Chapter 1

Introduction
John Stachurski, Alain Venditti, and Makoto Yano

Kazuo Nishimura has made outstanding contributions to three main fields of
economic theory: international trade, general equilibrium, and the theory of economic growth. With his characteristic mix of originality, technical clarity, elegance,and rigor, Kazuo’s insights into nonlinear dynamics have transformed our understanding of economic growth, business cycles, and the relationship between them.
For young scholars beginning work in these areas, Kazuo’s papers are essential andinspiring.
Kazuo Nishimura obtained his Master’s degree from Tokyo University in 1972.
In 1973, he entered the graduate program in economics at the University ofRochester. His supervisor was Lionel McKenzie—one of the giants of twentiethcentury neoclassical economics. After spending three years at Rochester, Kazuo
joined the Department of Economics at Dalhousie, Canada, in 1976. He obtained
his PhD in 1977. Subsequently, he returned to Japan with an appointment at theTokyo Metropolitan University. He spent 10 years there, during which time he alsotaught at the State University of New York at Buffalo, and at the University ofSouthern California. In 1987, he joined the Institute of Economic Research at Kyoto
University. He served as its director in 2006–2010.
Kazuo has had a highly distinguished career. He was elected fellow of the Econometric Society in 1992. He served as the vice president of the Japanese Economic
Association in 1999–2000, and as its president in 2000–2001. He was co-editor of
J. Stachurski
Australian National University, Canberra, ACT, Australia
e-mail: john.stachurski@anu.edu.au
A. Venditti
National Center for Scientific Research, CNRS-GREQAM and EDHEC, Marseille, France
e-mail: alain.venditti@univmed.fr
M. YanoInstitute of Economic Research, Kyoto University, Yoshida-Honmachi, Japane-mail: yano@kier.kyoto-u.ac.jp
This chapter has been made open access under a CC BY-NC-ND 4.0 license. For details on rights andlicenses please read the Correction https://doi.org/10.1007/ 978-3-642-22397-6_19© The Author(s) 2012, corrected publication 2021
J. Stachurski et al. (eds.), Nonlinear Dynamics in Equilibrium Models,
https://doi.org/10.1007/978-3-642-22397-6_1

1

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the Japanese Economic Review during 1995–1997, and its editor-in-chief from 1997
to 2000. In 2004, he established the International Journal of Economic Theory,
a new journal for the profession that is already highly regarded within its field.
He has served as a managing editor since its inception. He has also served on theeditorial board of numerous leading journals in economics and applied mathematics,
and he has edited a number of special issues for prestigious journals. He hasorganized successful conferences around the world.
Recently Kazuo has become an active and leading participant in the publicdebate in Japan about the science curriculum taught in high schools and universities.
Through the popular press and scholarly journals on science and mathematics
education, Kazuo has strongly advocated the need for improved training in scienceand mathematics for high school students and humanities students at universities.
To further promote scientific and mathematical education, he founded the International Society for Education. He currently serves as president of the society.
To say that Kazuo is an extremely busy person is still an understatement.
However, as his numerous co-authors, friends and colleagues all over the world willattest, he is always available to discuss research, provide insight, encouragement,
and the benefit of his remarkable energy. A perfect day for Kazuo is to work throughtill evening with some co-authors on a difficult research problem, then to break fordinner and fine sake together in a Kyoto restaurant, all the while still discussing new
ideas. If, late into night, a break for sleep appears to be necessary, there is no doubtthat Kazuo will be back the next day with some hand-written notes providing thesolution to the problem.
We, Kazuo’s friends, colleagues and co-authors, have been privileged to benefit
over many years from his loyalty, warmth and generosity. It is with great pleasurethat we present this collection of selected papers, containing his most important andenduring contributions.

1.1 Overview of the Papers
In this section we provide an overview of the papers selected for this volume. While
the optimal growth model provides a common framework linking these papers,
they have been classified according to the particular qualitative property of theequilibrium that is analysed:
1. Optimal growth and endogenous cycles
2. Optimal growth and chaotic dynamics
3. Global dynamics in optimal growth models
4. Dynamic models with non-concave technologies
5. Stochastic optimal growth models
6. Indeterminacy in exogenous growth models
7. Indeterminacy in endogenous growth models
The complete list of papers can be found in Sect. 1.2.

1 Introduction

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1.1.1 Introductory Survey
As the title of the book suggests, all the selected papers involve nonlinear dynamics,
a major field of research within economic theory. In collections such as this, it isusually desirable to provide the reader with a review of basic material and concepts
common to the field. This task has already been accomplished by Kazuo, in apaper titled “Nonlinear dynamics in the infinite time horizon model,” publishedwith Gerhard Sorger in the Journal of Economic Surveys in 1999. Starting fromthe standard aggregate formulation of the optimal growth model, in which theunique optimal equilibrium converges toward a long-run balanced growth path, thepaper then goes on to introduce the main features of nonlinear dynamics, including
periodic cycles, chaotic dynamics, stochastic shocks, non-concave technologies,
externalities, local indeterminacy, sunspot fluctuations and analysis of multisector
models. It provides the perfect lead article of the present collection. It remains
only for us to provide the reader with an overview of the selected papers, inorder that he or she might quickly locate those papers most relevant to his or her
field.

1.1.2 Optimal Growth and Endogenous Cycles
This section contains three important contributions on the existence of endogenous
fluctuations in the perfect competitive framework of the multisector optimal growthmodel. The first and second belong to the set of papers written during a long andfruitful collaboration between Kazuo and Jess Benhabib (New York University).
They deal with closed economies. The third paper is the first of many collaborations
with Makoto Yano of Keio University. It studies a model of trade between two
countries. These two collaborations have produced highly influential contributions
on the analysis of non-linear dynamics under perfect competition.
The first paper, “The Hopf bifurcation and the existence and the stability ofclosed orbits in multi-sector models of economic growth,” was published with Jess
Benhabib in the Journal of Economic Theory in 1979. It shows that, under very
general circumstances, the standard optimal growth model with two or more capital
goods can give rise to optimal trajectories that are limit cycles. This was a striking
result, as research on optimal growth in the 1960s and 1970s had typically foundthese models to be characterized by a strong form of stability, particularly when
the discount rate is small (see, for instance, Brock and Scheinkman, 1976; Cassand Shell, 1976; McKenzie, 1976; and Scheinkman, 1976). In contrast, this paper,by treating the discount rate as a parameter, establishes via bifurcation theory that
optimal paths can become closed orbits as the steady state loses stability.
Bifurcations for differential equation systems arise when, for some value of aparameter, the Jacobian of the function describing motion acquires eigenvalues withzero real parts at a stationary point. If a real eigenvalue becomes zero, this results in

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multiplicity of steady states. In this paper, the focus is on a bifurcation that resultsfrom pure imaginary roots: the Hopf bifurcation. In contrast to bifurcation froma real eigenvalue, no additional stationary points arise from a Hopf bifurcation.
Instead, closed orbits emerge around the stationary point. Nonjoint production
and a single, possibly composite, consumption good are assumed to assure theuniqueness of the steady state. Under some additional assumptions, it is shown that
the Jacobian of the functions describing the motion of the system cannot vanish
at a steady state, and, as a result, that only the Hopf bifurcation is possible. An
example is given of a totally unstable steady state, giving rise to optimal paths thatare closed orbits. The stability of orbits that arise from the optimal growth problem
is discussed.
The second paper, “Competitive equilibrium cycles,” was published in 1985
with Jess Benhabib in the Journal of Economic Theory, and provides a discretetime extension of the previous contribution. Considering a two-sector optimal
growth model with one pure consumption good and one investment good, it gives
sufficient conditions for generating robust period-two cycles in outputs, stocks,
and prices. These conditions are expressed in terms of the discount rate anddifferences in relative factor-intensities between the two industries.1 Finally, some
intuitive explanation for the existence of cyclical equilibria that arise in neoclassicaltechnologies are also given. The paper is notable for the sophisticated techniquesfrom differential topology used to prove the global existence of period-two cycles,
corresponding locally to the existence of a flip bifurcation.
In the last paper of this section, “Interlinkage in the endogenous real businesscycles of international economies,” published in 1993 with Makoto Yano in
Economic Theory, the transmission of business cycles between trading countries
is analyzed. In a perfect foresight model with many consumers, it is well-known
that competitive equilibrium paths behave like optimal growth paths.2 Therefore,
a perfect foresight equilibrium path may exhibit business cycles, even in a manyconsumer model such as a large-country trade model. In this paper, a two-country
optimal growth model in which the countries have independent technologies is
considered. First, the determinants of each country’s global accumulation patternin autarkic equilibrium are analyzed using the results of Benhabib and Nishimura
contained in the second paper of this section, providing conditions on the sectoraltechnologies that lead to existence of business cycles (in particular, period-two
cycles). Global accumulation under free-trade is then addressed. Conditions for theinternational transmission of business cycles are given in terms of the parameters ofeach country’s technologies.

1

In particular, the discount factor must be sufficiently large, and the consumption good sector must
be more capital intensive than the investment good sector.
2
See, for instance, Becker (1980), Bewley (1982), Epstein (1987), and Yano (1983, 1984).

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1.1.3 Optimal Growth and Chaotic Dynamics
The two papers by Jess Benhabib and Kazuo Nishimura contained in the previous
section had a strong impact on the group of researchers studying macroeconomic
dynamics and business cycles, initiating a common program of research that aimed
to provide detailed analysis of the qualitative properties of capital accumulation
paths in optimal growth models. For example, the landmark paper of Boldrin andMontrucchio (1986), which provides an anything-goes result for optimal paths
in standard multisector optimal growth models, was initiated at the beginning of
the 1980s after Luigi Montrucchio attended a presentation of the “Competitive
equilibrium cycles” paper.
The findings of Boldrin and Montrucchio indicated that equilibrium time paths
in deterministic equilibrium economies may exhibit complex—even chaotic—
dynamics. Building on the results of McKenzie (1976) and Scheinkman (1976), they
showed specifically that almost any capital accumulation path can be realized as asolution of an optimal growth model, provided that the discount rate is large enough.
However, this leaves open the question as to whether or not chaotic optimal paths
can appear under more reasonable rates of discounting. The two papers with Makoto
Yano in this section, titled “Nonlinear dynamics and chaos in optimal growth: an
example,” and “Chaotic solutions in dynamic linear programming,” published in
Econometrica in 1995 and in Chaos, Solitons and Fractals in 1996 respectively,
showed that they can.
The first paper demonstrates the possibility of ergodically chaotic optimal accumulation in a two-sector model with linear preferences and Leontief production.
A condition is provided under which the optimal transition function is unimodal andexpansive. It is then shown that the set of parameter values satisfying this condition
is nonempty, no matter how weakly future utilities are discounted. The second paper
is a complement to the first, as it provides a simpler and more intuitive proof of thesame result.
While these two papers provide strong conclusions on the compatibility between
chaotic optimal paths and low discounting of future utilities, their main limitation
concerns the fact that preferences and technologies are specified. This extra structure
inhibits the formation of a general picture. In consequence, the paper “On theleast upper bound of discount factors that are compatible with period-three cycles,”published in the Journal of Economic Theory in 1996 with Makoto Yano, studies
a class of general, reduced form optimal growth models. In these models time isdiscrete, and an indirect utility function summarizes both preferences and the social
production function. The authors derive the least upper bound of the set of discountfactors such that a cyclical optimal period-three path emerges in a standard model of
optimal growth.3 For this purpose, they characterize a cyclical optimal period-three3

The existence of a cyclical path of period three is a fundamental criterion for the emergence ofcomplex nonlinear dynamics, in particular of cyclical paths of any other periodicity (see Sarkovskii(1964) and Li and Yorke (1975)).

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path in terms of the dual path of prices. They then demonstrate that a condition
for the existence of a cyclical optimal period-three path can be expressed by theexistence of a solution to a system of linear equations. By solving this system,
they pfinally demonstrate that the least upper bound mentioned above is equal to.3  5/=2.

1.1.4 Global Dynamics in Optimal Growth Models
Although they focus on the particular case of chaotic optimal paths, the contributions contained in the previous section are concerned more generally with thefollowing question: “How is long-run optimal behavior affected by changes in therate at which the future is discounted?” The paper contained in the present section,“Intertemporal complementarity and optimality: a study of a two-dimensional
dynamical system,” published in International Economic Review in 2005 with Tapan
Mitra, provides additional insights on this point. It studies the underlying structureof the two-dimensional dynamical system generated by a class of optimal growthmodels that allow for intertemporal complementarity between adjacent periods, but
preserves the time-additively separable framework of standard Ramsey models. The
purpose of the article is to complete the program sketched in the contribution ofSamuelson (1971), providing both a complete local and global analysis of the model
under which the standard results of the Ramsey model continue to hold, even withdependence of tastes between periods. Global convergence results are establishedand related to the local analysis using the mathematical theory of two-dimensional
dynamical systems. The local stability property of the stationary optimal stock isalso related to the differentiability of the optimal policy function near the stationary
optimal stock, using the stable manifold theorem.

1.1.5 Dynamic Models with Non-Concave Technologies
Traditionally, optimal growth theory is concerned with the derivation of an optimal
consumption plan of a representative agent when the production function in theeconomy is concave and preferences of the consumer exhibit decreasing marginalutility. In these settings, there exists a unique long-run steady state capital stock thatis globally stable, in the sense that the optimal capital stock converges to the steadystate level independent of the initial condition. The paper contained in this section,“A complete characterization of optimal growth paths in an aggregated model with a
non-concave production function,” published in the Journal of Economic Theory in1983 with W. Davis Deckert, provides some extensions of this framework by considering a non-concave production function and thus the local existence of increasing
returns. It appeared as a path-breaking study of optimal one-sector growth under
non-convex technology. When the technology is convex, the Euler equation is1 Introduction

7

sufficient to prove the monotonicity property of the equilibrium path. This is nolonger the case when non-convexities are introduced. Deckert and Nishimura showthat the Euler equation in conjunction with the Principle of Optimality is sufficient toprove monotonicity for non-concave production functions. For certain interest rates,
they prove that the optimal path converges to a steady state only if the initial capitalstock is above a critical level, otherwise it converges to zero. They also demonstratethat the set of points for which the value function is differentiable is precisely theset of initial capital stocks from which there is a unique optimal path.

1.1.6 Stochastic Optimal Growth Models
In the first paper in this section, “Stochastic optimal growth with nonconvexities,”
published with Ryszard Rudnicki and John Stachurski in the Journal of Mathematical Economics in 2006, the problem of optimal growth with non-concave
production (i.e., the problem studied in the paper with W. Davis Deckert discussed
in the last section) is revisited, but now with random shocks to productivity.
A number of key results from the deterministic case are replicated, such as almost
everywhere differentiability of the value function, and monotonicity of optimal
policies. In addition, the authors establish a basic dichotomy in growth dynamics
when sufficient mixing is present: Either global stability is observed, or the economy
is globally asymptotically collapsing to the origin.
While the dichotomy result of this paper is useful in its own right, it does notprovide a set of conditions on the primitives under which global stability is shownto hold. For the concave case, this problem was essentially solved as early as the
1970s (see, e.g., Brock and Mirman (1972)), but the non-concave case proved
far less tractable. The third paper in this section, “Stability of stochastic optimal
growth models: a new approach,” published with John Stachurski in the Journal of
Economic Theory in 2005, provided a breakthrough. The paper gave a set of stability
conditions on the model primitives that could be viewed as natural analogues of theclassical deterministic conditions. The breakthrough resulted from a new method
developed by the authors for studying stability of optimal dynamic models, basedon viewing marginal utility of consumption as a Lyapunov function.

1.1.7 Indeterminacy in Exogenous Growth Models
A wide-spread perception among economists is that macroeconomic business-cycle
fluctuations can be driven by changes in expectations of fundamentals. A major
strand of the literature focusing on fluctuations derived from agents’ beliefs isbased on the concept of sunspot equilibria, dating back to the early works of
Shell (1977), Azariadis (1981), and Cass and Shell (1983). As shown by Woodford(1986), the existence of sunspot equilibria is closely related to the indeterminacy of8

J. Stachurski et al.

perfect foresight equilibrium (i.e., the existence of a continuum of equilibrium paths
converging toward one steady state from the same initial value of the state variable).
Following the seminal contribution of Benhabib and Farmer (1994), infinite
horizon Ramsey-type models augmented to include external effects in production
have been shown to exhibit multiple equilibria and local indeterminacy. Muchof the research in this area has been concerned with the empirical plausibilityof indeterminacy in markets with external effects, which exhibit some degree ofincreasing returns. While the early results on indeterminacy relied on relatively
large increasing returns, Benhabib and Farmer (1996) showed that indeterminacy
can also occur in two-sector models with small sector-specific external effects
and mild increasing returns. Nevertheless, a number of empirical researchers on
disaggregated US data (see for instance Basu and Fernald (1997)) have concluded
that returns to scale seem to be roughly constant.
The three papers contained in this section show how indeterminacy can occur
in standard growth models with sector-specific externalities, constant social returns,
decreasing private returns and small or negligible external effects. The first one,
“Indeterminacy and sunspots with constant returns,” published with Jess Benhabibin the Journal of Economic Theory in 1998, is an influential contribution that has
initiated many extensions and developments. Considering a two-sector economy
with Cobb-Douglas technology and a utility function that is linear in consumption,
Benhabib and Nishimura show that local indeterminacy arises if and only if thesector-specific externalities generate a capital intensity reversal between the private
and the social levels. Specifically, indeterminacy requires that the consumption good
sector is capital intensive at the private level but labor intensive at the social level.
This conclusion is fundamentally based on the breakdown of the duality between
the Rybczynski and Stolper-Samuelson effects, caused by introduction of market
imperfections.
The second paper, “Trade and indeterminacy in a dynamic general equilibrium
model,” published with Koji Shimomura in the Journal of Economic Theory in 2002,
introduces sector-specific externalities in a Heckscher-Ohlin, two-country, dynamic
general equilibrium model. In their model, factors are internationally immobile, andthe countries both use identical Cobb-Douglas technologies. Under the same capitalintensity reversal condition as in the previous paper, Nishimura and Shimomura
show that there are multiple equilibrium paths from the same initial distribution ofcapital in the world market, and that the distribution of capital in the limit differs
among equilibrium paths. One equilibrium path converges to a long-run equilibrium
in which the international ranking of factor endowment ratios differs from the initialranking, while another equilibrium path maintains the initial ranking and converges
to a different long-run equilibrium. Since the path realized is indeterminate, so is
the long-run trade pattern. Therefore, the long-run Heckscher–Ohlin prediction,that each country exports those goods such that the abundantly endowed factor of
production is intensively used for producing it, is vulnerable to the introduction of
externality.
In most papers focusing on the existence of indeterminacy within two-sector
models with constant social returns, the utility function is assumed to be linear,1 Introduction

9

which implies an infinite elasticity of intertemporal substitution in consumption.
However, as shown recently by Mulligan (2002) and Vissing-Jorgensen (2002)
for instance, this elasticity is estimated around unity or even below. The last
paper of this section, “Indeterminacy in discrete-time infinite-horizon models withnonlinear utility and endogenous labor,” published with Alain Venditti in the Journalof Mathematical Economics in 2007, considers a two-sector economy in which
preferences are given by a additively separable CES utility function, defined overconsumption and leisure. Three particular cases are considered:
1. Labor supply is infinitely elastic,
2. The elasticity of intertemporal substitution in consumption is infinite, and
3. Labor supply is inelastic.
In the first case, the steady state is a saddle-point for any value of elasticity ofintertemporal substitution in consumption, and any level of external effects. In thesecond case, local stability of the steady state is independent from elasticity oflabor supply, and local indeterminacy relies only on the loss of duality between
the Rybczynski and Stolper-Samuelson effects. In the third case, a geometricalmethod of Grandmont et al. (1998) is used to show that the steady state islocally indeterminate if and only if the elasticity of intertemporal substitution
in consumption is large enough. On this basis, the main result of the paper isestablished: the steady state is locally indeterminate if and only if the elasticity
of intertemporal substitution in consumption is sufficiently large, and that of labor
supply is sufficiently low. Moreover, period-two cycles always occur as the elasticityof labor supply is increased, and the steady state becomes saddle-point stable.

1.1.8 Indeterminacy in Endogenous Growth Models
The fundamental contributions of Romer (1986) and Lucas (1988) showed that if
preferences are homothetic and external effects in production generate constantreturns to scale for reproducible factors, then endogenous growth can occur in
infinite-horizon models. This means that the growth process is endogenously
determined, perpetual, and can be characterized by the dynamics of growth rates
for output and consumption. This fact, combined with some of the previous results
provided in this collection, suggests the possible existence of growth rate fluctuations. The three contributions contained in this section are concerned precisely withthis point.
The first, “Indeterminacy under constant returns to scale in multisector
economies,” published with Jess Benhabib and Qinglai Meng in Econometrica
in 2000, is an extension to the endogenous growth framework of the first paper
discussed in Sect. 1.1.7. The authors consider a multisector economy with one
consumable capital good and n pure capital goods produced without fixed factors
from Cobb-Douglas technologies. The economy is characterized by sector-specificexternalities and constant social returns. Instantaneous utility is homogeneous witha constant intertemporal elasticity of substitution. Under some conditions on the

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matrices of private and social input coefficients, local indeterminacy exists. In theparticular case of a two-sector endogenous growth model, these conditions imply
that the consumable capital good is intensive in the pure capital good from theprivate perspective, but it is intensive in itself from the social perspective. Contrary
to the case of exogenous growth, this result holds for any value of the elasticity ofintertemporal substitution in consumption.
Most of the papers in this collection considering multisector economies withproductive external effects assume sector-specific externalities, and show that acapital intensive consumption good sector at the private level is a necessary
condition for the occurrence of local indeterminacy. However, such a formulationis quite different from the one initially considered by Romer (1986) and Lucas(1988), where global externalities are assumed. In this case, some additional
inter-relationships between the sectors are introduced, and these new mechanisms
may provide different channels through which local indeterminacy can arise. The
second paper of this section, “Global externalities, endogenous growth and sunspot
fluctuations,” published with Harutaka Takahashi and Alain Venditti in Advanced
Studies in Pure Mathematics in 2009, considers a two-sector economy with CobbDouglas technologies, labor-augmenting global external effects, and increasingsocial returns. The paper focuses on the local indeterminacy of endogenous growth
paths. After proving the existence of a balanced growth path, existence of sunspot
fluctuations is shown to be compatible with a labor intensive consumption good
sector at the private level, provided that elasticity of intertemporal substitution in
consumption admits intermediary values. In particular, the existence of period-two
cycles requires precisely such a capital intensity configuration.
The last paper of this collection, “A homoclinic bifurcation and global indeterminacy of equilibrium in a two-sector endogenous growth model,” publishedwith Paolo Mattana and Tadashi Shigoka in the International Journal of Economic
Theory in 2009, also focuses on the consequences of global externalities, discussed
in the context of an endogenous growth model similar to that of Lucas (1988). In the
Lucas model, an external effect appears in the physical-goods sector. In this paper,
it appears in the educational sector. Contrary to the Lucas formulation, this external
effect yields multiple balanced growth paths. Moreover, the model undergoes a
global homoclinic bifurcation, and exhibits global indeterminacy of equilibrium.

1.1.9 Summary
This selection of Kazuo Nishimura’s papers, since the contribution published in
the Journal of Economic Theory in 1979, provides a precise description of theusefulness of nonlinear dynamics for the analysis of infinite-horizon intertemporal
optimization models. As co-authors and friends, it is our pleasure to show how
Kazuo’s works contributed to the development of nonlinear dynamics in economic
theory. We look forward to seeing even further progress along the lines of theseselected papers in the coming years.

1 Introduction

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1.2 List of Papers
The selected papers of Kazuo Nishimura chosen for this volume are as follows:

Introductory Survey:
1. “Nonlinear Dynamics in the Infinite Time Horizon Model,” (with Gerhard
Sorger), Journal of Economic Surveys, Vol. 13, 619–652, 1999

Optimal Growth and Endogenous Cycles:
2. “The Hopf Bifurcation and the Existence and the Stability of Closed Orbits in
Multi-Sector Models of Economic Growth,” (with Jess Benhabib), Journal of
Economic Theory, Vol. 21, pp. 421–444, 1979
3. “Competitive Equilibrium Cycles,” (with Jess Benhabib), Journal of Economic
Theory, Vol. 35, pp. 284–306, 1985
4. “Interlinkage in the Endogenous Real Business Cycles of International
Economies,” (with Makoto Yano), Economic Theory, Vol. 3, pp. 151–168, 1993

Optimal Growth and Chaotic Dynamics:
5. “Nonlinear Dynamics and Chaos in Optimal Growth: An Example,” (with
Makoto Yano) Econometrica, 63, 981–1001, 1995
6. “Chaotic Solutions in Dynamic Linear Programming,” (with M. Yano), Chaos,
Solitons and Fractals, 7, 1191–1953, 1996
7. “On the Least Upper Bound of Discount Factors that are Compatible with
Optimal Period-Three Cycles,” (with Makoto Yano), Journal of Economic
Theory, 66, 306–333, 1996

Global Dynamics in Optimal Growth Models:
8. “Intertemporal Complementarity and Optimality: A Study of a Two-Dimensional
Dynamical System,” (with Tapan Mitra), International Economic Review,Vol. 46, No. 1, pp. 93–131, 2005

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Dynamic Models with Non-concave Technologies:
9. “A Complete Characterization of Optimal Growth Paths in an Aggregated Modelwith a Non-concave Production Function,” (with W. Davis Dechert), Journal of
Economic Theory, Vol. 31, pp. 332–354, 1983

Stochastic Optimal Growth Models:
10. “Stochastic Optimal Growth with Nonconvexities,” (with Ryszard Rudnicki
and John Stachurski), Journal of Mathematical Economics, Vol. 42(1),pp. 74–96, 2006
11. “Stability of Stochastic Optimal Growth Models: a New Approach,” (with John
Stachurski), Journal of Economic Theory, Vol.22, pp. 100–118, 2005

Indeterminacy and Externalities in Exogenous Growth Models:
12. “Indeterminacy and Sunspots with Constant Returns,” (with Jess Benhabib),
Journal of Economic Theory, Vol. 81, pp. 58–96, 1998
13. “Trade and Indeterminacy in a Dynamic General Equilibrium Model,” (withKoji Shimomura), Journal of Economic Theory, Vol. 105, pp. 244–259, 2002
14. “Indeterminacy in discrete-time infinite- horizon models with nonlinear utility
and endogenous labor,” (with Alain Venditti), Journal of Mathematical Economics, Vol. 43(3–4), pp. 446–476, 2007

Indeterminacy and Externalities in Endogenous Growth Models:
15. “Indeterminacy Under Constant Returns to Scale in Multisector Economies,”(with Jess Benhabib and Qinglai Meng), Econometrica, Vol. 68, pp. 1541–48,
2000
16. “Global Externalities, Endogenous Growth and Sunspot Fluctuations,” (withHarutaka Takahashi and Alain Venditti), Advanced Studies in Pure Mathematics, Vol. 53, pp. 227–238, 2009
17. “A Homoclinic Bifurcation and Global Indeterminacy of Equilibrium in a
Two-Sector Endogenous Growth Model,” (with Paolo Mattana and Tadashi
Shigoka), International Journal of Economic Theory Vol. 5, pp. 25–60, 2009
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