Endogenous business cycles in a perpetual youth model with financial market imperfections**

• Date: 01 September 2019
• Author: Takuma Kunieda,Kazuo Nishimura
• Published date: 01 September 2019
• DOI: http://doi.org/10.1111/ijet.12233

doi: 10.1111/ijet.12233

Endogenous business cycles in a perpetual youth model

with financial market imperfections**

Takuma Kunieda* and Kazuo Nishimura

†

An economy in which entrepreneurs and financiers interact with each other through an

imperfect financial market is investigated by applying a dynamic general equilibrium theory.

In each period, there is a certain probability of each entrepreneur’s life ending, and a certain

number of entrepreneurs are newly born. Although entrepreneurs are potential capital pro-

ducers, they receive an idiosyncratic productivity shock in each period. Therefore, en-

trepreneurs who draw higher productivity become capital producers and those who draw

lower productivity become lenders. Financiers do not have an entrepreneurial talent for ca-

pital production, and thus they lend their assets in the financial market to acquire an interest

income. In equilibrium, deterministic endogenous business cycles can occur at the inter-

mediate level of financial constraints.

Key words financial constraint, endogenous business cycle, chaos, Mitra’ssufficient con-

dition, heterogeneous agent

JEL classification E32, E44, O41

Accepted 26 May 2019

1 Introduction

In this paper we study the perpetual youth model with agents facing financial constraints and

productive heterogeneity. Perpetual youths were introduced in the continuous‐time dynamic

general equilibrium model by Blanchard (1985). We consider a discrete‐time version of the

perpetual youth model and demonstrate that endogenous business cycles can occur in equili-

brium. In particular, chaotic equilibrium can be obtained under moderate parameter conditions.

In our model, one unit measure of entrepreneurs, a representative financier, and a representative

firm inhabit the economy. As in the model of Blanchard (1985), each entrepreneur’slifemayend

unexpectedly with probability

ν

in each period. According to the law of large numbers, a measure‐

(

ν

−1) continuum of entrepreneurs subsists in each period. Because we assume that a measure‐

ν

continuum of entrepreneurs is newly born in each period, the total population is consistently equal to

1

International Journal of Economic Theory 15 (2019) 231–248 © IAET 231

*School of Economics, Kwansei Gakuin University, Nishinomiya, Hyogo, Japan. Email: tkunieda@kwansei.ac.jp

†

Kobe University, RIEB, Nadaku, Kobe, Japan and RIETI, Tokyo, Japan.

The second author discussed various theoretical issues with his close friend and coauthor, the late Professor Tapan

Mitra. In this paper we apply Mitra’s result on topological chaos, which is one of the themes they discussed, to the

dynamic general equilibrium theory with heterogeneous agents facing financial constraints.

**We are grateful to an anonymous referee for his/her invaluable comments which helped us to improve this paper. Of

course, all remaining errors, if any, are ours. This work is financially supported by the Japan Society for the Promotion

of Science, Grants‐in‐Aid for Scientific Research (Nos. 15H05729, 16H02026, 16H03598, 16K03685).

in the economy.

1

Each entrepreneur becomes a worker only in the initial period of her lifetime, supplying

one unit of labor inelastically to earn a wage income. Although she grows up to be a potential capital

producer, she receives an idiosyncratic productivity shock in each period. Therefore, she actually pro-

duces capital only when she draws higher productivity; otherwise, she lends her net worth in the financial

market to earn an interest income. When she produces capital, she can borrow only up to a certain

proportion of her net worth in the financial market; that is, she faces financial constraints. The re-

presentative financier has an infinite lifetime without any inherent entrepreneurial talent and lends her

net worth in the financial market to earn an interest income, similarly to low‐productivity entrepreneurs.

The representative firm produces general goods by employing workers and capital.

In equilibrium, the economy is expressed by a two‐dimensional autonomous difference equation

system with respect to capital and a cutoffof the productivity that divides entrepreneurs into bor-

rowers and lenders. Our analysis focuses exclusively on the difference equation of the cutoffbecause it

is straightforward to investigate the difference equation of capital, and the difference equation of the

cutoffis independent of capital. The difference equation of the cutoffhas two steady states, which we

call a high steady state and a low steady state, under moderate parameter conditions. Whereas the high

steady state is unstable, the stability of the low steady state depends upon the parameters of the model.

In particular, the stability of the low steady state is subject to the distribution of productivity shocks.

More concretely, if the productivity distribution is very thin around the low steady state, the low steady

state is more likely to be unstable. When the low steady state is unstable, endogenous business cycles

can occur in the economy. In this situation, the presence of chaotic equilibrium is demonstrated by

applying Mitra’s (2001) sufficient condition for topological chaos. Whereas the Li–Yorke theorem (Li

and Yorke, 1975) is a well‐known sufficient condition for topological chaos, Mitra proposes a tractable

sufficient condition, which enables us to identify the presence of topological chaos without knowing

the occurrence of a period‐

3

cycle.

2

The current paper is related to the literature on deterministic endogenous business cycles in

dynamic general equilibrium theory. Among others, Grandmont (1985) employs overlapping

generations models to derive deterministic endogenous business cycles, and Benhabib and

Nishimura (1985) and Nishimura and Yano (1995) develop a model of an infinitely lived re-

presentative agent with two production sectors and derive deterministic endogenous business

cycles. Unlike these previous studies, we assume financial market imperfections in the perpetual

youth model and consider heterogeneous productivity between capital producers.

The rest of this paper is organized as follows. In the next section we develop a perpetual youth

growth model in which entrepreneurs and financiers interact with each other through a financial

market. In Section 3 we derive a dynamical system in equilibrium and investigate its local stability.

In Section 4 we demonstrate the presence of chaotic equilibrium by applying Mitra’s (2001)

sufficient condition. Section 5 concludes.

2 Model

An economy continues from time =t0to

∞

=+t

in discrete time. One unit measure of en-

trepreneurs, a representative financier, and a representative firm inhabit the economy in each

International Journal of Economic Theory 15 (2019) 231–248 © IAET232

1

See Kunieda and Shibata (2017) for an analysis of the case in which agents face financial constraints and productive

heterogeneity without perpetual youth.

2

Mitra (2001) applies his result to the models of Boldrin et al. (2001) and Matsuyama (1999). Subsequently, Nishimura

et al. (2004) use an alternative condition to prove the robustness of the chaos in the optimal growth model, and Deng

and Khan (2018) provide an alternative proof of Mitra’s result on Matsuyama’s model.

Perpetual youth model Takuma Kunieda and Kazuo Nishimura