\documentclass[12pt]{article}%
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}%
\setcounter{MaxMatrixCols}{30}
%TCIDATA{OutputFilter=latex2.dll}
%TCIDATA{Version=4.10.0.2345}
%TCIDATA{CSTFile=LaTeX article (bright).cst}
%TCIDATA{Created=Friday, May 16, 2003 16:35:59}
%TCIDATA{LastRevised=Thursday, January 08, 2004 14:40:41}
%TCIDATA{}
%TCIDATA{}
%TCIDATA{Language=American English}
\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}{Acknowledgement}
\newtheorem{algorithm}{Algorithm}
\newtheorem{axiom}{Axiom}
\newtheorem{case}{Case}
\newtheorem{claim}{Claim}
\newtheorem{conclusion}{Conclusion}
\newtheorem{condition}{Condition}
\newtheorem{conjecture}{Conjecture}
\newtheorem{corollary}{Corollary}
\newtheorem{criterion}{Criterion}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{exercise}{Exercise}
\newtheorem{lemma}{Lemma}
\newtheorem{notation}{Notation}
\newtheorem{problem}{Problem}
\newtheorem{proposition}{Proposition}
\newtheorem{remark}{Remark}
\newtheorem{solution}{Solution}
\newtheorem{summary}{Summary}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\begin{document}
\title{Factor Intensity Reversal and Chaos \thanks{We wish to thank seminar
participants at Delhi School of Economics for comments and suggestions. Odile
Poulsen acknowledges financial support from the Danish research Council (SSF),
\textquotedblleft project number 212.2269.01\textquotedblright. }}
\author{Aditya Goenka\thanks{Correspondence to A. Goenka}\\Department of Economics\\University of Essex\\Wivenhoe Park\\Colchester CO4 3SQ\\United Kingdom \\EMail: ecsadity@nus.edu.sg \& \\goenka@essex.ac.uk
\and Odile Poulsen\\Department of Economics\\Aarhus School of Business \\Silkeborgvej 2\\8000 Aarhus C\\Denmark\\EMail: odp@asb.dk}
\maketitle
\begin{center}
{\large Preliminary Version }
\end{center}
\textbf{Abstract}: We derive necessary and sufficient conditions for the
occurrence of ergodic oscillations and geometric sensitivity in a two-sector
model of economic growth with labor augmenting externalities. We transform the
Euler equation into a first order backward first order equation. Factor
intensity reversal is a necessary condition for the dynamics to be chaotic,
both in the sense of ergodic oscillations and geometric sensitivity when
utility is linear. Under reasonable assumptions on the economic fundamentals,
we show that a necessary and sufficient condition for the occurrence of
ergodic oscillations and geometric sensitivity is that the representative
consumer is sufficiently patient.\bigskip
\textbf{Keywords: }labor-augmenting externalities, backward dynamics, factor
intensity reversal, ergodic oscillations, geometric sensitivity\bigskip
JEL Classification: C61, D90, O41\newpage
\section{Introduction}
As is documented by a number of surveys (see for instance Baumol and Benhabib
(1989), Day and Panigiani (1991), Boldrin and Woodford (1990)) equilibrium
trajectories can exhibit chaotic behavior in optimal growth models where
agents have perfect foresight.
The contribution of this paper is to add further results to this literature.
We show that:
(i) Chaotic dynamics can arise in a model of economic growth for a wider class
of productions functions than the ones used in the literature so far.
(ii) Chaotic dynamics can occur for low levels of impatience in general
two-sector economic growth models.
(iii) When utility is linear \textit{factor intensity reversal} is a necessary
condition for chaotic motion.
(iv) When utility is concave, ergodic oscillations can occur for different
configuration of the factor intensity. We show that ergodic oscillations occur
both when the consumption good sector is capital intensive for all values of
the growth factor, or when factor reversal occurs.
We use the same analytical framework as in Drugeon and Venditti (1998),
Drugeon, Poulsen and Venditti (2003) and Goenka and Poulsen (2002). In a
two-sector model sector 1 produces a pure consumption good and sector 2
produces a pure investment good. In Sections 1 to 3 utility is linear and the
production functions in both sectors are homogenous of degree one with respect
to the private inputs. In addition to the labor and capital inputs provided by
the representative consumer, each sector uses the aggregate capital
stock.\ This provides\ a positive externality a la\ \textit{Harrod-Neutral} in
the production of both sectors through learning by doing. In Section 4 we
enlarge the class of utility function to the CES family.
When utility is linear, we show that for the law of motion of the growth
factor of capital to be \textquotedblleft tent-shaped\textquotedblright\ or
\textit{unimodal} it is necessary that the capital labor ratios between
sectors change at least once over the set of feasible growth factors (this is
called \textit{factor intensity reversal (}see Benhabib and Nishimura (1985)).
When utility is concave, a sufficient condition for unimodality is that the
consumption good sector is capital intensive.
Whether utility is linear or concave, we show that a necessary and sufficient
condition for the law of motion to be chaotic in the sense of \textit{ergodic}
oscillations and \textit{geometric sensitivity} (GS) is that the rate of
impatience of the representative consumer is not too high.
The result in (i) generalizes the results obtained by Boldrin et al. (2001),
Nishimura and Yano (1995,2000) and Nishimura et al.(1994). This is because in
these papers the production functions in both sectors are specific\footnote{In
Nishimura and Yano (1995,2000) and in Nishimura et.al. (1994) the production
functions in both sectors are Leontief. In Boldrin et.al. (2001) the
production function in the consumption good sector is Cobb-Douglas and the
production function in the investment good sector is linear.}, whereas in our
paper we work with general functional forms. All we need to assume is that the
production functions in both sectors are homogenous of degree one with respect
to private inputs and that external effects are labor augmenting.
The result in (ii) is similar to the results obtained by Nishimura and Yano
(1995,2000) and Nishimura et al. (1994). They derive sufficient conditions on
the parameters of a family of optimal growth models that give rise to an
ergodic chaotic optimal policy function for all discount factor inside the
unit interval. The differences between their results and ours are the
following. First, we derive an upper bound on the rate of impatience of the
representative consumer. The Transversality Condition (TVC) requires that the
discount factor\footnote{The discount factor can be rewritten as
$\beta=1/(1+\rho),$ where $\rho>0$ is the rate of time preference or the rate
of impatience of the representative consumer.} must be lower than the inverse
of the maximum feasible growth factor. If growth is bounded the model exhibits
chaos both in the sense of ergodic oscillations and GS for all discount
factors arbitrarily close to unity. If growth is unbounded, there exists, as
in Boldrin et al. (2001), an inverse relationship between discounting and
growth. As long as the discount factor is inside the range of values allowed
by the TVC, ergodic oscillations and GS can be obtained without further
restrictions on the level of impatience of the representative consumer. In
Boldrin et al. (2001) the upper bound on the discount rate can be lower than
the bound derived in our paper for certain range of the external effects.
Second, we derive an upper bound on the rate of impatience is related to the
underlying economic fundamentals. Complicated dynamics emerge in our model
even for levels of impatience at which the turnpike theory predicts convergence.
The result in (iii) and (iv) allows us to establish a link between the results
of Boldrin et al. (2001), Nishimura and Yano (1995,2000), Nishimura et al.
(1994) and the earlier literature on chaos. In all these papers the capital
intensity in both sectors are arbitrarily chosen. The unimodal map is also
derived from specific functional forms when no factor intensity reversal takes
place between sectors. In the earlier literature factor intensity reversal is
a necessary condition for the emergence of topological chaos, as shown by
Deneckere and Pelikan (1986), Boldrin (1989), Boldrin and Deneckere (1990).
However, the so-called "minimum impatience theorems" (Mitra (1996) and
Nishimura and Yano (1996), Sorger (1992, 1994)) show that topological chaos is
not feasible in optimal growth models unless the representative consumer
discounts future utilities very heavily. In our paper we establish that factor
intensity reversal is a necessary condition for ergodic oscillations when
utility is linear or when the investment good sector is capital intensive for
some values of the growth factor. Furthermore, chaotic motion occurs in our
model if the representative consumer does not discount future utilities too
heavily. Hence ergodic oscillations and geometric sensitivity can not be
theoretically ruled out if factor reversal takes place and the representative
consumer is sufficiently patient.
The rest of the paper is organized as follows. In Section 2 we present the
model and give some mathematical definitions. Section 3 states the results on
chaos known in the litearture. In Section 4 we derive sufficient conditions
for the existence of chaos in our model. Section 5 analyzes the case with
non-linear utility. We summarize the results in section 6. The proofs of the
main results can be found in Section 7.
\section{The Model}
Our model is a two--sector growth model (Uzawa (1964), Srinivasan (1964)) with
Harrod-Neutral externalities (Uzawa (1961)). Indeterminacy in this has been
examined by Drugeon and Venditti (1998), Drugeon, Poulsen and Venditti (2003)
and Goenka and Poulsen (2003). Consumers are indexed by $h$ and are
distributed along the unit interval. The utility function of the
representative consumer is linear. \bigskip
\noindent$\mathbf{Assumption}$ $\mathbf{1:}$ $u(c_{t})=c_{t},$ where $c_{t}$
is per capita consumption at time $t$.
\bigskip Consumers discount lifetime utility by the discount factor $\beta$,
where, $0<\beta<1$. The discount factor can also be expressed in terms of the
rate of time preference, $\rho$, as $\beta= \displaystyle\frac{1}{1 + \rho}$.
A consumer with a higher rate of time preference, thus, discounts the future
at a higher rate or is more impatient. Each consumer $h\in\lbrack0,1]$ is
initially endowed with an equal fraction of the aggregate capital stock
$\overline{k}.$ Consumers also supply a single unit of labor inelastically.
This labor is allocated between the two productive sectors of the economy.
Given the normalization assumptions, the aggregate labor supply in the economy
is unity.
There are two productive sectors in the economy. Sector 1 produces the
consumption good, $c_{t}.$ Sector 2 produces the investment good, $y_{t}$.
Inputs, capital, $k$, and labor, $l$, are freely mobile between sectors.
Capital is assumed to depreciate fully in each period. Thus, $k_{t+1} = y_{t}%
$. Market clearing in the two sectors is given by:%
\begin{align*}
k_{t} & =k_{t}^{1}+k_{t}^{2},\\
1 & =l_{t}^{1}+l_{t}^{2},
\end{align*}
where $k_{t}^{i}$ and $l_{t}^{i}$ denotes the amount of private inputs used in
sector, $i=1,2.$ We will omit the time subscripts whenever they are not
necessary. The stationary production functions in the two sectors are given
by:
\begin{equation}
c=F^{1}(k^{1},l^{1},X),\text{ }y=F^{2}(k^{2},l^{2},X). \label{2}%
\end{equation}
The production of output in sector $i$ requires the use of non negative
amounts of private inputs. The productivity of the private inputs is affected
by the aggregate capital stock $X,$ where $X=\int_{0}^{1}k(h)dh$. Thus, it is
a two-sector version of the learning-by-doing model (Arrow (1961), Sheshinski
(1967), Romer (1986)). We make the standard assumption that for a given level
of the aggregate capital stock the marginal productivities of both inputs are
positive and there are diminishing marginal productivities in private inputs.
Furthermore, we assume that both inputs are necessary. \bigskip
\noindent$\mathbf{Assumption\ 2}$: For $i = 1, 2$, $F^{i}: \Re_{+}%
^{3}\rightarrow\Re_{+}$, are continuous functions. For a given $X\in\Re_{+}$:
\medskip
$(i)$ $F^{i}(.,.,.)$ is $C^{3}$ on $\Re_{++}\times\Re_{++}\times\Re_{+}$;
$(ii)$ $F^{i}(.,.,X)$ is homogenous of degree one and increasing over
$\Re_{++}\times\Re_{++};$
$(iii)$ $F_{11}^{i}(.,l^{i},X)<0,$ for all $k^{i}\in\Re_{++}$ and
$\underset{k^{i}\rightarrow0}{\lim}F_{1}^{i}(k^{i},l^{i},X)=\infty;$
$(iv)$ $F_{22}^{i}(k^{i},.,X)<0,$ for all $l^{i}\in(0,1]$ and $\underset
{l^{i}\rightarrow0}{\lim}F_{2}^{i}(k^{i},l^{i},X)= \infty.$\bigskip
While we work with general production functions, the key restriction in our
model is the nature of the externality.$\bigskip$
\noindent$\mathbf{Assumption\ 3}$ $(Harrod-Neutrality):$ External effects in
sector $i $ are Harrod-Neutral (labor augmenting):
\[
F^{i}(k^{i},l^{i},X)=\mathcal{F}^{i}(k^{i},l^{i}X),\text{ where }i=1,2,
\]
\noindent where $\mathcal{F}^{i}(.,.)$ is homogenous of degree 1 in
$\mathcal{\ }k^{i}$ and $l^{i}X.$\bigskip
We define the growth factor of capital $\gamma_{t}$ as $\gamma_{t}%
=k_{t+1}/k_{t}.$ Under Assumption 3, it follows that the maximum feasible
growth rate $\overline{\gamma}$ is equal to $\mathcal{F}^{2}(1,1).$ Because
capital depreciates fully in every period, the minimum feasible growth rate
$\underline{\gamma}$ is $0.$ Hence for all $0\leq k_{t+1}\leq\mathcal{F}%
^{2}(k_{t},X_{t}),$ $\gamma_{t}\in\lbrack0,\overline{\gamma}].$ To insure that
the TVC is satisfied we impose the following restriction\bigskip
\noindent$\mathbf{Assumption}$ $\mathbf{5}$: $\beta\mathcal{F}^{2}(1,1)
\equiv\beta\overline{\gamma}<1.$\bigskip
For unbounded growth we need to assume that $\overline{\gamma}>1.$ In that
case we see that the TVC will be satisfied for all $\beta\in(0,1/\overline
{\gamma}).$ For bounded growth $\overline{\gamma}<1$. In this case the TVC
holds for all $\beta\in(0,1).$
In order to guarantee that there exists a non trivial interior growth rate we
make the following assumption:\bigskip
\noindent$\mathbf{Assumption}$ $\mathbf{4}$:
$\beta\mathcal{F}_{1}^{2}(k^{2}(1,0,1),l^{2}(1,0,1))>(<)1$ and $\beta
\mathcal{F}_{1}^{2}(k^{2}(1,\overline{\gamma},1),l^{2}(1,\overline{\gamma
},1))<(>)1$\bigskip
The representative consumer's behavior is described by the following
optimization problem:%
\[
\underset{\{c_{t},,k_{t}^{1},l_{t}^{1},k_{t}^{2},l_{t}^{2},\}}{\max}\sum
_{t=0}^{\infty} \beta^{t}c_{t}
\]
subject to:%
\begin{align}
c_{t} & =\mathcal{F}^{1}(k_{t}^{1},l_{t}^{1}X_{t})\nonumber\\
y_{t} & =\mathcal{F}^{2}(k_{t}^{2},l_{t}^{2}X_{t}),\\
k_{t+1} & =y_{t}\\
k_{t} & =k_{t}^{1}+k_{t}^{2},\\
1 & =l_{t}^{1}+l_{t}^{2},\nonumber\\
k_{t}^{i} & \geq0,\text{ }l_{t}^{i}\geq0,\text{ }\{X_{t}\}_{t=0}^{\infty
}\text{, }k_{0}\text{ given}.\nonumber
\end{align}
The constraints can be collapsed using the production possibility frontier
(PPF from here on), $T(k,y,X)$, given by $c_{t}=\mathcal{F}^{1}(k_{t}%
-k^{2}(k,y,X),X_{t}(1-l^{2}(k,y,X))).$ Since utility is linear the above
maximization problem reduces to:\bigskip%
\begin{align}
& \underset{\{k_{t}\}}{\max}\sum_{t=0}^{\infty} \beta^{t}T(k_{t}%
,k_{t+1},X_{t})\label{PPF}\\
& \text{subject to}\nonumber\\
0 & \leq k_{t+1}\leq\mathcal{F}^{2}(k_{t},X_{t}),\nonumber\\
& \text{ }\{X_{t}\}_{t=0}^{\infty}, \text{ }k_{0}\;\text{given}.\nonumber
\end{align}
\
Let $k_{t}\{X_{t}\}_{t=0}^{\infty}$ denote the solution to this problem$.$ If
$\{k_{t}\}_{t=0}^{\infty}$ satisfies $k_{t}\{X_{t}\}_{t=0}^{\infty}=X_{t}$ for
all $t\geq0,$ then the path $\{k_{t}\}_{t=0}^{\infty}$ will be referred to as
equilibrium path. This fixed point problem is solved in a different framework
in detail by Romer (1983) and Mitra (1998). We do not address this issue here.
We assume for simplicity that there exists an equilibrium path $\{k_{t}%
\}_{t=0}^{\infty}$ such that $k_{t}\{X_{t}\}_{t=0}^{\infty}=X_{t}$ for all
$t\geq0.$
As we are working with general technologies, we make the following regularity
assumption\footnote{It is satisfied in the case of Cobb-Douglas production
functions, among others.}$\bigskip$
\noindent$\mathbf{Assumption\ 6}:T(k_{t},y_{t},X_{t})$ is \textbf{\ }$C^{3}$
on $\Re_{++}\times\Re_{++}\times\Re_{+}.\bigskip$
Following Boldrin et. al (2001), $\{k_{t}\}_{t=0}^{\infty}$ is an equilibrium
path if and only if the following conditions are satisfied
\begin{equation}
T_{2}(k_{t},k_{t+1},k_{t})+\beta T_{1}(k_{t+1},k_{t+2},k_{t+1})=0,
\label{Euler Equation 1}%
\end{equation}%
\begin{equation}
\underset{t\rightarrow\infty}{\lim}\beta^{t}k_{t}T_{1}(k_{t},k_{t+1},k_{t})=0,
\label{TVC 1}%
\end{equation}
%
\begin{equation}
\sum_{t=0}^{t}\beta^{t}T(k_{t},k_{t+1},k_{t})<\infty. \label{SUM}%
\end{equation}
Conditions (\ref{Euler Equation 1}) to (\ref{SUM}) are identical to conditions
(3.1)-(3.3) given in Boldrin et. al (2001).
The simplest dynamics in the model arise in a situation when capital grows at
a constant rate. The issue is whether there exists an interior solution with
such a property.
As the dynamics of the model are given by (\ref{Euler Equation 1}) to
(\ref{SUM}) we transform these to obtain a one dimensional difference equation
which will be more amenable to analysis.
For this to be the case, and that unbounded growth is possible, i.e., the
maximal feasible growth rate is greater than unity, we assume the
following\footnote{If $\mathcal{F}_{1}^{2}(k^{2}(1,1),l^{2}(1, 1)) < 1$ there
will still exist a BGP but there is only bounded growth}:\bigskip
\noindent$\mathbf{Assumption}$ $\mathbf{7:}$
\begin{enumerate}
\item $\beta\left[ \mathcal{F}_{1}^{2}(k^{2}(1,1),l^{2}(1, 1)) \right] >1,$
\item $\mathcal{F}^{2}(1,1)> 1$.
\end{enumerate}
\bigskip
An equilibrium path $\{k_{t}\}$ is a balanced growth path (BGP) if there
exists a growth factor $\gamma\in\lbrack1,\overline{\gamma}]$ such that for
all $t\geq0,$ $k_{t}=\gamma^{t}k_{0},$ where $k_{0}\neq0.$
\begin{proposition}
Let Assumptions 1-7 be satisfied. Then there exists an interior BGP with
unbounded growth, $\widetilde{\gamma}\in(1,\overline{\gamma}).$
\end{proposition}
\begin{proof}
See Goenka and Poulsen (2003).
\end{proof}
\bigskip
Under Harrod-Neutrality, Drugeon and Venditti (1998) show that the input
demand functions $k^{i}(k_{t},y_{t},X_{t})$ and $l^{i}(k_{t},y_{t},X_{t}),$
$i=1,2,$ are homogenous of degree $1$ and $0$ respectively and that
$T(k_{t},y_{t},X_{t})$ is homogenous of degree 1. Then using the homogeneity
properties of the production functions, Drugeon, Poulsen and Venditti (2003)
establish the following:
\begin{lemma}
\label{T21}Let Assumptions 2-3 be satisfied. Then, function of $T(k,y,X),$ is
homogenous of degree one. Furthermore if Assumption 6 holds then,
\begin{align}
T_{21} & =\frac{\mathcal{F}_{12}^{1}\mathcal{F}_{12}^{2}q\mathcal{F}%
^{2}l^{1}}{\Omega k^{2}k^{1}}\left( \frac{k^{1}}{l^{1}}-\frac{k^{2}}{l^{2}%
}\right) ,\label{t21}\\
T_{23} & =-T_{21}\frac{k^{1}}{l^{1}X}+\frac{2l^{2}}{\mathcal{F}_{1}^{2}%
}\left( \mathcal{F}_{12}^{1}+q\mathcal{F}_{12}^{2}\right) ,\label{t23}\\
T_{22} & =\frac{l^{2}}{\mathcal{F}^{2}}\left( \frac{k^{1}}{l^{1}}%
-\frac{k^{2}}{l^{2}}\right) T_{21}, \label{t22}%
\end{align}
where
\[
\Omega=-\frac{\mathcal{F}_{12}^{1}(\mathcal{F}^{1})^{2}(\mathcal{F}_{1}%
^{2})^{2}}{(\mathcal{F}_{1}^{1})^{2}k^{1}l^{1}X}-\frac{\mathcal{F}_{12}%
^{2}(\mathcal{F}^{2})^{2}\mathcal{F}_{1}^{1}}{\mathcal{F}_{1}^{2}k^{2}l^{2}%
X}<0.
\]
\end{lemma}
\begin{proof}
See Drugeon and Venditti (1998) and Drugeon, Poulsen and Venditti (2003).
\end{proof}
\bigskip
We see from this Lemma that the sign of $T_{21}$ depends on the factor
intensity. With factor intensity reversal, $T_{21}$ will change sign. This
result can also be found in\ Benhabib and Nishimura (1985) in a two-sector
optimal growth model with no external effects. As we shall see below factor
intensity reversal is a necessary and sufficient condition for the law of
motion of the growth factor to be unimodal. Deneckere and Pelikan (1986),
Boldrin (1989) and Boldrin and Deneckere (1990) establish that factor
intensity reversal is a necessary condition for the occurrence of topological chaos.
In the model, there can in general be several capital intensities where factor
intensity reversal takes place. We make the simplifying assumption that there
exists a unique value of the growth factor $\widehat{\gamma},$ where
$0<\widehat{\gamma}<\overline{\gamma},$ at which factor reversal takes
place\footnote{One can relax this assumption and still show that $\theta$ is
chaotic in the sense of ergodic oscillations and GS. To do so, one would need
to (i) establish that $\theta$ is piecewise strictly monotonic, piecewise
$C^{2}$and expansive on the following intervals $(0,\gamma_{1}),(\gamma
_{1},\gamma_{2}),...$and $(\gamma_{n},\overline{\gamma}),$ where $\gamma_{j}$
$,j=1,...n,$ denotes a growth factor at which no factor reversal occurs$.(ii)$
Show that $\theta(\widehat{\gamma_{j}}),$ $,j=1,...n,$ is a turning point.
Each $\widehat{\gamma_{j}}$ $,j=1,...n,$ is the growth factor at which the
$ith$ factor reversal occurs and such that $\gamma_{j-1}<\widehat{\gamma_{j}%
}<\gamma_{j+1}.$ For more details for this case, see Theorem 8.5 and
Corollaries 8.3 and 8.4 in Day (1994) .}. \bigskip
$\mathbf{Assumption}$ $\mathbf{8}$: \textit{(Unicity of factor intensity
reversal)} There exists a unique $\widehat{\gamma}\in(0,\overline{\gamma}) $
such that%
\[
\frac{k^{1}(1,\widehat{\gamma},1)}{l^{1}(1,\widehat{\gamma},1)}=\frac
{k^{2}(1,\widehat{\gamma},1)}{l^{2}(1,\widehat{\gamma},1)}.
\]
\bigskip
Under \textit{Harrod -Neutrality}, the set of feasible growth factors can be
reduced to $[0,\overline{\gamma}]$ with $\overline{\gamma}=\mathcal{F}%
^{2}(1,1).$ We want to represent the general forward dynamics as a first order
nonlinear difference equation,%
\begin{equation}
\gamma_{t+1}=\tau(\gamma_{t}) \label{FOD1}%
\end{equation}
or, in backward dynamics:%
\begin{equation}
\gamma_{t}=\theta(\gamma_{t+1}). \label{FOD2}%
\end{equation}
The homogeneity properties of the PPF imply that the left hand side of
equation (\ref{Euler Equation 1}) can be rewritten as
\begin{equation}
\psi(\gamma_{t},\gamma_{t+1})\equiv T_{2}(1,\gamma_{t},1)+\beta T_{1}%
(1,\gamma_{t+1},1) \label{Euler Equation 3}%
\end{equation}
or, as%
\begin{equation}
\psi(\gamma_{t},\gamma_{t+1})=0. \label{EE4}%
\end{equation}
From (\ref{Euler Equation 3}), we can define the following two functions:%
\begin{align}
f(\gamma_{t+1}) & =\beta T_{1}(1,\gamma_{t+1},1),\label{DEFF}\\
g(\gamma_{t}) & =-T_{2}(1,\gamma_{t},1)\nonumber
\end{align}
and hence, (\ref{Euler Equation 3}) can be written as
\begin{equation}
-g(\gamma_{t})+f(\gamma_{t+1})=0. \label{Euler Equation 4}%
\end{equation}
To obtain a first order difference equation such as (\ref{FOD1}) or
(\ref{FOD2}) either $g$ or $f$ has to have a well defined one-to-one inverse
on $[0,\overline{\gamma}].$ As we can see from (\ref{t21}), $T_{12}$ changes
sign if factor intensity reversal takes place at $\widehat{\gamma}$. Hence $f$
does not have a well defined one-to-one inverse. The same is not true for the
existence of $g^{-1}.$
\begin{proposition}
\label{theta} Let Assumptions 1-6 and 8 be satisfied. Then there exists a
$1-1$ map $\theta:[0,\overline{\gamma}] \rightarrow[0,\overline{\gamma}]$ such
that
\begin{equation}
\gamma_{t}=\theta(\gamma_{t+1}). \label{FOD3}%
\end{equation}
\end{proposition}
\begin{proof}
See the Appendix \ \bigskip
\end{proof}
Grandmont (1985) also studies backward foresight dynamics in an overlapping
generations model. He faces the same problem about the non-existence of the
inverse of $f$. In his model, the excess demand function of the young
consumers is a unimodal function if the Arrow-Pratt relative degree of risk
aversion of the old is a non-decreasing function of their wealth.
\section{Results on chaos}
To make the discussion self contained, some standard results and definitions
on the theory of chaos in dynamical systems are collected here (see Collet and
Eckmann (1980), Day (1994) and Eckmann and Ruelle (1985) for more details). We
focus on chaos in the sense of ergodic oscillations and geometric sensitivity
(GS). In other words, law of motion is generated by a nonlinear difference
equation with no random terms such that \textit{(i)} the long run behavior of
the system is characterized by extremely complicated aperiodic dynamics,
\textit{(ii)} small differences in the initial conditions are magnified at a
geometric rate for arbitrary finite lengths of time (GS) and \textit{(iii)}
the time average of the orbit can be replaced by the space average (ergodic chaos).
A dynamical system is a pair $(I,\theta)$ where $I$ is a compact interval on
the real line and is called the state space and $\theta$ a function describing
the law of motion of the state variable $x\in I$. Thus, if $x_{t}$ is the
state of the system in period $t,$ then $x_{t+1}=\theta(x_{t})$ is the state
of the system in period $t+1$. If we denote by $x_{0}$\ the initial state of
the system, and $\theta^{0}(x)=x$\ for all $x\in I$\, then $\theta
^{t+1}(x)=\theta(\theta^{t}(x))$\ for all $t\geq0$\ and all $x\in I $\ where
$\theta^{t}$\ is the $t-th$\ iterate of $\theta$, $t = 0,1,2,...$
The notion of chaos that is often used in the economics literature is that of
topological chaos. In many models this is easy to establish using the theorem
of Li and Yorke (1975) that a ``cycle of period 3 implies chaos.'' (also see
Mitra (2001) for a different approach to establishing topological chaos).
\begin{definition}
Let $\theta: I \rightarrow I$ define a dynamical system. We say that $\theta$
exhibits topological chaos if:
\begin{enumerate}
\item For every period $N$, there exist points $x_{N} \in I$ such that
$\theta^{N}(x_{N}) = x_{N}$.
\item There exists an uncountable set $S \subset I$ and an $\epsilon> 0$ such
that every pair $x, y$ in $S$ with $x\neq y$:\newline
\begin{enumerate}
\item $\lim_{n\rightarrow\infty} \sup\left\vert \theta^{n}(x) - \theta
^{n}(y)\right\vert \geq\epsilon$.
\item $\lim_{n\rightarrow\infty} \inf\left\vert \theta^{n}(x) - \theta
^{n}(y)\right\vert = 0.$
\item For every periodic point $z$ and $x \in S$: $\lim_{n\rightarrow\infty}
\sup\left\vert \theta^{n}(x) - \theta^{n}(z)\right\vert \geq\epsilon$.
\end{enumerate}
\end{enumerate}
\end{definition}
While topological chaos can often be easy to show, $S$ may have Lebesgue
measure zero. For example, for the quadratic map $\theta(x) = \mu x(1-x)$ with
$\mu\in[1,4]$, there is topological chaos if $\mu= 3.828427$, but almost all
initial conditions lead to a cycle of period 3.
A different notion of chaos is the following.
\begin{definition}
(Nishimura and Yano (2000)). The dynamical system $(I,\theta),$ exhibits
Geometric Sensitivity (GS) if there exists a constant $h>1$ such that for any
$\tau\geq0$ there exists $\varepsilon>0$ such that for all $x$ and $x^{\prime
}\in I$ with $\left\vert x_{t}-x_{t}^{\prime}\right\vert <\varepsilon$ and for
all $t\in\left\{ 0,1,...,\tau\right\} $%
\[
\left\vert \theta^{t}(x)-\theta^{t}(x^{\prime})\right\vert \geq h^{t}%
\left\vert \gamma-\gamma^{\prime}\right\vert .
\]
\end{definition}
As $I$ is bounded, the geometric magnification of the effects of a small
perturbation cannot last indefinitely. Furthermore, the dynamical system
$(I,\theta)$ has no locally stable cyclical path.
There is also the notion of \textit{ergodic chaos}. Ergodic chaos is a
stronger property than topological chaos in the sense that it is
\textquotedblleft observable chaos\textquotedblright.
Let $\Upsilon$ be a $\sigma$-algebra on I. \footnote{A $\sigma-$algebra is a
collection of subsets $\Upsilon$ of $I$ such that $(i)$ $I$ is inside
$\Upsilon,$ $(ii)$ the complement of any set $Y$ included in $\Upsilon$ is
also in $\Upsilon,$ $(iii)$ the union of any countable collection of subsets
in $\Upsilon$ is inside $\Upsilon.$} Define a probability measure
$\mu:\Upsilon\rightarrow\Re^{+}$ such that $(i)$ $\mu(\varnothing)=0$ and
$(ii)$ $\mu(\cup_{n=0}^{\infty}Y_{n})=\Sigma_{n=0}^{\infty}\mu(Y_{n}),$
$(iii)$ $\mu(I)=1$, where $\{Y_{t}\}_{n=0}^{\infty}$ is a countable collection
of disjoint sets in $\Upsilon$. The probability measure $\mu$ is said to be
invariant with respect to $\theta$ if $\mu(\theta^{-1}(Y))=\mu(Y)$ for all
$Y\in\Upsilon.$ Invariant measures have an important property.
\begin{theorem}
\textbf{(Poincar\'{e} Recurrence Theorem:)} Let $\{I, \Upsilon, \mu\}$ be a
probability space and let $\mu$ be invariant under $\theta$. Let $Y$ be any
measurable set of positive measure. Then all points of $Y$ return to $Y$
infinitely often.
\end{theorem}
Consider $x \in Y$. Then, $x \in\theta^{-1}(x) \Rightarrow\theta(x) \in Y$.
Thus, the trajectory will stay in $Y$ forever. The invariant probability
measure $\mu$ is said to be ergodic if for some $Y\in\Upsilon$ then
$\theta^{-1}(Y)=Y$ implies either $\mu(Y)=0$ or $\mu(Y)=1.$ In other words,
the system cannot split into non-trivial parts. For ergodic measures, a
fundamental result is:
\begin{theorem}
(\textbf{The Birkhoff-von Neuman Mean Ergodic Theorem:)} Let $(I,\theta)$ be a
dynamical system. If $\mu$ is an invariant and ergodic probability measure
then, for any $\mu-$integrable function $g,$%
\begin{equation}
\underset{t\rightarrow\infty}{\lim}\frac{1}{t}\sum_{\tau=0}^{t-1}%
g(\theta^{\tau}(x))=\int_{I}gd\mu\label{MET}%
\end{equation}
for almost all $x\in I.$
\end{theorem}
The left hand side of (\ref{MET}) is the average value of $g\ $along the orbit
$\left\{ x,\text{ }\theta(x),...\right\} .$ The right hand side of
(\ref{MET}) is the expected value of $g$ evaluated on $I$. In other words, the
time averages along an orbit $\left\{ x,\text{ }\theta(x),...\right\} $ can
be replaced by space averages. Furthermore, according to this theorem, for
almost all $x\in I,$ an orbit $\left\{ x,\text{ }\theta(x),...\right\} $
will visit every measurable set proportionally to its measure. The system is
``chaotic'' if the support of the measure is a ``large set.'' However, the
measure could be concentrated on a point as in the case of a fixed point, or
on a finite subset of points in the case of a cycle. Absolute continuity of a
measure avoids this problem.
\begin{definition}
The probability measure $\mu$ is \textit{absolutely continuous} with respect
to the Lebesgue measure $\mathcal{L}$ on $I$ if there exists an integrable
function $f$ such that
\[
\mu(Y)=\int_{Y}fdm
\]
for all measurable sets $Y$ in $\sigma-$algebra $\Upsilon.$
\end{definition}
Absolute continuity implies that for all $Y\in\Upsilon,$ if $\mathcal{L}(Y)=0$
then $\mu(Y)=0.$ Thus, the support of $\mu$ cannot be a set of measure zero.
We can now define the concepts of ergodic chaos in the following way:
\begin{definition}
The dynamical system $(I,\theta)$ exhibits ergodic chaos if there exists a
probability measure $\mu$ on $I$ which is absolutely continuous, invariant and ergodic.
\end{definition}
Lasota and Yorke (1973) establish that if $\theta$ is a \textit{piecewise}
$C^{2}$ and \textit{expansive} mapping then there exists an absolutely
continuous invariant measure
\begin{definition}
A mapping $\theta$ defined on $[a,b]$ is piecewise $C^{2}$ and expansive if:
\begin{enumerate}
\item There exists a finite set $x_{0}=a1$ for all $x\in
(x_{j,}x_{j+1}).$
\end{enumerate}
\end{definition}
Li and Yorke (1978) show that if $\theta$ is also a unimodal map then this
measure is ergodic.
\begin{definition}
Assume there exits a constant $c\in\lbrack a,b],$ $a0$ for all $\gamma_{t}\in(1,\overline{\gamma}).$\newline
\end{lemma}
\begin{proof}
See the Appendix.\bigskip
\end{proof}
As we can see from (\ref{t21}) that when the consumption good sector is more
capital intensive for all $\gamma_{t}\in\lbrack0,\overline{\gamma}],$ then
$T_{12}<0.$ So, if $V_{21}$ changes sign at some value of $\gamma\in
\lbrack0,\overline{\gamma}],$ then $f$ does not have a well defined inverse.
The same is true if one assumes that factor intensity reversal takes place. We
show that $g^{-1}$ exists since $V_{2}$ is one to one.
\begin{lemma}
\label{thetanl}Let Assumptions 1b-5b be satisfied. Then there exists a map
$\theta$ defined from $[0,\overline{\gamma}]$ onto $[0,\overline{\gamma}],$
such that
\begin{equation}
\gamma_{t}=\theta(\gamma_{t+1}). \label{FOD3 nl}%
\end{equation}
\end{lemma}
\begin{proof}
See the Appendix.\bigskip
\end{proof}
The next step is to check that $\theta$ is unimodal.
\subsubsection{Unimodality of $\theta$}
To establish the unimodality of $\theta$ we need to show $\theta$ is
continuous on $[0,\overline{\gamma}],$ strictly monotonically increasing on
some open interval $(a,b)\subset\lbrack0,\overline{\gamma}]$ and strictly
monotonically decreasing on some interval $(b,c)\subset\lbrack0,\overline
{\gamma}].$ We first make the following assumption\footnote{Assumption 9 holds
for a simple economic example. If $c=(k^{1})^{\sigma}(l^{1}X)^{1-\sigma}$ and
$y=(k^{2})^{1/2}(l^{2}X)^{1/2}$ then $\widehat{\gamma}=1.$ In this case it can
be shown that if $k^{1}/l^{1}>k^{2}/l^{2}$ there exists $\overset{0}{\gamma
}\in\lbrack0,\overline{\gamma}]$ such that V$_{21}(1,\overset{0}{\gamma
},1)=0.$ Full request of this statement can be obtained on request.}%
:$\bigskip$
$\mathbf{Assumption}$ $\mathbf{9:}$ There exists $\overset{0}{\gamma},$ where
$0<\overset{0}{\gamma}<\overline{\gamma},$ such that $V_{21}(1,\overset
{0}{\gamma},1)=0.\bigskip$
It can then be shown that the sign of $\theta^{^{\prime}}(\gamma_{t})$ depends
on the sign of $V_{12}.$
\begin{lemma}
\label{unimodality nl}Let Assumptions 1b-6 and 9 be satisfied. Let
$\theta:[0,\overline{\gamma}]\rightarrow\lbrack0,\overline{\gamma}]$ be
defined as in (\ref{FOD3 nl}). Then, $\theta$ is unimodal if one of the
following holds.\newline(i) Factor reversal occurs at least once.\newline(ii)
For all $\gamma_{t}\in(0,\overline{\gamma})$ the consumption good sector is
capital intensive$.$
\end{lemma}
\begin{proof}
See the Appendix.\bigskip
\end{proof}
In other words, under \textit{unicity of factor intensity reversal, }%
$\theta^{^{\prime}}(\gamma_{t})$ is no longer a necessary and sufficient
condition for the unimodality of $\theta.$ It is sufficient for unimodality
that the consumption good sector is capital intensive for all $\gamma_{t}%
\in\lbrack0,\overline{\gamma}]$ and that Assumption 10 is satisfied$.$ This
result is the same as the one obtained in Nishimura and Yano (1995,2000) and
Nishimura et.al. (1994).
We show next that $\theta$ is expansive.
\subsubsection{Expansiveness of $\theta,$ Ergodic oscillations and Geometric
Sensitivity}
In this section we show that if the discount factor is bounded below then the
slope of $\theta$ is everywhere greater than unity for all feasible values of
the growth factor$.$ We then establish the main result on ergodic oscillations
and geometric sensitivity.
\begin{lemma}
\label{Expantheta nl}Let Assumptions 1b--5b and 9 be satisfied. Then a
necessary condition for $\theta:[0,\overline{\gamma}]\rightarrow
\lbrack0,\overline{\gamma}]$ to be expansive is
\begin{equation}
\left\vert \frac{V_{21}(1,\theta(\gamma_{t+1}),1)+V_{23}(1,\theta(\gamma
_{t+1}),1)}{\gamma_{t}^{\alpha}V_{21}(1,\gamma_{t+1},1)}\right\vert <1\text{
for all }\gamma_{t+1}\in(0,\widehat{\gamma})\cup(\widehat{\gamma}%
,\overline{\gamma}). \label{Exp1 nl}%
\end{equation}
A necessary and sufficient condition for $\theta:[0,\overline{\gamma
}]\rightarrow\lbrack0,\overline{\gamma}]$ to be expansive is that $\beta
\in(\beta_{\min},\beta_{\max})$, where%
\begin{align*}
\beta_{\min} & =\underset{\gamma_{t+1}\in(0,\widehat{\gamma})\cup
(\widehat{\gamma},\overline{\gamma})}{\max}\left\vert \frac{V_{21}%
(1,\theta(\gamma_{t+1}),1)+V_{23}(1,\theta(\gamma_{t+1}),1)}{V_{21}%
(1,\gamma_{t+1},1)}\right\vert ,\\
\beta_{\max} & =\frac{1}{\left[ \mathcal{F}^{2}(1,1)\right] ^{\alpha}}.
\end{align*}
\end{lemma}
\begin{proof}
See the Appendix.\bigskip
\end{proof}
At the BGP a necessary condition for condition (\ref{Exp1 nl}) to be satisfied
is%
\begin{equation}
\frac{V_{23}(1,\widetilde{\gamma},1)}{V_{21}(1,\widetilde{\gamma},1)}<0.
\label{CBGP}%
\end{equation}
If either:
(i) the consumption good sector is capital intensive or,
(ii) factor intensity reversal takes place at $\widehat{\gamma},$%
\ $0<\widehat{\gamma}<\overline{\gamma},$ and the investment good sector is
capital intensive for $\gamma_{t}\in(0,\widehat{\gamma})$
then $T_{21}(1,\widetilde{\gamma},1)<0$ and $T_{23}(1,\widetilde{\gamma
},1)>0.$ Looking at (\ref{V21nl}) and (\ref{V23nl}) we see that (\ref{CBGP})
reduces to%
\[
(\alpha-1)T_{2}T_{1}<-TT_{21}\text{.}%
\]
\begin{proposition}
\label{EOGS2 nl}Let Assumptions 1b-5b and 6-10 be satisfied. $\theta
:[0,\overline{\gamma}]\rightarrow\lbrack0,\overline{\gamma}]$ is chaotic both
in the sense of ergodic oscillations and geometric sensitivity for all $t\leq
T,$ $T\in\Re_{+}^{\ast}$ if the following three conditions hold:\newline(i)
$\theta$ is unimodal,,\newline(ii) (\ref{Exp1 nl}) is satisfied for all
$\gamma_{t+1}\in(0,\widehat{\gamma})\cup(\widehat{\gamma},\overline{\gamma})$
and\newline(iii) The discount factor $\beta$ satisfies $\beta\in(\beta_{\min
},\beta_{\max}),$\newline\newline.
\end{proposition}
\begin{proof}
The results follows from the application of Theorem 2. We have established the
unimodality and expansiveness of $\theta$ in Lemmata \ref{unimodality nl} and
\ref{Expantheta nl}. \bigskip
\end{proof}
\section{Summary and Conclusion}
In our paper we establish first that factor intensity reversal is a necessary
condition for ergodic oscillations if utility is linear. We then show that, if
for some values of the growth factor the investment good sector is capital
intensive then factor intensity reversal is a necessary condition for ergodic
oscillations. With both preferences' specifications, chaotic motion is
feasible if the representative consumer does not discount future utilities too heavily.
We have established that factor intensity reversal is a necessary condition
for the occurrence of ergodic oscillations as in Deneckere and Pelikan (1984).
Boldrin (1989), Boldrin and Deneckere (1990). However, in contrast to a
so-called "minimum impatience theorems" (Mitra (1996) and Nishimura and Yano
(1996), Sorger (1992, 1994)) we show that chaos are feasible if the
representative consumer is sufficiently patient. Propositions \ref{EOGS2} and
\ref{EOGS2 nl} establish an upper and a lower bound on the discount factor.
The upper bound is imposed by the satisfaction of the TVC. In an economy with
endogenous growth, $\beta_{\max}$ is lower than one$.$ In an economy with no
endogenous growth, $\beta_{\max}=1.$ This result is consistent with the
results of Nishimura and Yano (1995, 2000) and Nishimura, et al. (1994). The
TVC imposes in this case a lower bound on the level of impatience of the
representative consumer only if growth is unbounded. This upper bound on
$\beta$ is however, lower than the one obtained in Boldrin et al. (2001) in a
model with endogenous growth.
Proposition \ref{EOGS2} also states that if the representative consumer is not
too impatient then the map $\theta$ is chaotic in the sense of ergodic
oscillations and for all $t\leq T.$
\section{Appendix\bigskip}
\textbf{Proof of Lemma \ref{theta}:}$\bigskip$
Let us define%
\begin{align}
f(\gamma_{t+1}) & =\beta T_{1}(1,\gamma_{t+1},1),\label{A1}\\
g(\gamma_{t}) & =-T_{2}(1,\gamma_{t},1), \label{A2}%
\end{align}
where $f:[0,\overline{\gamma}]\rightarrow J\sqsubseteq\Re_{+}^{\ast}$ and
$g:[0,\overline{\gamma}]\rightarrow I\sqsubseteq\Re_{+}^{\ast}.$\newline Using
(\ref{A1}) and (\ref{A2}), the Euler equation (\ref{Euler Equation 3}) can be
written as $-g(\gamma_{t})+f(\gamma_{t+1})=0.$ Assumptions 2, 3 and the
\textit{unicity of factor intensity reversal (}Assumption 7) imply that
$g^{\prime}>0$ for all $\gamma_{t}\in(0,\widehat{\gamma})\cup(\widehat{\gamma
},\overline{\gamma}).$ Furthermore, Assumption 7 implies there is a unique
$\widehat{\gamma},$ $0<\widehat{\gamma}<\overline{\gamma}$ such that
$g^{\prime}(\widehat{\gamma})=0$. Hence $g$ is one to one. From the definition
of $g$ we see that range$(g)=I.$ So $g$ has a well defined inverse $g^{-1}.$
We can rewrite (\ref{EE4}) as $\gamma_{t}=\theta(\gamma_{t+1}),$ where
$g^{-1}\left[ f(\gamma_{t+1})\right] .$ From this definition, it follows
that $\theta:[0,\overline{\gamma}]\rightarrow\lbrack0,\overline{\gamma
}].\blacksquare\bigskip$
\textbf{Proof of Proposition \ref{unimodality}:}$\bigskip$
(i) Continuity of $\theta$. From the definition of $\theta$ given in Lemma
\ref{theta}, $\theta$ is continuous as the composite function of two
continuous functions.\newline(ii) Monotonicity of $\theta.$ Let Assumptions
1-9 be satisfied. Then using the inverse function theorem we can derive
$\theta^{^{\prime}}(\gamma_{t+1})$ on $(0,\widehat{\gamma})$ and
$(\widehat{\gamma},\overline{\gamma})$ as
\begin{equation}
\theta^{^{\prime}}(\gamma_{t+1})=-\dfrac{\beta T_{21}(1,\gamma_{t+1}%
,1)}{T_{22}(1,\gamma_{t},1)}. \label{theta0}%
\end{equation}
Upon inspection of (\ref{theta0}) we see that the sign of the denominator
depends on the sign of $-T_{22}$. Looking at (\ref{t22}), we see that under
Assumption 7 $-T_{22}<0$ for all $\gamma_{t}\in(1,\widehat{\gamma}%
)\cup(\widehat{\gamma},\overline{\gamma}).$ Hence $\theta^{^{\prime}}%
(\gamma_{t+1})\gtrless0$ if and only if $T_{12}(1,\gamma_{t+1},1)\gtrless0.$
Assumptions 7 and 9, together with the results of Lemma \ref{T21}, imply that%
\[
\theta^{^{\prime}}(\gamma_{t+1})>0\text{ for all }\gamma_{t+1}\in
(1,\widehat{\gamma})\text{and }\theta^{^{\prime}}(\gamma_{t+1})<0\text{ for
all }\gamma_{t+1}\in(\widehat{\gamma},\overline{\gamma}).
\]
$\blacksquare$\bigskip
\textbf{Proof of Proposition \ref{EOGS2}:}$\bigskip$
From Assumption 2 and the definition of $\theta$ it follows that $\theta$ is
twice continuously differentiable on both $(0,\widehat{\gamma})$ and
$(\widehat{\gamma},\overline{\gamma}).$ Then, $\theta$ is expansive if
$\left\vert \theta^{^{\prime}}(\gamma_{t+1})\right\vert >1$ for all
$\gamma_{t+1}\in(1,\widehat{\gamma})\cup(\widehat{\gamma},\overline{\gamma}).$
Or, equivalently, if%
\[
\left\vert -\frac{l^{2}}{\mathcal{F}^{2}}\left( \frac{k^{1}}{l^{1}}%
-\frac{k^{2}}{l^{2}}\right) \frac{T_{21}(1,\theta(\gamma_{t+1}),1)}%
{T_{21}(1,\gamma_{t+1},1)}\right\vert <\beta\text{ for all }\gamma_{t+1}%
\in(1,\widehat{\gamma})\cup(\widehat{\gamma},\overline{\gamma}).
\]
\newline It follows that%
\[
\beta_{\min}=\underset{\gamma_{t+1}\in(1,\widehat{\gamma})\cup(\widehat
{\gamma},\overline{\gamma})}{\max}\left\vert -\frac{l^{2}}{\mathcal{F}^{2}%
}\left( \frac{k^{1}}{l^{1}}-\frac{k^{2}}{l^{2}}\right) \frac{T_{21}%
(1,\theta(\gamma_{t+1}),1)}{T_{21}(1,\gamma_{t+1},1)}\right\vert
\]
Since, by assumption, we have $\beta<1$, a necessary condition for
expansiveness is%
\[
\left\vert -\frac{T_{22}(1,\theta(\gamma_{t+1}),1)}{T_{21}(1,\gamma_{t+1}%
,1)}\right\vert <1.
\]
$\blacksquare$\bigskip
\textbf{Proof of Lemma \ref{Vijnl}: }By definition $V(k_{t},k_{t+1}%
,X_{t})=\left[ T(k_{t},k_{t+1}-(1-\delta)k_{t},X_{t})\right] ^{\alpha
}/\alpha.$ Under Assumption 4 we can compute the following derivative:
$V_{2}=T^{\alpha-1}T_{2},$ $V_{21}=T^{\alpha-2}[(\alpha-1)T_{2}T_{1}+TT_{21}]$
and $V_{23}=T^{\alpha-2}\left[ (\alpha-1)T_{2}T_{3}+TT_{23}\right] .$ Using
the expressions of $V_{21}$ and $V_{23},$ derived in Lemma \ref{Vij}, we get:%
\[
V_{21}+V_{23}=T^{\alpha-2}[(\alpha-1)T_{2}(T_{1}+T_{3})+T(T_{21}T_{23})].
\]
Under Assumptions 1-7 we can show that $T_{3}=\mathcal{F}_{1}^{1}%
\frac{\partial k^{1}}{\partial X}+X\mathcal{F}_{2}^{1}\frac{\partial l^{1}%
}{\partial X}+l^{1}\mathcal{F}_{2}^{1}.$ By definition, $y=\mathcal{F}%
^{2}(k^{2},l^{2}X).$ So,
\begin{equation}
0=\mathcal{F}_{1}^{2}\frac{\partial k^{2}}{\partial X}+\mathcal{F}_{2}%
^{2}\frac{\partial l^{2}}{\partial X}+l^{2}\mathcal{F}_{2}^{2}. \label{fx nl}%
\end{equation}
Under the full employment of productive resources we have%
\begin{align}
\frac{\partial k^{1}}{\partial X} & =-\frac{\partial k^{2}}{\partial
X},\label{fer1 nl}\\
\frac{\partial l^{1}}{\partial X} & =-\frac{\partial l^{2}}{\partial X}.
\label{fer2 nl}%
\end{align}
Furthermore, the envelope theorem tells us that $T_{2}(k,y,X)=-q,$ and by
definition,%
\begin{equation}
q=\frac{\mathcal{F}_{1}^{1}}{\mathcal{F}_{1}^{2}}=\frac{\mathcal{F}_{2}^{1}%
}{\mathcal{F}_{2}^{2}}. \label{q nl}%
\end{equation}
Substituting (\ref{fer1 nl}), (\ref{fer2 nl}) and (\ref{q nl}) into
(\ref{T3b nl}), and using (\ref{fx nl}), we get%
\begin{equation}
T_{3}=q\mathcal{F}_{2}^{2}>0. \label{T3b nl}%
\end{equation}
We now determine the sign of $T_{21}+T_{23}.$ Using (\ref{T23}), we have%
\[
T_{21}+T_{23}=T_{21}\left( 1-\frac{k^{1}}{l^{1}X}\right) +\frac
{2l^{2}(\mathcal{F}_{12}^{1}+q\mathcal{F}_{12}^{2})}{\mathcal{F}_{1}^{2}}.
\]
Under Assumption 2, $2l^{2}(\mathcal{F}_{12}^{1}+q\mathcal{F}_{12}%
^{2})/\mathcal{F}_{1}^{2}>0.$ It follows that $V_{21}+V_{23}>0$ if
$T_{21}+T_{23}>0.$ And $T_{21}+T_{23}>0$ if $T_{21}\left( 1-k^{1}%
/l^{1}X\right) \geq0.$ So, if $T_{21}\left( 1-k^{1}/l^{1}X\right) \geq0,$
then $V_{21}+V_{23}>0.$ Along a BGP,%
\[
1-\frac{k^{1}}{l^{1}X}=\frac{l^{1}k-k^{1}}{l^{1}k}=\frac{l^{2}}{k}\left(
\frac{k^{2}}{l^{2}}-\frac{k^{1}}{l^{1}}\right) .
\]
Hence, on using (\ref{t21}), we obtain%
\[
T_{21}\left( 1-\frac{k^{1}}{l^{1}X}\right) =\frac{\mathcal{F}_{12}%
^{1}\mathcal{F}_{12}^{2}q\mathcal{F}^{2}l^{1}l^{2}}{\Delta k^{1}k^{2}k}\left(
\frac{k^{2}}{l^{2}}-\frac{k^{1}}{l^{1}}\right) ^{2}\geq0.
\]
$\blacksquare$\bigskip
\textbf{Proof of Lemma \ref{thetanl}:}$\bigskip$
Let us define%
\begin{align}
f(\gamma_{t+1}) & =V_{1}(1,\gamma_{t+1},1)/\beta,\label{A1 nl}\\
g(\gamma_{t}) & =-\gamma_{t}^{1-\alpha}V_{2}(1,\gamma_{t},1), \label{A2 nl}%
\end{align}
where $f:[0,\overline{\gamma}]\rightarrow J\sqsubseteq\Re_{+}^{\ast}$ and
$g:[0,\overline{\gamma}]\rightarrow I\sqsubseteq\Re_{+}^{\ast}.$\newline Using
(\ref{A1 nl}) and (\ref{A2 nl}), the Euler equation (\ref{Euler Equation 1nl})
can be written as $-g(\gamma_{t})+f(\gamma_{t+1})=0.$
\begin{equation}
g^{\prime}=(1-\alpha)\gamma_{t}^{-\alpha}V_{2}(1,\gamma_{t},1)+\gamma
^{1-\alpha}V_{22}(1,\gamma_{t},1). \label{EEL1 nl}%
\end{equation}
From the Euler theorem on homogenous functions we have%
\[
(\alpha-1)V_{2}(1,\gamma_{t},1)=V_{21}(1,\gamma_{t},1)+V_{22}(1,\gamma
_{t},1)\gamma_{t}+V_{23}(1,\gamma_{t},1).
\]
Hence%
\[
g^{\prime}=V_{21}(1,\gamma,1)+V_{23}(1,\gamma,1)>0
\]
$g$ is one to one. From the definition of $g$ we see that range$(g)=I.$ So $g
$ has a well defined inverse $g^{-1}.$ We can rewrite (\ref{EEL1 nl}) as
$\gamma_{t}=\theta(\gamma_{t+1}),$ where $g^{-1}\left[ f(\gamma
_{t+1})\right] .$ From this definition, it follows that $\theta
:[0,\overline{\gamma}]\rightarrow\lbrack0,\overline{\gamma}].\blacksquare
\bigskip$
\textbf{Proof of Proposition \ref{unimodality nl}:}$\bigskip$
(i) Continuity of $\theta$. From the definition of $\theta$ given in Lemma
\ref{thetanl}, $\theta$ is continuous as a composite function of two
continuous functions.\newline(ii) Monotonicity of $\theta.$ Let Assumptions
1b--5b and 6-9 be satisfied. Then using the inverse function theorem we can
express $\theta^{^{\prime}}(\gamma_{t+1})$ on $(0,\overset{0}{\gamma})$ and
$(\widehat{\gamma},\overset{0}{\gamma})$ as
\begin{equation}
\theta^{^{\prime}}(\gamma_{t+1})=\dfrac{\beta\gamma_{t}^{\alpha}%
V_{21}(1,\gamma_{t+1},1)}{(\alpha-1)V_{2}(1,\gamma_{t},1)-\gamma_{t}%
V_{22}(1,\gamma_{t}1,)}. \label{theta0 nl}%
\end{equation}
From the Euler theorem on homogenous function we have
\[
(\alpha-1)V_{2}(1,\gamma_{t},1)=V_{21}(1,\gamma_{t}1,)+V_{22}(1,\gamma
_{t}1,)\gamma_{t}+V_{23}(1,\gamma_{t}1,).
\]
Hence it follows that%
\[
\theta^{^{\prime}}(\gamma_{t+1})=\dfrac{\beta\gamma_{t}^{\alpha}%
V_{21}(1,\gamma_{t+1},1)}{V_{21}(1,\gamma_{t},1)+V_{23}(1,\gamma_{t}1,)}.
\]
Upon inspection of (\ref{theta0}) we see that the sign of the denominator
depends on the sign of $V_{21}+V_{23}$. Under the results of Lemma
\textbf{\ref{unimodality nl}}\ it follows that $\theta^{^{\prime}}%
(\gamma_{t+1})\gtrless0$ if and only if $V_{12}(1,\gamma_{t+1},1)\gtrless0.$
$\blacksquare$\bigskip
\textbf{Proof of Proposition \ref{EOGS2 nl}:}$\bigskip$
From Assumption 2 and the definition of $\theta$ it follows that $\theta$ is
twice continuously differentiable on both $(0,\widehat{\gamma})$ and
$(\widehat{\gamma},\overline{\gamma}).$ $\theta$ is expansive if $\left\vert
\theta^{^{\prime}}(\gamma_{t+1})\right\vert >1$ for all $\gamma_{t+1}%
\in(1,\widehat{\gamma})\cup(\widehat{\gamma},\overline{\gamma}).$ Or,
equivalently, if%
\[
\left\vert -\frac{V_{21}(1,\theta(\gamma_{t+1}),1)+V_{23}(1,\theta
(\gamma_{t+1}),1)}{\gamma_{t}^{\alpha}V_{21}(1,\gamma_{t+1},1)}\right\vert
<\beta\text{ for all }\gamma_{t+1}\in(1,\widehat{\gamma})\cup(\widehat{\gamma
},\overline{\gamma}).
\]
\newline It follows that%
\[
\beta_{\min}=\underset{\gamma_{t+1}\in(1,\widehat{\gamma})\cup(\widehat
{\gamma},\overline{\gamma})}{\max}\left\vert -\frac{V_{21}(1,\theta
(\gamma_{t+1}),1)+V_{23}(1,\theta(\gamma_{t+1}),1)}{\gamma_{t}^{\alpha}%
V_{21}(1,\gamma_{t+1},1)}\right\vert .
\]
Since, by assumption, we have $\beta<1$, a necessary condition for
expansiveness is%
\[
\left\vert -\frac{V_{21}(1,\theta(\gamma_{t+1}),1)+V_{23}(1,\theta
(\gamma_{t+1}),1)}{\gamma_{t}^{\alpha}V_{21}(1,\gamma_{t+1},1)}\right\vert
<1.
\]
$\blacksquare$\bigskip
\section*{References}
Arrow, K.J., (1962): \textquotedblleft The Economic Implications of Learning
by Doing\textquotedblright, \textit{Review of Economic Studies,} 29: 155-173.
Benhabib, J., and K. Nishimura, (1985): \textquotedblleft Competitive
equilibrium cycles\textquotedblright, \textit{Journal of Economic Theory} 35: 284-306.
Baumol, W.J. and J. Benhabib, (1989): \textquotedblleft Chaos: Signifiance,
Mechanisms and Economic Applications\textquotedblright, \textit{Journal of
Economic Perspectives}, 3: 77-105.
Boldrin, M., (1989): \textquotedblleft Paths of Optimal Accumulation in
Two-Sector Models\textquotedblright, in Barnett, W., J. Geweke and K. Shell
(Eds.): \textquotedblleft Economic Complexity, Chaos, Sunspots, Bubbles and
Nonlinearity\textquotedblright, Cambridge University Press.
Boldrin, M., and R. Deneckere, (1990): \textquotedblleft Sources of Complex
dynamics in Two-Sector Growth Models\textquotedblright, \textit{Journal of
Economic Dynamics and Control}, 14: 627-653.
Boldrin, M., Nishimura K., Shigoka, T. and M. Yano, (2001): \textquotedblleft
Chaotic Equilibrium Dynamics in Endogenous Growth Models\textquotedblright,
\textit{Journal of Economic Theory,} 96: 9-132.
Boldrin, M., and M. Woodford, (1990): \textquotedblleft Equilibrium Models
Displaying Endogenous Fluctuations and Chaos\textquotedblright,
\textit{Journal of Monetary Economics}, 25: 189-222.
Collet, P., and J.P. Eckman, (1980): \textquotedblleft Iterated Maps on the
Interval as Dynamical Systems\textquotedblright, Birkhauser.
Deneckere, R., and S. Pelikan, (1986): \textquotedblleft Competitive
Chaos\textquotedblright, \textit{Journal of Economic Theory}, 40: 13-25.
Day, R.H., (1994): \textquotedblleft Complex Economic
Dynamics\textquotedblright, MIT Press.
Day, R.H.,. and G. Panigiani, (1991): \textquotedblleft Statistical Dynamics
and Economics\textquotedblright, \textit{Journal of Economic Behavior and
Organizations}, 16: 37-83.
Drugeon, J.P. and A. Venditti, (1998): \textquotedblleft Intersectoral
External Effects, Multiplicities and Indeterminacies II: The Long -Run Growth
Case\textquotedblright, GREQAM Working Paper 98A20.
Drugeon, J.P., Poulsen O. and A. Venditti, (2003): \textquotedblleft On
Intersectoral Allocations, Factor Substitutability and Multiple Long-Run
Growth Paths\textquotedblright,\ \textit{Economic Theory}, 21\textbf{: }175-183.
Goenka, A. and O. Poulsen, (2002): \textquotedblleft Indeterminacy and Labor
Augmenting Externalities\textquotedblright, Aarhus School of Business
Department of Economics, Working Paper 02-09.
Grandmont, J.M., (1985): \textquotedblleft On endogenous Competitive Business
Cycles\textquotedblright, \textit{Econometrica}, 5, 995-1045.
Lasota, A. and J. Yorke, (1973): \textquotedblleft On the Existence of
Invariant Measures for Piecewise Monotonic Transformations\textquotedblright,
\textit{Transaction of the American Mathematical Society}, 186: 481-488.
Li, T., and J. Yorke, (1978): \textquotedblleft Ergodic Transformations from
an Interval into Itself\textquotedblright, \textit{Transaction of the American
Mathematical Society}, 235: 183-192.
Lucas, R, (1988): \textquotedblleft On the Mechanics of Economic
Development\textquotedblright, \textit{Journal of Monetary Economics}, 22: 3-42.
Mitra, T., (1996): \textquotedblleft An Exact Discount Factor Restriction for
Period-Three Cycles in Dynamics Optimization Models\textquotedblright,
\textit{Journal of Economic Theory}, 69: 281-305.
Mitra, T., (1998): \textquotedblleft On Equilibrium Dynamics under
Externalities in a Model of Economic Development\textquotedblright,
\textit{Japanese Economic Review}, 49\textit{: }85-107.
Nishimura, K., Sorger G., and M. Yano, (1994): \textquotedblleft Ergodic Chaos
in Optimal Growth Models with Low Discount Rates\textquotedblright,
\textit{Economic Theory}, 4: 705-717.
Nishimura, K., and M. Yano, (1995): \textquotedblleft Nonlinear Dynamics and
Chaos in Optimal Growth: An Example\textquotedblright, \textit{Econometrica},
63, 981-1001.
Nishimura, K., and M. Yano, (1996): \textquotedblleft On the Least Upper Bound
of Discount Factors that are Compatible with Optimal Period-Three
Cycles\textquotedblright, \textit{Journal of Economic Theory}, 66: 306-333.
Nishimura, K., and M. Yano, (2000): \textquotedblleft Non-Linear Dynamics and
Chaos in Optimal Growth: A Constructive Exposition\textquotedblright, in
Majumdar, M, T. Mitra and K. Nishimura (Eds.) \textquotedblleft Optimization
and Chaos\textquotedblright, Springer.
Romer, P., (1983): \textquotedblleft Dynamic Competitive Equilibria with
Externalities, Increasing Returns and Unbounded Growth\textquotedblright,
Unpublished Ph.D. Thesis, University of Chicago.
Romer, P., (1986): \textquotedblleft Increasing Returns and Long Run
Growth\textquotedblright, \textit{Journal of Political Economy,} 94: 1002-1037.
Sheshinski, E., (1967): \textquotedblleft Optimal Accumulation with Learning
by Doing\textquotedblright.\ In \textit{Essays on the theory of optimal
economic growth}, edited by K. Shell. Cambridge: MIT Press, 31-52.
Sorger, G., (1992): \textquotedblleft On the Minimum Rate of Impatience for
Complicated Optimal Growth Paths\textquotedblright, \textit{Journal of
Economic Theory}, 56, 160-179.
Sorger, G., (1994): \textquotedblleft Period Three Implies Heavy
Discounting\textquotedblright, \textit{Mathematical Operations Research}, 19: 1-16.
Uzawa, H., (1961): \textquotedblleft Neutral Inventions and the Stability of
Growth Equilibrium\textquotedblright, \textit{Review of Economic Studies,} 28: 117-124.
\end{document}