Chaotic dynamics of a piecewise linear model of credit cycles
Introduction
The overlapping generations (OLG, hereafter) model has been widely used as a general equilibrium model in many fields of macroeconomics. As one of the major research concerns in macroeconomic dynamics, the endogenous business cycles have also been investigated using the OLG framework, initiated by Benhabib and Day (1982) and Grandmont (1985). They find that even without external shocks, nonlinearities inherent in the underlying economic systems can cause complicated, in particular chaotic, dynamics.
From a technical viewpoint, however, it is not necessarily easy to detect and characterize chaotic behaviors in a given deterministic nonlinear dynamic economic model, especially when the long-run observability of irregular perpetual fluctuations is concerned, even if the model is described by a single one-dimensional difference equation. One apparent exception appears when the economic model is piecewise linear. A piecewise linear dynamic model has much to offer. Among other features, it facilitates analytical tractability and dynamic richness at the same time. Even if piecewise linear modeling seems to be an extreme simplification, it is very beneficial in the above sense as long as it is a tolerable approximation of the real world.
Furthermore, there are empirical affinities of piecewise linear models to the threshold autoregressive (TAR) models used in nonlinear time series analysis. See Tong and Lim (1980) and Tong (1983) for TAR models and Chan and Tong (1986) for the smooth threshold autoregressive models.1 That is, a theoretical piecewise linear economic model free from external shocks can be thought of as a deterministic counterpart of the TAR model. In this sense, the development of deterministic piecewise linear models in the general equilibrium framework2 will give some sound microeconomic foundations for the TAR models.
In the literature of optimal growth, which is thought of as another endogenous business cycle theory in the general equilibrium framework, there are some two-sector models with Leontief technology in which optimal transition functions exhibit piecewise linearity, giving rise to ergodic chaos (that is, complex dynamics with observability in the long run). This piecewise linearity allows us to characterize complicated optimal growth paths in depth. See e.g. Nishimura et al. (1994) and Nishimura and Yano (1995) for more details.
Recent studies have investigated complex dynamics of piecewise smooth growth cycle models intensively. Piecewise smooth modeling can be regarded as an intermediate modeling between the two extremes: piecewise linear modeling and smooth modeling. See Gardini et al. (2008)3 and Matsuyama et al. (2016),4 who employ the relatively new theory of border-collision bifurcation to show that their macroeconomic models can exhibit complicated transition to chaotic behavior. Admittedly, the complex dynamics of the piecewise linear map of the interval has been well understood since long before at least by mathematical experts, and therefore we have seemingly little to add to the literature.5 On the other hand, piecewise linear modeling is recognized to give analytical results of complex dynamics in a sharp and clear way. Taking this fact into account and regarding that there seems to be no counterpart6 to the chaotic piecewise linear optimal growth model in the OLG literature, at least to the best knowledge of the authors, it must be still worth filling the gap and developing a piecewise linear OLG model exhibiting chaotic behavior. Therefore, the purpose of this paper is to explore this task.
Among recent studies related to this paper, Matsuyama (2007) proposes an OLG model with endogenous technology switch caused by financial imperfection, and shows that the model can generate several growth patterns. In Matsuyama’s (2007) model, agents face borrowing constraints due to financial imperfection, and each agent can choose to be either an entrepreneur or a lender. Furthermore, multiple investment technologies are assumed to be available. The market interest rate affects entrepreneurs’ choice of technology and the market rate varies over time depending on the level of capital. This implies that the entrepreneurs’ choice of technology changes endogenously, which gives rise to richer dynamics compared to other models in the literature on endogenous business cycles. Although the model proposed by Matsuyama (2007) is relatively simple, it leads to various phenomena, such as credit traps, credit collapses, leapfrogging, credit cycles, and growth miracles. As such, Asano et al. (2012) analyze the dynamic property of the macroeconomic model proposed by Matsuyama (2007) in depth, and show that the model can be analyzed within the framework of the neuron model studied by Hata (1982).7
Furthermore, Asano et al. (2012) show that the model can exhibit either periodic fluctuations or fluctuations which are chaotic in some sense.8 It is important to notice that chaos in Asano et al. (2012) occurs only on the set of parameter values of measure zero.9 In the very sense, chaos is virtually not observable in the models of Asano et al. (2012) and, consequently, Matsuyama (2007). Once such a pathological parameter value for the occurrence of Hata’s chaos is somehow chosen, however, any initial condition leads to a complicated long-run behavior. In the latter sense, Hata’s chaos is observable, but such a case hardly occurs. Observability of chaos in terms of both sets of parameter values and initial conditions is important because it can be thought to capture the “recurrent but not periodic” nature of business cycles in the deterministic framework. Therefore, when we talk about the observability of chaos in the long run, we have to pay attention to both the state space and the parameter space. To avoid confusion, when we talk about observability in terms of the parameter values, we sometimes use closely related or synonymous terms such as abundance or robustness.10
Some existing studies need mentioning from the viewpoint of analytical techniques. Ishida and Yokoo (2004) develop a macroeconomic model in which firms face a binary choice problem in investment, and show that due to piecewise linearity, the model exhibits periodic cycles. Yokoo and Ishida (2008) modify the model by introducing imperfect observability,11 and provide a mechanism by which observation errors lead to chaotic fluctuations. That is, Yokoo and Ishida (2008) show that observation errors or what they call misperception can be a source of observable chaos in economic systems.
Since we hardly know the true state of the world with precision, especially when aggregate amounts are concerned, it is natural to assume that such insufficient information affects decision making, in particular, about whether an agent chooses to become an entrepreneur or not. Therefore, it is of interest to incorporate such information imperfection or imperfect observability into Matsuyama’s (2007) or equivalently Asano et al.’s (2012) framework under perfect observation to see how the dynamic patterns change.
The model proposed in this paper can be thought of as an extension of the model in Asano et al. (2012), which is a special case of Matsuyama’s (2007), along the line of Yokoo and Ishida (2008). As a result, we transform Matsuyama’s (2007) original model, together with misperception or observation errors, into a piecewise linear model, which is tractable enough to investigate the dynamics in depth by using the techniques of the Frobenius–Perron operators to find invariant measures (i.e., observable chaos) as in Yokoo and Ishida (2008).12 Indeed, by specifying the set of parameters for some kind of Markov properties,13 we can easily establish and characterize chaotic dynamics with observability in terms of initial conditions in more detail. Imposing Markov properties on models seems rather restrictive at first glance, however, this will be relaxed to the extent that such chaos is shown to be abundant in terms of the set of parameter values.
The organization of this paper is as follows. Based on Matsuyama (2007) and Asano et al. (2012), Section 2 provides a benchmark model, in which the productivity of agents is perfectly observable. Section 3 provides the main model of this paper, in which the productivity of agents is imperfectly observable. Section 4 considers further specifications of our model discussed in Section 3. Section 5 analyzes chaotic dynamics in detail. Section 6 gives concluding remarks. Some derivations are relegated to the Appendix.
Section snippets
The model under perfect observation
In this section, based on Asano et al. (2012), directly following Matsuyama (2007), we consider the situation in which the returns generated by entrepreneurs’ projects are perfectly observable. In the following sections, we extend the perfectly observable framework to an imperfectly observable one in which the returns generated by entrepreneurs’ projects are observed with some noise.
A final good is produced by the following constant returns to scale technology: where and
The model under imperfect observation
The main model in this paper builds on that with no uncertainty described in the previous section. To do this, we suppose that a project with a higher rate of return is riskier. To capture this idea in an easier way, we suppose, à la Yokoo and Ishida (2008), that entrepreneurs perceive the rate of return from project 2, which earns higher returns than project 1, with some “noise”. For simplicity, we assume that entrepreneur perceives to be , which we formulate as17
Piecewise-linearization of the model
As the form of (11) is still too general to characterize its dynamics in detail, we need to further specify its functional form. First, for small enough, we can define and by solving where is given by (9) for . Further computations show that for It is easy to check that and . Note that
Chaotic dynamics
In the following, we show that the model given by (22) is capable of generating chaotic behaviors in some sense. First, note that the study of Asano et al. (2012) shows that for any integer , there is a such that the limiting map, , given by (23) exhibits a periodic attractor of period . On the other hand, by a variation of the Li–Yorke Theorem (Li and Yorke, 1975), it is known that if a continuous map on the interval has a periodic point whose period is not 2 to the power of
Concluding remarks
Based on the model analyzed by Asano et al. (2012), which is essentially equivalent to the original credit cycle model proposed by Matsuyama (2007) as long as permanent fluctuations are concerned, we developed an OLG model of endogenous business cycles. By specifying the distribution of “noise” representing imperfect observability, we obtained a continuous piecewise linear model, for which we showed that, using the Markov property, observable chaos is detected and described by its invariant
Acknowledgments
We acknowledge an anonymous reviewer whose comments improve this paper. We are grateful to Masaki Narukawa and Yosuke Umezuki for their comments and discussions on this work. This research is financially supported by the JSPS KAKENHI Grant Numbers 26380240, 17K03806, 16H02026, 16H03619, 16K03558, and the Joint Research Program of KIER.
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