Acta Polytechnica Hungarica
Vol. 8, No. 3, 2011
A Dynamic Model of a Small Open Economy
Under Flexible Exchange Rates
Petra Medveďová
Department of Quantitative Methods and Information Systems
Faculty of Economics
Matej Bel University
Tajovského 10, 975 90 Banská Bystrica, Slovakia
petra.medvedova@umb.sk
Abstract: In the paper a three dimensional dynamic model of a small open economy,
describing the development of net real national income, real physical capital stock and the
expected exchange rate of the near future, which was introduced by T. Asada in [1], is
analysed under flexible exchange rates. We study the question of the existence of business
cycles. Sufficient conditions for the existence of a pair of purely imaginary eigenvalues with
the third one negative in the linear approximation matrix of the model are found. For the
existence of business cycles and their properties the structure of the bifurcation equation of
the model is very important. Formulae for the calculation of the bifurcation coefficients in
the bifurcation equation of the model are derived. Theorem on the existence of business
cycles in a small neighbourhood of the equilibrium point is presented.
Keywords: dynamical model; matrix of linear approximation; eigenvalues; bifurcation
equation; business cycle
1
Introduction
Toichiro Asada formulated in [1] a Kaldorian business cycle model in a small
open economy. He studied both the system of fixed exchange rates and that of
flexible exchange rates with the possibility of capital mobility. In this article we
investigate Asada’s model which was introduced in [1] under flexible exchange
rates. In this case Asada’s model has the form
Y = α (C + I + G + J − Y ), α > 0,
K = I ,
π
e
(1)
(
= γ π −π
e
), γ > 0,
where
– 13 –
P. Medved’ová
A Dynamic Model of a Small Open Economy Under Flexible Exchange Rates
C = c(Y − T ) + C0 ,0 < c < 1, C0 > 0,
T = τY − T0 ,0 < τ < 1, T0 > 0,
I = I (Y , K , r ), I Y =
∂I
∂Y
> 0, I K =
∂I
∂K
< 0, I r =
M
∂L
∂L
= L (Y , r ), LY =
> 0, Lr =
< 0,
p
∂Y
∂r
∂I
∂r
< 0,
(2)
∂J
∂J
< 0, J π =
> 0,
∂Y
∂π
e
⎛
π −π ⎞
⎟, β > 0,
Q = β ⎜⎜ r − r f −
π ⎟⎠
⎝
J = J (Y , π ), J Y =
A = J + Q,
A = 0,
M = constant, (M = 0).
The meanings of the symbols in (1) and (2) are as follows: Y - net real national
income, C - real private consumption expenditure, I - net real private
investment expenditure on physical capital, G - real government expenditure
(fixed), T - real income tax, K - real physical capital stock, M - nominal
money stock, p - price level, r - nominal rate of interest of domestic country,
e
r f - nominal rate of interest of foreign country, π - exchange rate, π -
expected exchange rate of near future, J - balance of current account (net export)
in real terms, Q - balance of capital account in real terms, A - total balance of
α - adjustment speed in goods market, β - degree of
capital mobility, α , β , γ - positive parameters, and the meanings of other
dY dK e dπ e
, K=
, π =
symbols are as follows, Y =
, t - time.
dt
dt
dt
payments in real terms,
In the whole paper we assume as well as Asada in [1] a fixed-price economy, so
that p is exogenously given and normalized to the value 1. Asada assumed that
(
)
the equilibrium on the money market M = L Y , r is always preserved, which
enables using the Implicit-function theorem to express interest rate r as the
function of Y , so
– 14 –
Acta Polytechnica Hungarica
r = r (Y ), rY =
Vol. 8, No. 3, 2011
∂r
> 0.
∂Y
⎡
π e −π ⎤
A = J (Y , π ) + β ⎢r (Y ) − rf −
= 0 with respect
π ⎥⎦
⎣
Solving equation
to π , we have
π = π (Y , π e ), π Y =
∂π − J Y − βrY
∂π
β
=
, ππe =
=
> 0.
e
e
βπ
βπ e
∂Y
∂π
Jπ + 2
Jπ π +
π
π
Further we suppose that r f is also given exogenously because of the assumption
of a small open economy. Under these assumptions, taking into account (2) and
the explicit expression for r , the model (1) takes the form
[
(
(
)) ]
Y = α c (1 − τ )Y + cT0 + C 0 + G + I (Y , K , r (Y )) + J Y , π Y , π e − Y
K = I (Y , K , r (Y ))
(3)
π e = γ [π (Y , π e ) − π e ].
In the whole paper we suppose that:
(i)
the
model
(
(3)
E = Y , K ,π
e∗
), Y
has
a
unique
> 0, K > 0, π
triple of positive parameters (α , β , γ ) .
∗
∗
∗
∗
∗
e∗
equilibrium
point
> 0, to an arbitrary
(ii)
0 < IY + I r rY < 1 − c(1 − τ ) − J Y at the equilibrium point.
(iii)
π π − 1 < 0 at the equilibrium point.
(iv)
The functions in the model (3) have the following properties: the
function I is linear in the variable K and r . The function π is
e
linear in the variable π . The function J is nonlinear in the
variable π . In the variable Y the functions I , J , r , π are
nonlinear, and have continuous partial derivatives with respect to
Y up to the sixth order in a small neighbourhood of the equilibrium
point.
e
In [1] Asada found sufficient conditions for local stability and instability of the
equilibrium point. He studied how changes of the parameter β affect the dynamic
characteristics of the model.
– 15 –
P. Medved’ová
A Dynamic Model of a Small Open Economy Under Flexible Exchange Rates
We analyse the question of the existence of business cycles analytically. Stable
business cycles can arise only in the case when the linear approximation matrix of
the model (3) has at the equilibrium point a pair of purely imaginary eigenvalues
with the third one negative. In Section 2, Theorem 1 gives sufficient conditions for
the existence of a pair of purely imaginary eigenvalues with the third one negative.
The bifurcation equation of the model (3) is very important for the existence of
business cycles. In Section 3, Theorem 2 gives the formulae for the calculation of
the bifurcation coefficients in the bifurcation equation. Theorem 3 speaks about
the existence of business cycles in a small neighbourhood of the equilibrium point.
Such an analytical approach was applied to study similar models in [6], [7], [8],
[9], [10].
2
The Analysis of the Model (3)
Consider an isolated equilibrium poin E
∗
π e > 0,
∗
(
∗
)
= Y ∗ , K ∗ , π e , Y ∗ > 0, K ∗ > 0,
of the model (3).
After the transformation
∗
Y1 = Y − Y ∗ , K1 = K − K ∗ , π 1e = π e − π e ,
the equilibrium point E
the model (3) becomes
[
(
∗
(
)
= Y ∗ , K ∗ , π e goes into the origin E1∗ = (0,0,0), and
∗
) (
(
))
Y1 = α c (1 − τ ) Y1 + Y ∗ + I Y1 + Y ∗ , K1 + K ∗ , r Y1 + Y ∗ + cT0 +
(
(
∗
))
(
+ C0 + G + J Y1 + Y ∗ , π Y1 + Y ∗ , π 1e + π e − Y1 + Y ∗
(
(
K 1 = I Y1 + Y ∗ , K1 + K ∗ , r Y1 + Y ∗
[(
∗
)(
))
∗
)]
)]
(4)
π1e = γ π Y1 + Y ∗ , π 1e + π e − π 1e + π e .
Performing the Taylor expansion of the functions on the right-hand side of this
system at the equilibrium point E1 = (0,0,0 ) the model (4) obtains the form
∗
Y1 = α [c(1 − τ )Y1 + I Y Y1 + I K K1 + I r rY Y1 + J Y Y1 +
~
+ J π π Y Y1 + J π π π e π 1e − Y1 + Y1
]
~
K 1 = I Y Y1 + I K K 1 + I r rY Y1 + K 1
(5)
– 16 –
Acta Polytechnica Hungarica
Vol. 8, No. 3, 2011
π1e = γ [π Y Y1 + π π π 1e − π 1e ] + π~1e ,
e
where
( )
( )
( )
( )
( )
( )
( )
∂I E ∗
∂I E ∗
∂I E ∗
∂r E ∗
∂J E ∗
, IK =
, Ir =
, rY =
, JY =
,
∂Y
∂r
∂Y
∂Y
∂K
∂J E ∗
∂π E ∗
∂π E ∗
~ ~
Jπ =
, πY =
, ππe =
, and the functions Y1 , K 1 , π~1e
e
∂π
∂Y
∂π
IY =
( )
are nonlinear parts of the Taylor expansion.
The linear approximation matrix
following form
A = A(α , β , γ ) of the system (5) has the
⎛ α [c(1 − τ ) + I Y + I rrY + J Y + J π π Y − 1] αI K
⎜
A=⎜
I Y + I r rY
IK
⎜
γπ Y
0
⎝
The characteristic equation of
3
αJ π π π ⎞
⎟
⎟.
− 1 ⎟⎠
e
0
γ (π π
e
)
(6)
A(α , β , γ ) is given by
2
λ + b1λ + b2 λ + b3 = 0,
(7)
where
b1 = − trace A(α , β , γ ) =
= −{α [c (1 − τ ) + I Y + I r rY + J Y + J π π Y − 1] + I K + γ (π π e − 1)}
b2 = sum of all principal second – order minors of A(α , β , γ ) =
(
)
= αI K [c(1 − τ ) + J Y + J π π Y − 1] + γI K π π e − 1 +
{
(
)
+ αγ [c (1 − τ ) + I Y + I r rY + J Y − 1] π π e − 1 − J π π Y
}
b3 = − det A(α , β , γ ) =
= −αγ I K
{(π
πe
)
}
− 1 [c (1 − τ ) + J Y − 1] − J π π Y . As we are interested in the existence and stability of limit cycles we need to find
such values of parameters α , β , γ at which the equation (7) has a pair of purely
imaginary eigenvalues and the third one is negative. We will call such values of
parameters α , β , γ the critical values of the model (3). We denote these critical
values by
α0 , β0 , γ 0.
The mentioned types of eigenvalues are ensured by the
Liu’s conditions [5].
– 17 –
P. Medved’ová
A Dynamic Model of a Small Open Economy Under Flexible Exchange Rates
b1 > 0, b3 > 0, b1b2 − b3 = 0.
(8)
For an arbitrary α exists γ such that the first inequality is satisfied under the
condition (iii). The second inequality is satisfied under the condition
(π
πe
− 1)[c(1 − τ ) + J Y − 1] > J π π Y . This condition is satisfied when β is
sufficiently small.
The equation b1b2 − b3 = 0 gives
{α [c(1 − τ ) + I + I r + J + J π − 1] + I + γ (π − 1)}.
.{γI (π − 1) + αI [c(1 − τ ) + J + J π − 1] +
+ αγ [(c (1 − τ ) + I + I r + J − 1)(π − 1) − J π ]}−
Y
K
r Y
πe
π
Y
K
Y
Y
r Y
Y
Y
πe
K
π
Y
π
πe
Y
− αγI K {(π π e − 1)[c(1 − τ ) + J Y − 1] − J π π Y } = 0.
π Y < 0 and for π Y = 0. The
= 0 is satisfied under the condition π Y > 0. This inequality
Asada indicated in [1] that b1b2 − b3 > 0 for
equation b1b2 − b3
is satisfied if
The
β<
equation
− JY
.
rY
b1b2 − b3 = 0
can
be
expressed
in
the
f 1 (α , β )γ + f 2 (α , β )γ + f 3 (α , β ) = 0,
2
where
f1 (α , β ) = J π2π 2 [I K − α (J Y − R ) − αβ rY ]
f 2 (α , β ) = α 2 β 2TJ π πrY + α 2 β J π π ( J Y − R )(T − J π πrY ) −
− α 2 J π2π 2 (R − J Y ) − 2αβ I K J π πT + 2αI K J π2π 2 (J Y − R ) −
2
− βI K2 J π π − I K2 J π2π 2
f 3 (α , β ) = α 2 β 2 I K TU + α 2 βI K J π π [T (P − J Y ) + U (R − J Y )] +
+ α 2 I K J π2π 2 (R − J Y )(P − J Y ) + αβ 2 I K2 U +
+ αβ I K2 J π π (P − J Y + U ) + αI K2 J π2π 2 (P − J Y )
where
– 18 –
form
Acta Polytechnica Hungarica
Vol. 8, No. 3, 2011
P = c(1 − τ ) + J Y − 1 < 0,
R = P + IY + I r rY < 0,
T = R − J π πrY < 0,
U = P − Jπ πrY < 0.
We see that for an arbitrary
γ exists α̂ such that f1 (αˆ ,0) = 0, αˆ > 0.
Put F (α , β , γ ) = f 1 (α , β ) + f 2 (α , β )
Instead of
γ introduce ϑ =
1
γ
1
γ
+ f 3 (α , β )
1
γ2
and consider an equation
= 0.
Φ(α , β ,ϑ ) = 0 when
f1 (α , β ),
ϑ =0
⎧
Φ(α , β ,ϑ ) = ⎨
2
⎩ f1 (α , β ) + f 2 (α , β )ϑ + f 3 (α , β )ϑ , ϑ ≠ 0.
We see that
Φ(α , β ,ϑ ) = 0 is equivalent to F (α , β , γ ) = 0. Analyze
Φ(α , β ,ϑ ) = 0.
It holds:
1.
Φ(αˆ , β = 0,ϑ = 0) = 0
2.
∂Φ(αˆ , β = 0,ϑ = 0)
= − J π2π 2 (J Y − R ) ≠ 0.
∂α
α = f (β ,ϑ ) in a small
αˆ = f (0,0) and
that
By the implicit function theorem there exists a function
neighbourhood
of
Φ( f (β ,ϑ ), β ,ϑ ) = 0.
(β = 0,ϑ = 0)
γ and sufficiently small β 0
of parameter β there exists value α 0 of parameter α such that the triple
is (α 0 , β 0 , γ 0 ) the critical triple of the model (3). The following theorem gives
We see that for a sufficiently large
γ0
such
of parameter
sufficient conditions for the existence of a critical triple of the model (3).
– 19 –
P. Medved’ová
A Dynamic Model of a Small Open Economy Under Flexible Exchange Rates
Theorem 1. Let the condition
π π − 1 < 0 is satisfied. If parameter β <
e
is sufficiently small and parameter
γ is sufficiently large, then there exists a
(α 0 , β 0 , γ 0 ) of model (5).
critical triple
3
− JY
rY
Existence of Limit Cycles and their Stability
According to the assumption (iv) the model (5) can be itemized in the form
{
}
Y1 = α [I Y + I r rY − (1 − c(1 − τ ) − J Y ) + J π π Y ]Y1 + I K K1 + J π π π e π 1e +
(
)
( )
(
)
2
1
1
1
+ α I Y(2 ) + J Y(2 ) Y12 + αJ π(2e ) π 1e + α I Y(3) + J Y3 Y13 +
2
2
6
( )
(
)
( )
(
3
4
1
1
1
+ αJ π(3e) π 1e + α IY(4 ) + J Y(4 ) Y14 + αJ π(4e ) π 1e + O5 Y1 , π 1e
6
24
24
)
1
1
1 (4 ) 4
K 1 = (I Y + I r rY )Y1 + I K K1 + I Y(2 )Y12 + I Y(3)Y13 +
I Y Y1 +
2
6
24
+ O5 (Y1 )
(9)
π1e = γπ Y Y1 + γ (π π − 1)π 1e + γπ Y(2 )Y12 + γπ Y(3)Y13 +
1
2
e
1
6
1
γπ Y(4 )Y14 +
24
+ O5 (Y1 ),
where
I Y(2 ) =
( )
( )
( )
( )
( )
( )
∗
∗
∗
3
4
2
∂2I E∗
(3 ) ∂ I E
(4 ) ∂ I E
(2 ) ∂ J E
,
I
=
,
I
=
,
J
=
,
Y
Y
Y
∂Y 2
∂Y 3
∂Y 4
∂Y 2
( )
( )
( )
( )
∂ J (E ) ( ) ∂ π (E ) ( ) ∂ π (E ) ( ) ∂ π (E )
J( ) =
,π =
,π =
,π =
.
∂Y
∂Y
∂Y
∂ (π )
J Y(3) =
4
π
e
4
∗
∗
∗
2
3
∂3 J E∗
(4 ) ∂ J E
(2 ) ∂ J E
(3 ) ∂ J E
J
J
,
J
=
,
,
,
=
=
e
e
Y
2
3
π
π
∂Y 3
∂Y 4
∂ πe
∂ πe
4
∗
e 4
2
Y
∗
2
2
3
Y
– 20 –
∗
3
3
4
Y
∗
4
4
Acta Polytechnica Hungarica
Vol. 8, No. 3, 2011
Consider a critical triple
(α 0 , β 0 , γ 0 )
of the model (3). Let us investigate the
Y1 , K1 and π 1e around the equilibrium E1∗ = (0,0,0) with respect to
the parameter α ,α ∈ (α 0 − ε ,α 0 + ε ), ε > 0, and the fixed parameters
β = β0 , γ = γ 0.
behavior of
After the shifting
α0
into the origin by relation
α1 = α − α 0 ,
the model (9)
becomes
Y1 = α 0 [I Y + I r rY − (1 − c (1 − τ ) − J Y ) + J π π Y ]Y1 + α 0 I K K1 +
+ α 0 J π π π e π 1e + [IY + I r rY − (1 − c(1 − τ ) − J Y ) + J π π Y ]Y1α1 +
(
)
( )
2
1
1
+ I K K1α1 + J π π π e π 1eα1 + α 0 I Y(2 ) + J Y(2 ) Y12 + α 0 J π(2e ) π 1e +
2
2
+
(
)
( )
(
)
2
1 (2 )
1
1
I Y + J Y(2 ) Y12α1 + J π(2e ) π 1e α1 + α 0 I Y(3 ) + J Y(3 ) Y13 +
2
2
6
( )
(
)
( )
3
3
1
1
1
+ α 0 J π(3e ) π 1e + I Y(3 ) + J Y(3 ) Y13α1 + J π(3e ) π 1e α1 +
6
6
6
+
(
)
( )
(
4
1
1
α 0 I Y(4 ) + J Y(4 ) Y14 + α 0 J π(4e ) π 1e + O5 Y1 , π 1e , α1
24
24
(10)
)
1
1
1 (4 ) 4
K 1 = (I Y + I r rY )Y1 + I K K1 + I Y(2 )Y12 + I Y(3 )Y13 +
I Y Y1 + O5 (Y1 )
2
6
24
π1e = γ 0π YY1 + γ 0 (π π − 1)π 1e + γ 0π Y(2 )Y12 + γ 0π Y(3)Y13 +
e
1
2
1
6
1
γ 0π Y(4 )Y14 +
24
+ O5 (Y1 ).
Denote the eigenvalues of (6) as
λ1 = ξ (α , β , γ ) + iω (α , β , γ ), λ2 = ξ (α , β , γ ) − iω (α , β , γ ),
λ3 = λ3 (α , β , γ )
and let
λ1 = iω0 , λ2 = −iω 0 , ω0 = ω (α 0 , β 0 , γ 0 ), λ30 = λ3 (α 0 , β 0 , γ 0 ).
Express the model (10) in the form
~
x = A(α 0 , β 0 , γ 0 )x + Y(x,α1 ),
– 21 –
P. Medved’ová
A Dynamic Model of a Small Open Economy Under Flexible Exchange Rates
where
~
⎛Y ⎞
⎛ Y1 ⎞
⎜ ⎟ ~ ⎜ ~1 ⎟
x = ⎜ K1 ⎟, Y = ⎜ Y2 ⎟.
⎜⎜ ~ ⎟⎟
⎜π e ⎟
⎝ 1⎠
⎝ Y3 ⎠
( )
Consider a matrix M = mij ,
i, j = 1,2,3, which transfers the matrix
( )
A(α 0 , β 0 , γ 0 ) into its Jordan form J, and its inverse matrix M −1 = mij−1 .
⎛ Y2 ⎞
⎜ ⎟
By the transformation x = My , y = ⎜ K 2 ⎟ we obtain
⎜π e ⎟
⎝ 2⎠
y = Jy + F(y,α1 ),
(11)
Where
⎛ iω 0
⎜
J =⎜ 0
⎜ 0
⎝
0 ⎞
⎟
− iω0 0 ⎟,
λ30 ⎟⎠
0
−1 ~
−1 ~
−1 ~
⎛ F1 (y,α1 ) ⎞ ⎛⎜ m11 Y1 + m12 Y2 + m13 Y3 ⎞⎟
⎜
⎟
−1 ~
−1 ~
−1 ~
F(y,α1 ) = ⎜ F2 (y,α1 )⎟ = ⎜ m21
Y1 + m22
Y2 + m23
Y3 ⎟,
⎜ F (y,α )⎟ ⎜⎜ m −1Y~ + m −1Y~ + m −1Y~ ⎟⎟
32 2
33 3 ⎠
1 ⎠
⎝ 3
⎝ 31 1
0
K 2 = Y2 , F2 = F1, and F3 is real function (the symbol "−" means complex
conjugate expression).
Theorem 2. There exists a polynomial transformation
Y2 = Y3 + h1 (Y3 , K 3 ,α 1 )
K 2 = K 3 + h2 (Y3 , K 3 ,α 1 )
(12)
π 2e = π 3e + h3 (Y3 , K 3 ,α 1 ) ,
where h j (Y3 , K 3 , α 1 ), j = 1,2,3, are nonlinear polynomials with constant
coefficients of the kind
– 22 –
Acta Polytechnica Hungarica
h j (Y3 , K 3 ,α 1 ) =
Vol. 8, No. 3, 2011
4 − 2 m3
∑ h( {
m1 , m2 , m3 ) m1
j
3
m1 + m2 + m3 ≥ 2 , m3∈ 0 ,1}
which transforms the model
(
Y2 = iω0Y2 + F1 Y2 , K 2 , π 2e ,α 1
Y K 3m2 α 1m3 , j = 1,2,3, h2 = h1 ,
)
(
K 2 = −iω0 K 2 + F2 Y2 , K 2 , π 2e , α1
)
(11)
π 2e = λ30π 2e + F3 (Y2 , K 2 , π 2e , α1 )
into its partial normal form on a center manifold
(
)
(
Y3 = iω 0Y3 + δ 1Y3α 1 + δ 2Y32 K 3 + U D Y3 , K 3 , π 3e , α 1 + U ∗ Y3 , K 3 , π 3e , α 1
K 3 = −iω 0 K 3 + δ 1 K 3α 1 + δ 2Y3 K 32 + U D Y3 , K 3 , π 3e ,α 1 +
(
+ U ∗ Y3 , K 3 , π 3e ,α 1
(
)
)
)
(13)
π 3e = λ30π 3e + V D (Y3 , K 3 , π 3e , α 1 ) + V ∗ (Y3 , K 3 , π 3e , α 1 ),
where U (Y3 , K 3 ,0, α 1 ) = V (Y3 , K 3 ,0, α 1 ) = 0,
D
D
(
( αY,
) ( α ) U~(Y , K ,π ,α ),
~
,α ) = ( α ) V (Y , K , π ,α ),
5
U ∗ α1 Y3 , α1 K 3 , α1 π 3e ,α1 =
V∗
1 3
α1 K 3 , α1 π 3e
1
3
5
1
1
3
e
3
3
3
e
3
1
1
~ ~
and U ,V are continuous functions.
The resonant coefficients δ 1 and
formulae
δ2
in the model (13) are determined by the
δ1 = m11−1 {[I Y + I r rY + c(1 − τ ) − 1 + J Y + J π π Y ]m11 + I k m21 + J π π π m31 }
e
δ 2 = m11−1 ⎢ α 0 (I Y(3) + J Y(3) )m112 m12 + α 0 J π(3)m312 m32 +
⎡1
⎣2
1
2
e
]
+ α 0 (I Y(2 ) + J Y(2 ) )A + α 0 J π(2e ) B +
⎡1
⎤
⎡1
⎤
+ m12−1 ⎢ I Y(3 )m112 m12 + I Y(2 ) A⎥ + m13−1 ⎢ γ 0π Y(3 )m112 m12 + γ 0π Y(2 ) A⎥,
⎣2
⎦
⎣2
⎦
where
– 23 –
P. Medved’ová
A Dynamic Model of a Small Open Economy Under Flexible Exchange Rates
(Cm
1
h1(1,1, 0 ) = −
−1
11
iω 0
+ Dm12−1 + Em13−1
(
h1(2, 0, 0 ) =
1
Fm11−1 + Gm12−1 + Hm13−1
2iω 0
h2(1,1, 0 ) =
1
iω 0
h2(2, 0, 0 ) =
−1
21
−1
−1
+ Dm22
+ Em23
)
)
(
1
−1
−1
−1
Fm21
+ Gm22
+ Hm23
6iω 0
h3(1,1, 0 ) = −
h3(2, 0, 0 ) =
(Cm
1
λ30
(Cm
−1
31
1
)
)
−1
−1
)
+ Dm32
+ Em33
2(2iω 0 − λ30
(Fm
)
−1
31
)
−1
−1
+ Gm32
+ Hm33
,
and
(
)
(
)
A = m112 h1(1,1,0 ) + m122 h2(2, 0,0 ) + m11m12 h2(1,1, 0 ) + h1(2,0, 0 ) +
+ m11m13 h3(1,1, 0 ) + m12 m13 h3(2, 0, 0 )
2 (1,1, 0 )
2 (2, 0, 0 )
B = m31
h1
+ m32
h2
+ m31m32 h2(1,1, 0 ) + h1(2 , 0 , 0 ) +
+ m31 m33 h3(1,1, 0 ) + m32 m33 h3(2 , 0 , 0 )
C = m11m12α 0 (I Y(2 ) + J Y(2 ) ) + m31m32α 0 J π(2e )
D = m11m12 I Y(2 )
E = m11m12γ 0π Y(2 )
(
)
2
F = m112 α 0 I Y(2 ) + J Y(2 ) + m31
α 0 J π(2e )
G = m112 I Y(2 )
H = m112 γ 0π Y(2 ) ,
while
all
partial
derivatives
are
Y1 = K1 = π = 0,α 1 = 0.
e
1
– 24 –
calculated
at
the
values
Acta Polytechnica Hungarica
Vol. 8, No. 3, 2011
Proof. The unknown terms
h (jm1 ,m2 ,m3 ) , j = 1,2,3 and the resonant terms δ1 , δ 2
can be found by the standard procedure which is described for the example in [2].
As the whole process of finding them is rather elaborate, we do not present it here.
iϕ
In polar coordinates Y3 = re , K 3 = re
(
)
(
− iϕ
the model (13) can be written as
)
(
)
r = r ar 2 + bα 1 + R D r , ϕ , π 3e , α 1 + R ∗ r , ϕ , π 3e , α 1
1
ϕ = ω 0 + cα 1 + dr 2 + Φ D r ,ϕ , π 3e ,α 1 + Φ ∗ r ,ϕ , π 3e ,α 1
r
[ (
)
(
)]
(14)
π 3e = λ30π 3e + W D (r ,ϕ , π 3e ,α 1 ) + W ∗ (r ,ϕ , π 3e ,α 1 ),
where
a = Re δ 2 , b = Re δ1 . The equation
ar 2 + bα1 = 0
(15)
is the bifurcation equation of the model (14). It determines the behaviour of
solutions in a neigbourhood of the equilibrium point of the model (5). Utilizing the
results from the bifurcation theory [3], [11] we can formulate the following
theorem.
Theorem 3. Let the coefficients a, b in the bifurcation equation (15) exist.
a < 0 then there exists a stable limit cycle for every small enough
α1 > 0, if b is positive and for every small enough α1 < 0, if b is
1) If
negative.
a>0
α1 < 0,
2) If
then there exists an unstable limit cycle for every small enough
if
b is positive and for every small enough α1 > 0, if b is
negative.
Conclusions
The main contributions of this paper are the results in Theorem 1and in Theorem
2. Theorem 1 gives sufficient conditions for the existence of a critical triple of the
model (3). Theorem 2 gives the formulae for the calculation of the first two
resonant coefficients of the model. These theorems are important for the
investigation of the existence of limit cycles which are interpreted as business
cycles in economics. We intend to show an application of both the model of
flexible exchange rates and the model of fixed exchange rates on selected
countries.
Acknowledgement
The work was supported by the Slovak grant agency VEGA No. 1/0828/10.
– 25 –
P. Medved’ová
A Dynamic Model of a Small Open Economy Under Flexible Exchange Rates
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– 26 –