\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 15, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2005/15\hfil
Hopf bifurcation in the IS-LM business cycle model]
{Hopf bifurcation in the IS-LM business cycle model with time delay}
\author[J. P. Cai\hfil EJDE-2005/15\hfilneg]
{Jianping Cai}
\address{Jianping Cai \hfill\break
Department of Mathematics\\
Zhangzhou Teachers College\\
Zhangzhou 363000, China}
\email{mathcai@hotmail.com}
\date{}
\thanks{Submitted November 17, 2004. Published February 1, 2005.}
\subjclass[2000]{37G15, 91B62}
\keywords{Hopf bifurcation; business cycle; IS-LM model; time delay}
\begin{abstract}
The distinction between investment decisions and
implementation leads us to formulate a new IS-LM business
cycle model. It is shown that the dynamics depends crucially
on the time delay parameter - the gestation time period of
investment. The Hopf bifurcation theorem is used to predict
the occurrence of a limit cycle bifurcation for the time
delay parameter. Our analysis shows that the limit cycle
behavior is independent of the assumption of nonlinearity
of the investment function. An example is given to verify
the theoretical results.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
The Hopf bifurcation theorem \cite{mm} as a tool for establishing
the existence of closed orbits in dynamical systems seems to have
been originally introduced to economics by Torre \cite{t}, who
studied the standard IS-LM model
\begin{gather*}
\dot {Y} = \alpha (I(Y,r) - S(Y,r)) \\
\dot {r} = \beta (L(Y,r) - \bar {M})
\end{gather*}
with $Y$ as the gross product, $I$ as the investment, $S$ as the
saving, $L$ as the demand for money, and $\bar {M}$as the constant
money supply. Here $\alpha $ and $\beta $ are respectively the
adjustment coefficients in the markets of goods and money. Other
applications of the Hopf bifurcation theorem can be found in,
Benhabib and Nishimura \cite{bn}, Medio \cite{me}, Krawiec
and Szydlowski \cite{ks}, and Asada and Yoshida \cite{ay}. In the
two-dimensional case the use of bifurcation theory actually
provides no new insight into known models. The real domains of
bifurcation theory are dynamical systems of dimension greater than
or equal to three because the Poincare-Bendixson theorem cannot be
applied anymore. Gabisch and Lorenz considered an augmented IS-LM
business cycle model \cite[p. 168]{g},
\begin{equation} \label{eq1}
\begin{gathered}
\dot {Y} = \alpha (I(Y,K,r) - S(Y,r)) \\
\dot {r} = \beta (L(Y,r) - \bar {M})\\
\dot {K} = I(Y,K,r) - \delta K
\end{gathered}
\end{equation}
with $K$ as the capital stock and $\delta $ as the
depreciation rate of the capital stock. The model seems to be one
of the simplest complete business cycle models in the Keynesian
tradition. A similar model has been studied by Boldrin \cite{b}.
In the Kalecki business cycle model \cite{k}, Kalecki assumed that
the saved part of profit is invested and the capital growth is due
to past investment decisions. There is a gestation period or a
time delay, after which capital equipment is available for
production. Similar time lag has been introduced and discussed in
\cite{ay} and \cite{ks}. In this paper, Kalecki's idea is
introduced into the IS-LM model (\ref{eq1}) to formulate a
generalized IS-LM business cycle model as follow
\begin{equation} \label{eq2}
\begin{gathered}
\dot {Y} = \alpha (I(Y,K,r) - S(Y,r)) \\
\dot {r} = \beta (L(Y,r) - \bar {M}) \\
\dot {K} = I(Y(t - T),K,r) - \delta K
\end{gathered}
\end{equation}
with $T$ as the time delay parameter.
Investment depends on income at the time investment decisions are make and
on capital stock at the time investment is finished. The latter is a
consequence of the fact that at time $t - T$ there are some investments
which will be finished between $t - T$ and $t$. We assume that capital stock
produced in this period is taken into consideration when new investments are
planned. The Hopf bifurcation theorem is applied to predict the occurrence
of a limit cycle bifurcation for the time delay parameter. The crucial role
in the creation of the limit cycle is Kalecki's time delay parameter, rather
than the assumption of the S-shaped investment function. An example is given
to show that a Hopf bifurcation can occur in the present IS-LM model -- a
linear system with time delay.
\section{Linear IS-LM model}
Assume that the investment function $I$, the saving function $S$, and the
demand for money $L$ depend linearly on their arguments, that is
\begin{gather*}
I = \eta Y - \delta _1 K - \beta _1 r \\
S = l_1 Y + \beta _2 r\\
L = l_2 Y - \beta _{3 } r
\end{gather*}
with $\eta ,\delta _{1} ,l_1 ,l_2 ,\beta _1 ,\beta _2,\beta _3$
positive constants. Now system (\ref{eq2}) becomes
\begin{equation} \label{eq3}
\begin{gathered}
\dot {Y} = \alpha ((\eta - l_1 )Y - (\beta _1 + \beta _2 )r - \delta _1 K)\\
\dot {r} = \beta (l_2 Y - \beta _3 r - \bar {M}) \\
\dot {K} = \eta Y(t - T) - \beta _1 r - (\delta + \delta _1) K
\end{gathered}
\end{equation}
The characteristic equation of equation (\ref{eq3}) has the form
\[
\det \begin{pmatrix}
\alpha (\eta - l_1 ) - \lambda & - \alpha (\beta _1 + \beta _2 )
& - \alpha \delta _1 \\
\beta l_2 & - \beta \beta _3 - \lambda & 0 \\
\eta e^{ - \lambda T} & - \beta _1 & - (\delta +\delta _1 ) - \lambda
\end{pmatrix} = 0
\]
that is
\begin{equation} \label{eq4}
\lambda ^3 + A\lambda ^2 + B\lambda + C + D\lambda e^{ - \lambda T} +
E e^{ - \lambda T} = 0
\end{equation}
where
\begin{gather*}
A = \delta + \delta _1 + \beta \beta _3 - \alpha (\eta - l_1 ),\\
B = (\delta + \delta _1 )(\beta \beta _3 - \alpha (\eta - l_1 ))
+ \alpha \beta l_2 (\beta _1 + \beta _2 ) - \alpha \beta
\beta _3 (\eta - l_1 ),\\
C = - \alpha \beta \beta _{1 } l_2 \delta _1 - (\delta +
\delta _1 )\alpha \beta (\beta _3 (\eta - l_1 ) - l_2 (\beta _1
+ \beta _2 )),\\
D = \alpha \eta \delta _1, \quad
E = \alpha \beta \beta _3 \eta \delta _1
\end{gather*}
Generally speaking, transcendental equation (\ref{eq4}) cannot be solved
analytically and has indefinite number of roots. In essence, we have two
main tools besides direct numerical integration; firstly, the linear
stability analysis, especially in the case of small time delay, and
secondly, the Hopf bifurcation theorem. In the following sections, we
discuss both approaches.
\section{ Linear stability analysis}
For small time delay $T$, the method of linear stability analysis is much
convenient to find the bifurcation point. To this end, let $e^{ - \lambda
T} \approx 1 - \lambda T$, then the eigenvalue equation (\ref{eq4})
becomes
\begin{equation} \label{eq5}
\lambda ^3 + (A - DT)\lambda ^2 + (B + D - ET)\lambda + C + E = 0
\end{equation}
By the Hopf bifurcation theorem and the Routh-Huwitz criteria, a Hopf
bifurcation occurs at a value $T = T_0 $ where [4, pp166],
\begin{equation} \label{eq6}
A - DT_0 > 0, \quad
B + D - ET_0 > 0, \quad
C + E > 0
\end{equation}
and
\begin{equation} \label{eq7}
(A - DT_0 )(B + D - ET_0 ) = C + E
\end{equation}
Let
\[
g(\lambda , T) = \lambda ^3 + (A - DT)\lambda ^2 + (B + D -
ET)\lambda + C + E
\]
Evaluating $g$ at $T = T_0 $ yields
\[
g(\lambda , T_0 ) = \lambda ^3 + s\lambda ^2 + k^2\lambda +
k^2 s
\]
where $s = A - DT_0 $, $k^2 = B + D - ET_0 $. The eigenvalues of (\ref{eq5})
at $T_0$ are
\begin{gather*}
\lambda _0 (T_0 ) = - s = - (A - DT_0 ) \\
\lambda _{1 , 2} (T_0 ) = \pm i k = \pm i (B + D -
ET_0 )^{\frac{1}{2}}
\end{gather*}
where $i$ is the imaginary unit. Differentiating implicitly
$g(\lambda (T),T)$ yields
\[
\frac{d\lambda }{dT}
= - \frac{\partial g}{\partial T}
\big/ \frac{\partial g}{\partial \lambda }
= - \frac{ - D\lambda ^2 - E\lambda }{3\lambda ^2 + 2(A -
DT)\lambda + B + D - ET}
\]
Evaluating the required derivatives of $g$ at $T_0 $, we have
\begin{equation} \label{eq8}
\frac{d\lambda _1 (T_0 )}{dT} = - \frac{(Dk^2 - Ek i)( - 3k^2 + B + D -
ET_0 - 2k(A - DT_0 ) i )}{P^2 + R^2}
\end{equation}
where $P = - 3k^2 + B + D - ET_0 $, $R = 2(A - DT_0 )k$.
The real part of (\ref{eq8}) is
\[
\mathop{\rm Re}\big(\frac{d\lambda _1 (T_0 )}{dT}\big)
= - \frac{Dk^2( - 3k^2 + B + D -
ET_0 ) - 2Ek^2(A - DT_0 )}{P^2 + R^2}
\]
and $\mathop{\rm Re}(\frac{d\lambda _1 (T_0 )}{dT}) > 0$ is equivalent to
\begin{equation} \label{eq9}
- D(B + D - ET_0 ) < Ek^2(A - DT_0 )
\end{equation}
Noting that $D$ and $E$ are positive, inequality (\ref{eq9}) holds if the
following conditions are fulfilled
\[
A - DT_0 > 0 \quad\mbox{and}\quad B + D - ET_0 > 0
\]
So inequality (\ref{eq6}) is sufficient to have positive slope of the real
part of
the eigenvalue $\lambda _1 (T)$. This fact guarantees the bifurcation to a
limit cycle for $T = T_0 $ according to the Hopf bifurcation theorem.
\section{ Hopf bifurcation analysis}
For larger time delay $T$, the linear stability analysis of above section is
no longer effective and another approach is needed. Let $\lambda = \sigma +
i \omega $ and rewrite (\ref{eq4}) in terms of its real and imaginary parts
as
\begin{gather*}
\sigma ^3 - 3\sigma \omega + A\sigma ^2 - A\omega ^2 + B\sigma +
C + e^{ - \sigma T}(D\sigma \cos \omega T + D\omega \sin
\omega T + E\cos \omega T) = 0\\
3\sigma ^2 \omega - \omega ^3 + 2A\sigma \omega + B\omega + e^{
- \sigma T}(D\omega \cos \omega T - D\sigma \sin \omega
T - E\sin \omega T) = 0
\end{gather*}
To find the first bifurcation point, we set $\sigma = 0$. Then the above two
equations reduce to
\begin{gather} \label{eq10}
- A\omega ^2 + C + D\omega \sin \omega T + E\cos \omega T = 0 \\
\label{eq11}
- \omega ^3 + B\omega + D\omega \cos \omega T - E\sin \omega T = 0
\end{gather}
These two equations can be solved easily numerically. If the first
bifurcation point is $(\omega _{\rm bif} , T_{\rm bif} )$, then the other
bifurcation points $(\omega , T)$ satisfy
\[
\omega T = \omega _{\rm bif} T_{\rm bif} + 2n\pi , \quad n = 1, 2, \dots
\]
By squaring (\ref{eq10}) and (\ref{eq11}), and then adding them, it
follows that
\begin{equation} \label{eq12}
\omega ^6 + (A^2 - 2B)\omega ^4 + (B^2 - 2AC - D^2)\omega ^2 + C^2 - E^2 = 0
\end{equation}
This is a cubic equation in $\omega ^2$ and the left side is
positive for large values of $\omega ^2$ and negative for
$\omega = 0$ if $C^2 < E^2$. Hence, if the above condition is met,
then (\ref{eq12}) has at least one positive real root.
Moreover, we have the following lemma \cite{kh}.
\begin{lemma} \label{lem4.1}
Define
\[
\Delta = \frac{4}{27}a_2^3 - \frac{1}{27}a_1^2 a_2^2 +
\frac{4}{27}a_1^3 a_3 - \frac{2}{3}a_1 a_2 a_3 + a_3^3
\]
and suppose that $a_3 > 0$. Then necessary and sufficient
conditions for the cubic equation
\[
z^3 + a_1 z^2 + a_2 z + a_3 = 0
\]
to have at least one single positive root for $z$ are
\\
(1) either
$(a) a_1 < 0 , a_2 \ge 0$, and $a_1^2 > 3a_2 $, or
$(b) a_2 < 0 $; and
\\
(2) $\Delta < 0$.
\end{lemma}
Denote
\[
G(\lambda , T) = \lambda ^3 + A\lambda ^2 + B\lambda + C + D\lambda e^{
- \lambda T} + E e^{ - \lambda T}
\]
then
\begin{equation} \label{eq13}
\frac{d\lambda }{dT} = - {\frac{\partial G}{\partial T}}
\big/ {\frac{\partial G}{\partial
\lambda }} = \frac{(D\lambda ^2 + E\lambda )e^{ - \lambda T}}{3\lambda ^2 +
2A\lambda + B + (D - DT\lambda - ET)e^{ - \lambda T}}
\end{equation}
Evaluating the real part of this equation at $T = T_{\rm bif} $ and setting
$\lambda = i \omega _{\rm bif} $ yield
\[
{\frac{d\sigma }{dT}} \big|_{T = T_{\rm bif} } = Re
{(\frac{d\lambda }{dT})} \big|_{T = T_{\rm bif} } = \frac{\omega
_{\rm bif}^2 (3\omega _{\rm bif}^4 + 2\omega _{\rm bif}^2 (A^2 - 2B) + B^2 -
2AC - D^2)}{P_1^2 + Q_1^2 }
\]
where
\begin{gather*}
P_1^ = - 3\omega _{\rm bif}^2 + B + T_{\rm bif} ( - A\omega _{\rm bif}^2 + C)
+ D\cos \omega _{\rm bif} T_{\rm bif} \\
Q_1^ = 2A\omega _{\rm bif} + T_{\rm bif} ( - \omega _{\rm bif}^3 + B\omega
_{\rm bif} ) - D\sin \omega _{\rm bif} T_{\rm bif}
\end{gather*}
Let $x = \omega _{\rm bif}^2 $, then (\ref{eq12}) reduces to
\[
f(x) = x^3 + (A^2 - 2B) x^2 + (B^2 - 2AC - D^2) x + C^2 - E^2
\]
then
\[
{f}'(x) = 3x^2 + 2(A^2 - 2B) x + B^2 - 2AC - D^2
\]
If $\omega _{\rm bif} $ is the least positive simple root of equation
(\ref{eq12}),
unless this is a double root when we must take $\omega _{\rm bif} $ as the next
root, then
\[
{{f}'(x)} \big|_{T = T_{\rm bif} } > 0
\]
Hence,
\[
{\frac{d\sigma }{dT}} \big|_{T = T_{\rm bif} } = \frac{\omega
_{\rm bif}^2 {f}'(\omega _{\rm bif}^2 )}{P_1^2 + Q_1^2 } > 0
\]
According to the Hopf bifurcation theorem, we come to the main result of
this paper.
\begin{theorem} \label{thm1}
Assume that the conditions of Lemma \ref{lem4.1} are satisfied and
$\omega_{\rm bif} $ is the least positive root of equation (\ref{eq12})
unless this is a double root when we must take $\omega _{\rm bif} $ as the
next root which is simple, then a Hopf bifurcation occurs as $T$
passes through $T_{\rm bif}$.
\end{theorem}
A similar phenomenum appeared in the model of multiparty
political system studied in \cite{kh}.
\subsection*{Example}
When $\alpha = 3, \beta = 2$, $\delta = 0.1, \delta _1 = 0.5$, $\eta =
0.3, l_1 = 0.2, l_2 = 0.1$, $\bar {M} = 0.05$, $\beta _1 = \beta _2 =
\beta _3 = 0.2$, system (\ref{eq3}) becomes
\begin{gather*}
\dot {Y} = 0.3Y - 0.4r - 0.5 K \\
\dot {r} = 0.2 Y - 0.4 r - 0.1 \\
\dot {K} = 0.3Y(t - T) - 0.2 r - 0.6 K
\end{gather*}
The characteristic equation (\ref{eq4}) becomes
\[
\lambda ^3 + 0.7\lambda ^2 + 0.18\lambda + 0.012 + 0.45\lambda
e^{ - \lambda T} + 0.18 e^{ - \lambda T} = 0
\]
It is easy to verify that the conditions of Theorem \ref{thm1} are fulfilled,
so a limit cycle bifurcation occurs when the time delay parameter $T$ passes
through $T_{\rm bif} = 0.740471$ where the relative eigenvalues are
$\lambda _0 = - 0.382583$, $\lambda _{1, 2} = \pm 0.6993 i$. Moreover, we can
determine the approximate period of the closed orbit by
\[
\tilde {T} = \frac{2\pi }{| {\lambda (T_{\rm bif} )}|} =
\frac{2\pi }{0.6993} = 8.98496
\]
which implies the period of the economical system is about 9.
\subsection*{Conclusion}
When we take into account the distinction between investment decision and
expenditure, we come to the problem of gestation lags in investment. This
leads us to formulate the generalized IS-LM business cycle model with time
delay. The Hopf bifurcation theorem is used to predict the occurrence of a
bifurcation to a limit cycle for some values of the time delay parameter.
Our model admits the limit cycle behavior even for a linear investment
function instead of a S-shaped one. It is also shown that Kalecki's time
delay parameter plays the crucial role of existence of limit cycle behavior.
The example in section 5 verifies the analytical results.
\begin{thebibliography}{00}
\bibitem{ay} T. Asada, H. Yoshida; \emph{Stability, instability and complex
behavior in macrodynamic models with policy lag},
Discrete Dynamics in Nature and Society, \textbf{5}(2001), 281-295.
\bibitem{bn} J. Benhabib, K. Nishimura; \emph{The Hopf bifurcation and the
existence of closed orbit in multisector models of optimal economic growth},
Journal of Economic Theory, \textbf{21}(1979),
421-444.
\bibitem{b} M. Boldrin; \emph{Applying bifurcation theory: Some simple
results on Keynesian business cycles}, DP8403, University of Venice, 1984.
\bibitem{g} G. Gabisch, H.W. Lorenz; \emph{Business cycle theory - A survey
of methods and concepts} (Second Edition), Springer, New York, 1989.
\bibitem{k} M. Kalecki; \emph{A macrodynamic theory of business cycle},
Econometrica, \textbf{3}(1935), 327-344.
\bibitem{kh} Q. J. A. Khan; \emph{Hopf bifurcation in multiparty political
systems with time delay in switching}, Applied Mathematics Letters,
\textbf{13}(2000), 43-52.
\bibitem{ks} A. Krawiec, M. Szydlowski; \emph{The Kaldor-Kalecki business
cycle model}, Annals of Operations Research, \textbf{89}(1999),
89-100.
\bibitem{mm} J. E. Marsdem, M. Mckracken; \emph{The Hopf bifurcation and its
application}, Springer, New York, 1976.
\bibitem{me} A. Medio; \emph{Oscillations in optimal growth models}, Mimeo,
University of Venice, 1986.
\bibitem{t} V. Torre; \emph{Existence of limit cycles and control in
complete Keynesian systems by theory of bifurcations}, Econometrica,
\textbf{45}(1977), 1457-1466.
\end{thebibliography}
\end{document}