Chaotic dynamics in an overlapping generations model with myopic and adaptive expectations

https://doi.org/10.1016/j.jebo.2007.03.005Get rights and content

Abstract

In this paper, we study dynamic behavior of an overlapping generations model under three different expectations: perfect foresight, myopic expectations and adaptive expectations. We show that economic transition under myopic or adaptive expectations is very different from that under perfect foresight. When agents are perfectly foresighted, dynamics is simple, with a unique steady state that is globally attracting. However, cycles and chaotic motion can appear under myopic and adaptive expectations. We study the possibility of Li–Yorke chaos under myopic expectations and entropic chaos under adaptive expectations.

Introduction

One problem in an overlapping generations (OLG) model with capital accumulation is how agents form future expectations when making inter-temporal decisions. Traditional models usually assume that agents are perfectly foresighted. It is well-known that complex dynamics can easily arise in OLG models with perfect foresight. Galor and Ryder (1989) showed that in a standard OLG model with capital accumulation and perfect foresight, multiple equilibria may exist. Their study provided a sufficient condition for a unique, non-trivial steady state equilibrium by calculating the third order derivatives of the utility and production functions.2

In reality, however, the requirement of perfect expectation does not seem reasonable. Agents are more likely to construct their expectations based on current or past information, or simply nothing. We say that agents are myopic if their expectations are formed arbitrarily. That is, they neither take the future into consideration nor learn something from the past. Studies by Benhabib and Day (1982) and Michel and de la Croix (2000) exhibited that when agents are myopic, complex dynamics can arise in the OLG models. By introducing credit into a standard OLG model without capital accumulation, Benhabib and Day showed that a constant credit expansion rate can generate “erratic” dynamics in prices and credit real value. Michel and de la Croix provided an OLG model with capital accumulation and found that only when the myopic dynamics is monotonic can a study of the myopic dynamics characterize the perfect foresight dynamics. They gave a sufficient condition for the uniqueness of the non-trivial steady state by calculating the second order derivatives of the utility and production functions. If the myopic dynamics is not monotonic (for example, if the inter-temporal elasticity of substitution is high), then cyclical oscillations will emerge and chaotic dynamics can appear under myopic expectations. In this situation, the economic transitions under myopic and perfect foresight are very different.

Besides myopic expectations, some studies assume a learning process whereby agents develop their expectations (adaptive expectations); see Chiarella (1988), Hommes, 1994, Hommes, 2002, Brock and Hommes (1997), Grandmont, 1985, Grandmont, 1998, and Chiarella and He (2003). Grandmont (1985) used an OLG model without capital accumulation, but with elastic labor supply, to show that chaotic motion can arise when the curvature of a utility function is large due to strong income effects. By considering adaptive expectations of the real interest rate in an OLG model with capital accumulation, Longo and Valori (2001) studied the local properties around the steady states. Besides providing a uniqueness condition of the non-trivial steady state, they also showed that under certain conditions, period-doubling (saddle-node, Neimark–Hopf) bifurcation will occur and the dynamics will become complicated.

Following the literature of expectation formation in OLG models, we consider three different types of real interest rate expectations: perfect foresight, myopic expectations and adaptive expectations, in an OLG model with capital accumulation. Differing from previous studies, we focus on the global stability of equilibria and the possibility of chaotic dynamics. Our results indicate that economic dynamics is very different under different expectation formations of the real interest rate. We find that economic dynamics is simple under perfect foresight and can be rather complex under the other two types of expectation formations. Furthermore, we show that the inter-temporal elasticity of substitution is an important determinant to the complexity of economic dynamics under myopic expectations. Under adaptive expectations, besides the inter-temporal elasticity of substitution, the complexity of dynamics also depends on the total factor productivity and the learning process.

In an environment with perfect and myopic expectations, the economy can be represented by a one-dimensional (1D) dynamical system (a first order difference equation in capital per capita). Under perfect foresight, the dynamics is simple with a unique, non-trivial equilibrium that is a global attractor. However, under myopic expectations, economic dynamics becomes quite complex when the inter-temporal elasticity of substitution is high. The bifurcation diagram shown by Michel and de la Croix indicated that cycles and chaotic dynamics might emerge when the inter-temporal elasticity of substitution is sufficiently high, but they only provided numerical results without any rigorous proof. To complement the analysis of Michel and de la Croix, we provide another simple way to prove the uniqueness and stability of the non-trivial equilibrium under perfect foresight and prove the presence of chaotic dynamics in the sense of Li and Yorke when the inter-temporal elasticity of substitution is sufficiently high under myopic expectations.

When agents have adaptive expectations of the real interest rate, we obtain a second order difference equation in capital per capita to represent a two-dimensional (2D) dynamical system of the economy. Because it is much more difficult to study a 2D dynamical system than a 1D dynamical system, we apply a recent method developed by Juang et al. (2005) and Li and Malkin (2006) to approximate a 2D dynamical system by using a 1D dynamical system and identify the existence of entropic chaos (positive topological entropy) when both the inter-temporal elasticity of substitution and the total factor productivity are sufficiently high and when agents rely heavily on current information to form their expectations.

The remainder of this paper is organized as follows. The next section develops an OLG model with capital accumulation and shows that dynamics under perfect foresight is always simple. Section 3 modifies the model by assuming that agents have myopic expectations and shows that chaotic dynamics can appear. An OLG model with adaptive expectations is analyzed in Section 4. The final section concludes our study. Proofs of a proposition and theorems are given in the Appendices. Note that appendices are available on JEBO website (will not apear in har copy).

Section snippets

The model

We consider an infinite-horizon, discrete time OLG model. Agents live for two periods, each period covering approximately 30 years, corresponding to young agents and old agents. The population grows at the rate of n. Each agent is endowed with one unit of time in each period. Young agents use all the time for work to earn the real wage rates (wt) for consumptions (c1t) and savings (st).When young agents become old, they spend all the time for leisure and consume (c2t+1) their savings from the

Myopic expectations

Following Michel and de la Croix, we assume that when agents are myopic, they expect that the real interest rate in the next period will be the same as the one in the current period. That is,Rt+1e=Rt.Combining Eqs. (6) and (9), we get that given the initial value of k0, the myopic expectation equilibrium consists of the sequence {kt}t0 that satisfieskt+1=11+nst(wt,Rt)=wt(1+n)(1+βσRt1σ).Therefore, the dynamics of the economy can be expressed as the following equation:kt+1=fσ(kt),where fσ:(0,)

Adaptive expectations

Following Longo and Valori, we consider a first order autoregressive adaptive expectation of the real interest rate. That is, Rt+1e is constructed based on current and past information with decreasing weights. With λ(0,1), Rt+1e is represented byRt+1e=i=0λ(1λ)iRti=(1λ)Rte+λRt.Eq. (12) shows that Rt+1e can be represented by Rte and Rt with weights (1λ) and λ, respectively. An increase in λ indicates that agents rely more and more heavily on the current information (Rt) when forming their

Conclusion

In this paper, we study economic transition in an OLG model with capital accumulation under three different types of expectations: perfect foresight, myopic expectations and adaptive expectations. We show that only simple dynamics occurs under perfect foresight. However, the dynamics may become complex under myopic and adaptive expectations. Under myopic expectations, our results indicate that cycles will emerge as the inter-temporal elasticity of substitution increases and chaotic dynamics can

Acknowledgements

The authors appreciate comments provided by two anonymous referees and the Editor. The authors are also grateful to participants of 2007 CEANA/ASSA conference for their suggestions and comments. The first author (Chen) gratefully acknowledges financial support from the National Science Council (grant number: NSC 95-2918-I-002-015) and the Program for Globalization Studies, the Institute for Advanced Studies in Humanities, National Taiwan University. The second author (Li) was partially

References (17)

There are more references available in the full text version of this article.

Cited by (18)

  • Heterogeneous expectations and equilibria selection in an evolutionary overlapping generations model

    2023, Journal of Mathematical Economics
    Citation Excerpt :

    Moreover, heterogeneity makes the information requirement for the fully rational agents even more demanding, as they should also be able to have a precise knowledge of the distribution of expectations among agents, in particular when the agents are allowed to switch from time to time their decision rule. A large number of works have been studying the economic consequences of different expectation designs The related literature is vast and considers both pure exchange models (see e.g. Arifovic (1995), Grandmont (1985), Bullard (1994), Duffy (1994), Wenzelburger (2002) and Tuinstra (2003)) and frameworks with capital accumulation (see e.g. Böhm and Wenzelburger (1999), Michel and de la Croix (2000), Chen et al. (2008), Chen and Li (2008) and Cavalli and Naimzada (2016)). Most of the aforementioned papers focus on the stability properties of the equilibrium, often studying homogeneous settings with respect to the forecasting rule distribution.

  • A simple model of growth cycles with technology choice

    2019, Journal of Economic Dynamics and Control
    Citation Excerpt :

    In fact, the model provided in this study comprises only of the Cobb-Douglas utility and Cobb-Douglas production technologies. Furthermore, unlike Michel and de la (2000) and Chen et al. (2008), we do not assume imperfect foresight, such as set myopic expectations, or employ complicated learning mechanisms to achieve rich dynamic results for the model. Instead, we assume that, unlike in ordinary textbook-type OLG models with production, a representative firm (or its owner) can choose from multiple production technologies.2

  • Public health spending, old-age productivity and economic growth: Chaotic cycles under perfect foresight

    2011, Journal of Economic Behavior and Organization
    Citation Excerpt :

    However, in the neoclassical OLG growth model with production à la Diamond (1965) regular and complex cycles can emerge only either assuming not fully rational individuals, i.e. with either myopic or adaptive expectations about future factor prices (Benhabib and Day, 1982; Michel and de la Croix, 2000; de la Croix and Michel, 2002), or extending it with the assumptions of, e.g., endogenous labour supply (Medio and Negroni, 1996), production externality (Cazzavillan, 1996), market imperfections (Aloi et al., 2000), PAYG pensions depending on previous earnings (Wagener, 2003), or taking the accumulation of government debt into consideration (Yokoo, 2000), which result in higher dimensional systems than the one-dimensional Diamond's model and thus able to describe more complicated dynamical events such as the Neimark–Sacker bifurcations.2 In fact, the economic literature has definitely shown, by resorting to various mathematical tools, that the Diamond's model with rational individuals (perfect foresight) can never possess an unstable equilibrium, so that business cycle is prevented in such a case (Galor and Ryder, 1989; Longo and Valori, 2001; Wendner, 2003; Chen et al., 2008). Therefore, extending the basic OLG growth model with production with hypotheses that, however, preserve the feature of the one-dimensional map, may hardly transform it in a model suited to explain fluctuations in macroeconomic variables if individuals are rational.

  • Productive public expenditures, expectation formations and nonlinear dynamics

    2008, Mathematical social sciences
    Citation Excerpt :

    Longo and Valori (2001) develop an OLG model with different expectations to study local properties around steady states and derive stability conditions in each case. Global properties of dynamics in a standard OLG model with different expectations are studied by Chen et al. (in press). It is shown that the dynamics is simple under perfect foresight.

View all citing articles on Scopus
1

Tel.: +886 3 5712121x56463; fax: +886 3 5131223.

View full text