environment, which is endogenous, in such a manner so as to validate the initial perceptions of the existence of specific aggregate relationships. In particular, Lucas showed that aggregate relationships, e.g., π t ¼ απ t  1 þ βut ;

ð37Þ

are likely to be unstable as coefficients α and β are not invariant to the changes in the environment. In other words, Lucas argues that changes in the environment do affect the forms of aggregate relationships. In this paper, we make a related claim. Specifically, it turns out that in our model, if agents believe that an aggregate relationship of the form of (37) holds, then it may be the case that the environment – being endogenous in our model – will adjust itself so that indeed the postulated relationship, (37), will appear in the data. We can attempt to frame the key result in much stronger wording. Traditionally, it has been believed that the environment constitutes the key building block and any changes in the environment are eventually projected on human perceptions and eventually affect behavior. Here, we claim that the environment does not constitute a fundamental. Quite the contrary, we show that the existing environment is simply a reflection of the human mind. In other words, it need not be so much the case that economic agents rationally adapt to exogenous developments in the environment, but in fact the case that economic agents' beliefs shape the environment, making the environment endogenous and consistent with the initial beliefs. 5. Conclusions Normally, we consider economic agents to be submerged in a given, possibly stochastic, environment. Moreover, most of the time we simply assume that the environment is known to economic agents and economic agents behave rationally, taking the existing environment as given. Only occasionally we require economic agents to take the extra step and learn the – still given – existing environment. In this paper, we take on a related issue and offer a dual perspective to the existing theories. In fact, we take learning for granted. In our paper, economic agents know the model. However, we argue that the model itself, possibly to be learned, is not just given, but is endogenous and by itself does not constitute a constraint, but rather an outcome. In our dual framework, the model is shaped by the beliefs held by economic agents, and in turn macroeconomic outcomes constitute a reflection of agents' minds. Formally, we constructively reverse the standard causality. The environment is normally considered given, and agents, possibly through learning, adjust their beliefs so that the beliefs become consistent with the environment. In this paper, we treat the environment as endogenous and show that it is shaped by the beliefs, and in fact it can adjust sufficiently to match the initial – possibly misguided – beliefs. Our framework allows for a possibility that errors of perception persist indefinitely under rationality. In particular, we show that rational economic agents, despite a possession of a data-supported description of the economy, permanently err. Specifically, we argue that beliefs play a dual role. First of all, we illustrate that beliefs – comprising complete descriptions of the economy at the micro-level – of rational economic agents induce actual actions, which in turn determine the actual

222

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

allocation and form reality. At the same time, beliefs held by economic agents allow them to derive relevant relationships between economic variables and form descriptions of the economy, i.e., to build a perceived picture of the economy. In our framework reality is observable and is consistent with the perceived picture of the economy. Therefore, rational economic agents do not have an incentive to revise their beliefs and can consider themselves to be correct. However, we show that at the same time rational economic agents in our model are permanently wrong as the perceived picture does not correspond to the objective truth. In other words, it is the case that a perception of the economy, which fits into the observed data, is only a reflection of agents' beliefs: a reflection that depends on the beliefs and is consistent with the beliefs. In that sense the perceived world is a pure act of imagination of economic agents. However, at the same time this imaginary world is consistent with the observed data generated by an actual and endogenous process and is rationally considered to be real by economic agents despite the fact that the correct description of the economy is different. We do not consider our contribution to be final. In fact, there are possible extensions and modifications of our approach. First of all, the equilibrium nonlinearity present in our paper prevents us from employing a more sophisticated technical approach as that used by Hommes and Zhu (2014). However, it is possible, following Hommes and Zhu (2014), to examine how our results are affected by other learning rules than the simple OLS technique used in the paper. Furthermore, again following Hommes and Zhu (2014), it might be worthwhile to study how our approach fares in a more applied context. Nevertheless, we believe that our paper serves an important role and it can be considered complementary to other efforts as we effectively aim at reconciling the concept of CEE with REE whereas most researchers choose to adhere to the bounded rationality approach and examine the issues of stability and learning in an alternative framework.

Acknowledgments I would like to thank the participants of the 2011 Midwest Theory Meeting, and the Warsaw Economic Seminar for their remarks. I have benefited from numerous conversations with and the comments of Elzbieta Adamowicz, Andrei Barbos, Urszula Grzelonska, Eugeniusz Gurazdowski, Alain Jousten, Robert Kowalski, Michael Loewy, Gerhard Sorger, Konrad Walczyk, and Piotr Wysocki. This work was completed when I was visiting the Department of Economics at the University of South Florida. I thank Kwabena Gyimah-Brempong and Mark Herander for hospitability. I would like to extend my gratitude to the two anonymous referees and the former Editor - Cars Hommes - whose invaluable remarks have improved quality of the paper significantly. The remaining errors are my own responsibility. Appendix A In this Appendix we provide the key algebraic steps needed to establish that Eq. (29) listed in the main body of the text is valid. First let us assume that a given rational agent, i, believes that the total amount invested at time t by all rational agents is given by K Rtþ 1 ¼ m þ nK t þ MK αt ;

ð38Þ

where m, n, and M are constants. Recall that the total amount invested by naive agents at time t is given by Eq. (27). Furthermore, recall that R i Kt þ 1 ¼ KN t þ 1 þK t þ 1 and that E½εt þ 1 jΩt  ¼ ε hence we can write the following:  i      1  x 1 x  α AK t B ρ^ K t K þ K  ε þ m þ nK t þ MK αt ; ð39Þ E K t þ 1 Ωit ¼ 2 2 which reduces to





 i 1x 1 x Bð1  xÞ 1 x ð1  ρ^ ÞK  ρ^ þ n K t þ A þM K αt : ε þm þ  E K t þ 1 Ωit ¼  B 2 2 2 2 The actual amount invested by agent i is given by Eq. (15). Hence, using relationship (40) we can establish that







1 1x 1 x Bð1  xÞ 1 x AK αt  βit  B ð1  ρ^ ÞK  ρ^ þ n K t þ A þ M K αt B ε þm þ  : K R;i tþ1 ¼ 2 2 2 2 2

ð40Þ

ð41Þ

Recall that all remaining rational agents are assumed to share the mind set of agent i. Therefore, actual investment decisions of rational agents are dictated by relationships analogous to that represented with Eq. (41). Consequently, noting Rx that K Rtþ 1 ¼ 0 K R;j t þ 1 dj and recalling relationship (2) we can establish that







x 1 x 1x Bð1 xÞ 1x AK αt  β  B ð1  ρ^ ÞK  ρ^ þ n K t þ A þM K αt B ε þm þ  ; ð42Þ K Rtþ 1 ¼ 2 2 2 2 2 which upon rearranging terms reduces to





x 1x 1x x Bð1  xÞ x 1x β þ B B ð1  ρ^ ÞK þ ρ^  n K t þ AB A  BM K αt : ε m þ B K Rtþ 1 ¼ 2 2 2 2 2 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} m

n

M

ð43Þ

M.K. Dudek / Journal of Economic Dynamics & Control 41 (2014) 209–223

223

We require, by definition, that agent i be, in particular, internally consistent. Therefore, the coefficients in Eq. (43) must match those of Eq. (38). Hence, we must have m¼

^ x  2β þBð1  xÞε þ B2 ð1 xÞð1  ρÞK ; 2 2 þ xB

ð44Þ

n¼

x B2 ð1  xÞρ^ ; 2 2 þ xB

ð45Þ

and M¼

x ð2  Bð1 xÞÞA : 2 2 þxB

ð46Þ

Relationships (44)–(46) together with Eq. (38) allow us to establish that the actual amount invested by rational agents according to agent i is given by ! ^ x  2β þ Bð1  xÞε þ B2 ð1  xÞð1  ρÞK B2 ð1 xÞ ð2 Bð1 xÞÞ α R Kt þ 1 ¼ þ ρK ^ tþ AK t ; ð47Þ 2 2 þxB 2 þ xB 2 þxB which corresponds to Eq. (29) in the main body of the text. References Angeletos, G.M., La'O, J., 2013. Sentiments. Econometrica 81, 739–779. Brock, W.A., Dindo, P., Hommes, C.H., 2006. Adaptive rational equilibrium with forward looking agents. Int. J. Econ. Theory 2, 241–278. Cass, D., Shell, K., 1983. Do sunspots matter? J. Polit. Econ. 91 (2), 193–227. Diamond, P.A., 1965. National debt in a neoclassical growth model. Am. Econ. Rev. 55, 1126–1150. Dudek, M.K., 2010. A consistent route to randomness. J. Econ. Theory 145, 354–381. Eusepi, S., Preston, B., 2011. Expectations, learning and business cycle fluctuations. Am. Econ. Rev. 101 (6), 2844–2872. Grandmont, J.-M., 1998. Expectations formation and stability of large socioeconomic systems. Econometrica 66, 741–781. Grossman, S., Stiglitz, J., 1976. Information and Competitive Price Systems. Am. Econ. Rev. 66, 246–253. Hommes, C.H., 1998. On the consistency of backward-looking expectations: the case of the cobweb. J. Econ. Behav. Org. 33, 333–362. Hommes, C.H., Sorger, G., 1998. Consistent expectations equilibria. Macro. Dyn. 2, 287–321. Hommes, C.H., Sorger, G., Wagener, F.O.O., 2013. Consistency of linear forecasts in a nonlinear stochastic economy. In: Bischi, G.I., Chiarrela, C., Sushko, I. (Eds.), Global Analysis of Dynamic Models in Economics and Finance. Essays in Honour of Laura Gardini, Springer Verlag, Berlin, pp. 229–287. Hommes, C., Zhu, M., 2014. Behavioral learning equilibria. J. Econ. Theory. 150, 778–814. Lucas Jr., R.E., 1972. Expectations and the neutrality of money. J. Econ. Theory 4, 103–124. Lucas Jr., R.E., 1976. Econometric policy evaluation: a critique. Carnegie-Rochester Conf. Ser. Public Policy 1. Matsuyama, K., 1999. Growing through cycles. Econometrica 67, 335–347. Radunskaya, A., 1994. Comparing random and deterministic time series. Econ. Theory 4, 765–776. Sorger, G., 1998. Imperfect foresight and chaos: an example of a self-fulfilling mistake. J. Econ. Behav. Org. 33, 363–383. Townsend, R., 1983. Forecasting the forecasts of others. J. Polit. Econ. 91, 546–588. Wolfram, S., 2002. A New Kind of Science. Wolfram Media, Champaign, IL, USA.