Ergodic chaos for non‐expansive economic models**

• Author: Kenji Sato,Makoto Yano
• Date: 01 September 2019
• DOI: http://doi.org/10.1111/ijet.12232
• Published date: 01 September 2019

doi: 10.1111/ijet.12232

Ergodic chaos for non‐expansive economic models**

Makoto Yano*and Kenji Sato†

Ergodic chaos is one of the most important concepts in the theory of nonlinear economics.

Due to the lack of appropriate sufficient conditions, its reach has not been fully understood. In

this paper a sufficient condition for ergodic chaos that covers non‐expansive dynamical sys-

tems is examined. By analyzing an endogenous growth model as an example, we discuss the

breadth of models in which ergodic chaos can emerge.

Key words ergodic chaos, endogenous fluctuations

JEL classification E32, O39, C02

Accepted 26 May 2019

1 Introduction

Chaotic dynamics is a possible explanation for business cycles. Ergodic chaos is one of the most

important concepts in the theory of nonlinear economic dynamics. Its definition ensures that the

complex behaviors are observable in the sense that the set of initial conditions from which

complex trajectories start has a positive Lebesgue measure. Despite its theoretic appeal, it seems to

be, more or less, perceived to be a very rare phenomenon. In the present paper we will try to

challenge this preconceived notion.

One reason for the belief that ergodic chaos does not emerge very often is presumably the lack of

appropriate sufficient conditions. A well‐known sufficient condition for a dynamical system to be

ergodically chaotic is that (1) the (one‐dimensional) dynamical system has a single‐peaked shape

(unimodality) and (2) the absolute value of its derivative function is strictly greater than 1 (ex-

pansivenss); see Grandmont (1986, 2008). Simple and effective as it is for some economic models, this

conventional characterization is somewhat too stringent for others. This is because, while many

economic models possess the unimodality property, expansiveness often fails.Anexampleisgivenby

Nishimura and Yano (1995a). The optimal policy of their model goes along with the lower boundary

of the feasible region that is determined by a depreciation factor; the slope must be less than 1. Another

example is the endogenous growth model by Matsuyama (1999). The chaotic behavior of this model

was investigated by Yano et al. (2011) by applying the conventional sufficient condition. Although

their analysis shows that the model exhibits ergodic chaos under a certain parameter condition, the

region was very limited. For most parameter values, the system is non‐expansive.

In the present paper we will make use of the result of Sato and Yano (2010, 2012) that covers

non‐expansive dynamical systems. A dynamical system is characterized by ergodic chaos if (1) it is

International Journal of Economic Theory 15 (2019) 311–320 © IAET 311

*

RIETI, Kasumigaseki, Chiyoda, Tokyo, Japan. Email: yano@kier.kyoto-.ac.jp

†

Graduate School of Economics, Osaka Prefecture University, Sakai, Osaka, Japan.

**We are grateful to an anonymous referee for insightful comments. Makoto Yano gratefully acknowledges financial

support from JSPS science grants #23000001 and 16H02015. Kenji Sato gratefully acknowledges financial support from

JSPS science grants 16K03552 and 19K13650.