Hopf Bifurcation and Structural Instability in the Open-Economy
With Keynesian Rigidity*
by Edgardo Jovero** (Universidad Complutense de Madrid)
This paper attempts to contribute to the debate in macroeconomic dynamics by presenting the
neoKeynesian challenge. Proof is presented regarding the behavior of an open-economy two-sector growth
model in the neoKeyenesian tradition of non-market clearing. It has been shown that there possibly exists a
Hopf-bifurcation type of structural instability in a nonlinear dynamical model of the macroeconomy by which a
stable region is connected to an unstable region situated in a center manifold in the state space of the resulting
dynamical system. The Keynesian view that structural instability globally exists in the aggregate economy is put
forward, and therefore the need arises for policy to alleviate this instability in the form of dampened fluctuations
is presented as an alternative view for macroeconomic theorizing.
MSC (2000) : 91B62 (mathematical economics), 37F45 (complex dynamical systems)
PACS code: 89.67.Gh (economics, econophysics)
JEL code : F41 (growth in open economies)
Keywords: Hopf bifurcation, structural instability, neoKeynesian economics
In macroeconomics, the neoclassicals present the dominant view that there exists a unique and
globally stable equilibrium, which describes the long term limiting behavior of a macroeconomy,
endowed with rational expectations.
This paper aims to present the neoKeynesian challenge by analyzing a two-good dependent
open-economy model endowed with endogenous growth and which engenders an indeterminate
solution (in the form of short-term “chaotic fluctuations” and long-term unpredictability). The
dependent open-economy assumes that the economy is small, that is, it is subject to a given
international interest rate which it cannot influence.
Assume a Robinson Crusoe-type of economy where there are only 2 types of goods – a
consumption good (C ) and a capital good (K). The representative agent consumes C to derive a level
of satisfaction or utility which he maximizes in an iso-elastic CRRA utility function. K is saved to be
used in the next period using a production function of the form: Y = AK. Y is output; A is a fixed
production coefficient determining the level of technology held constant through time. Further assume
an intertemporal budget constraint expressed as: Y = C + I , where I is the change in the capital stock
or real investment such that I = K .
Given a situation of Keynesian rigidity, many authors in international trade and finance or in
what is termed the new open-economy macroeconomics [9] have reformulated the intertemporal
budget constraint of an open-economy to highlight price-setting behavior. A risk-premium parameter
is added to the intertemporal budget constraint such that:
D  rD  R  D / K   C  I 1   I K    AK .
The term R(D/K) represents the elasticity of debt to the capital stock. Capital is viewed here
as a collateral, and a high (D/K) ratio represents a high risk of default in the future repayment of the
principal and the interest on debt. R is assumed constant, but its effect is influenced by (D/K).
Now formulate the model:
* Paper prepared for the conference in complexity theory and applications in economics and econophysics .
**
corresponding email: [email protected]
2
t 
 C1
Max U   
1
t 0 
s.t.
with initial values at:
  t
 e dt

(1)
D  rD  R( D / K )  C  I 1   I K    AK
(2)
KI
(3)
K (0)  K0 ; D(0)  D0 .
Applying optimal control theory, formulate the current-value Hamiltonian, with m and q as
multipliers:
H c   C1 1     m  rD  R( D / K )  C  I 1   I K    AK   q  K   K 
(4)
The first of the Pontryagin maximum conditions is:
H c C  0  C   m
(5)
or this is simplified as: C    m . Taking logs:  log C   log m ; and time derivatives:


 C C   m m . Or simplifying: C C  1   m m ; for   0 .
The second condition requires that:
Hc D  m   m  m  r  R( D / K ) 
This can be simplified as:  m m    r  R / K
(6)
(7)
Assuming strict consumption smoothing (which is logical since the model is situated in an openeconomy environment, where access to international borrowing and lending results in strict
consumption smoothing) such that ( r   ) , then equation (7) may be re-written as:
m m  R / K
(8)
This can combined with equation (6) such that:
C C  R /  K
(9)
In equilibrium,  m m  is equal to zero so that the first of the transversality conditions is fulfilled.


Therefore, C C must necessarily be equal to zero, which is also an equilibrium condition.
Then to continue with the maximum first-order conditions:
H c K  q   q    mRD K 2    mI 2 K 2   mA

Simplifying: q q    q  R K  D K     q  K K
(10)
    q  A  r  0 . In equilibrium,
2
 q q  should be equal to zero to fulfill the transversality conditions.
Another condition is that in equilibrium, investment is zero such that a constant level of capital
stock (K) is maintained. Therefore: I  K K  0 so that K needs to be supplied in full in every
period so that the production of the consumption good C continues. Equation (10) is further simplified
as:
3
K K    R K  D K   A  r  q m 
1
2
0
(11)
The dynamical system is written as follows:
C C  R /  K  0
K K    R K  D K   A  r  q m 
(12)
1
2
0
(13)
D D  r  ( R / K )  (C / D)  A( K / D)  0
(14)
m m  R / K  0
(15)
q q   m q  R K  D K    m q   K K    m q  A  r  0
2
(16)
The transversality conditions are:
lim e   t m(t ) D(t )  0
t 
lim e   t q (t ) K (t )  0
t 
The above dynamical system shows an indeterminate path for the accumulation of K. This is
the indeterminacy situation --- where the transversality conditions are insufficient to guarantee
uniqueness nor the existence of an equilibrium balance growth path for both the state and control
variables. Then, apply a change of variable such that:
s(t )  C(t ) D(t ) ; d (t )  D(t ) K (t ) ; w(t )  m(t ) q(t ) ; Q  R K
where Q is assumed constant.
Taking logs and time derivatives:


s s  C C   D D
d d   D D   K K 
w w  m m  q q 
The new dynamical system is now expressed as:
s s  Q    r  Q  s   A d   0
d d  r  Q  s   A d    Qd  A   r w  
w w  Q  wQd  wA  r  0
(17)
1
2
0
(18)
(19)
with s, d and w not equated to 0.
Applying the Hartman-Grobman theorem on linearization, the result is the following:
*
s
s  ss 
 d   A  d    d  d *  for s*, d*, w* as equilibrium values.

 
  
 w
 w  w  w* 
A is the Jacobian of first derivatives where aij  Fi / x j , for i = 1rst,2nd,3rd equation and j = s,w,d.
Indeterminacy arises when one of the eigenvalues of A is equal to 0 completely or if the real part of a
complex root is zero. The dynamical system is non-hyperbolic in the language of dynamics, and
linearization fails.
4
This is easily proven by solving for the zeroes of the associated characteristic polynomial of
the canonical Jacobian matrix. A simple examination of the Jacobian shows that:
 F1
 s

F
A 2
 s

 F3
 s
F2
d
F2
d
F3
d
F3 
w   a

11
F2  
  a21
w 
 a
F3   13
w 
a13 
a23 
a33 
a12
a22
a32
The entries of the Jacobian matrix A are defined as follows:
a11    Q    r  Q  2s   A d 
a12  As d 2
a13  0
a21  d
a22  r  Q  s   dQ 2   Qd  A   r w  

a23   dr 2w2   Qd  A   r w  
1
1
2
1
  Qd  A   r w   2 

2
a31  0
a32   w2Q
a33  Q  2 wQd  2wA  r
Setting xi  A exp  t  ; for  = the associated eigenvalues of the dynamical system --- then
for det A  I   0 ; the characteristic polynomial is defined as follows:
p     det A  I   det
a11  
a12
a13
a21
a22  
a23
a31
a32
a33  
0
p      3   a11  a22  a33   2   a11a22  a12 a21    a11a33  a13a31    a22 a33  a23a32   
  a11a22 a33  a12 a23a31  a13a21a32  a13a22 a31  a11a23a32  a12 a21a33   0
p      3  Tr  2  M i   Det  0
This is further defined for the following variables:
Tr  trace of A   aii
Mi 
a11
a12
a21
a22

a11
a13
a31
a33

a22
a23
a32
a33
= sum of the diagonal minors of A
Det  det A
An eigenvector associated with the eigenvalue i is the vector  v1 , v2 , v3  satisfying the
equality i vi  Ai . An eigenvector may be complex if i is complex. In this case, there is a
complex conjugate pair of eigenvalues and eigenvectors in this eigensystem. Note also that the
following relationships hold: 1  2  3  Tr ; 12  13  23  M i ; 123  Det .
5
For aij all real, then Det, Tr and Mi are either all real; or it may be that only one eigenvalue is real and
the other 2 are a pair of complex conjugate eigenvalues (see [2], [6], and [7]).
From here, one can apply standard bifurcation theory by hypothesizing that the stability
condition of the eigensystem is a function of a bifurcation parameter --- in this model of Q (the
market-power parameter measuring the ability of the debt supplier to influence the price of debt).
Then one can analyze the limiting behavior as time approaches infinity to show structural stability.
This entails testing whether the system finally settles down to a unique and globally stable steady state
equilibrium as time goes to infinity --- irrespective of any small perturbation in any of the system’s
parameters.
Or alternatively, one can also test whether the dynamical system is capable of demonstrating
more complex behavior such as the presence of strange attractors or of Hopf bifurcation. Complex
dynamics arise when the system’s state-space geometry is completely changed for any minute change
in a bifurcation parameter. This characterizes “structural instability” in the state-space solution of the
dynamical system.
One way of testing this is to locate the roots of the associated characteristic polynomial
relatively with respect to the imaginary axis. If one root lies in the imaginary axis, then pure complex
roots in conjugates are at work. The most convenient way of root-location is by applying RouthHurwitz (R-H) theorem.1
Applying the theorem for the cubic characteristic polynomial with n=3, the dynamical system
above can be formulated in terms of the R-H array as follows:
ao
a2
3
2
a1
a3

1
  a1a2  a0 a3  a1 0
a3
0
1
Mi
Tr
Det
 Tr M i  Det 
Tr
0
 Det
The stability condition (as stated in [4]) for a cubic characteristic polynomial is as follows:
a1  0;
 a1a2  a0a3   0;
a3  0 which is the same as saying that
Tr  0; Det  0; Tr M i  Det . This would ensure that all the entries in the dominant column in the
R-H array are all positive. This means that all the roots are in the left-hand plane of the imaginary axis
(that is, the stable manifold) and the dynamical system possesses structural stability in the global
sense.
In the case of a cubic characteristic polynomial (n=3), the sufficient and necessary conditions
for a Hopf bifurcation to arise are:
p  ; Q 
 0 , and Tr<0, Det<0 and TrMi=Det. The theorem
Q
indicates that for an odd-numbered row in the R-H array containing all zeroes, then there is already a
necessary but not sufficient condition for pure complex roots to exist. This means that the roots are
1
Standard reference on Routh-Hurwitz is [1] which contains a complete discussion of the proofs.
6
located exactly in the imaginary axis itself with the real part being zero. In addition, the signs of the
column entries preceeding and succeeding this zero entry must be the same.
When these conditions are met, then there exists a non-hyperbolic root which means that
either det A  0 or a pair of pure complex roots exist, and the linearization theorem (HartmanGrobman) fails. For such non-hyperbolic systems, a center manifold exists2, which means one of the
roots contain either zero or pure complex numbers.
Furthermore, the role of Q as a bifurcation parameter can be expressed as follows:
p  ; Q    3  Tr  2  M i   Det  0
p  ; Q   p  ; Q    Tr   p  ; Q    M i   p  ; Q    Det






Q
 Tr   Q   M i   Q   Det   Q
(20)

  0 (21)

p  ; Q 
  2  Tr '  Q   M i '  Q   Det '  Q   0
Q
(22)
Therefore the bifurcation boundary can be defined in terms of the following:

M i '  Q   Det '  Q 
2Tr '  Q 
 0.
Complex dynamics in the open-economy with Keynesian rigidity: summarizing the results
The main topic of this paper is to rebuke the neo-classical emphasis on global uniqueness and
stability in economic growth models where the agent is endowed with rational expectations and
perfect foresight. In terms of modern dynamic analysis, neo-classical growth models exhibit
“structural stability” which does not allow for the existence of a center manifold, where short-term
fluctuations can have permanent and structural effects on the global behavior of the aggregate
economy.
The Lyapunov-Poincaré3 definition of structural stability places strong emphasis on the fact
that a slight change in a control or bifurcation parameter would not in any way modify the geometry of
the state-space of a dynamical system. Such global stability is not present in Keynesian models in
general. This is consistent with Keynes’s view that there exists fundamental uncertainty 4 in the
aggregate economy (particularly in the international financial markets), and by which policy (e.g.
fiscal or monetary) can have a stabilizing role to play. This view is the basis for the NeoKeynesian
dynamical model proposed in this paper to explain the dynamic non-linear relationship between
external debt and growth in a non-perfectly competitive environment where the debt supplier is
endowed with market power.
2
A center manifold allows for the existence of limit cycles as proposed by the Poincaré-Bendixson theorem. See
[2], [3] , [7] for details.
3
See [2], [7] for a more technical discussion on the theorem and [6], [8] for economic applications.
4
See chapter 12 of the General Theory.
7
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