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Stability and bifurcation of a model of the
price mechanism of export-import transactions
March 21, 2003
In modern literature on economics, there exists a rather detailed qualita-
tive description of a mechanism of the formation of home commodity prices,
based on an analysis of the dynamics of streams of gold and currency in
export-import transactions [1]. A special attention is paid to such an impor-
tant factor as an index of the trade balance whose surplus or deficit determine
the tendency of price changes.
Before considering a formalized mathematical model of the dynamics of
prices on the basis of the classical monetary theory, we must specify a number
of assumptions:
1) we consider a scheme of free trade without any influence of governments
and monopolistic agencies:
2) the national income level is given, whereas the level of prices is deter-
mined on the basis of a qualitative monetary theory;
3) during the considered period, changes in the money supply are deter-
mined only by a deficit (or a surplus) of the trade balance:
4) as currency exchange rates are assumed to be fixed, we can set them
equal to unity, which is equivalent to international transactions in gold;
5) transport expenses, insurance and other expenses are not taken into
account both for commodity and financial streams.
Below, we use the following notation: @ is the money supply; V is the
speed of the money turnout {a constant); Y is the national income level (a
constant); P is the level of domestic prices; Py is the level of foreign prices
(a constant); M is the volume of import; X is the volume of export,
The basic equation of the model has the form
QV = PY. (1)
The function of the volume of export is a decreasing function of the domestic
price
X=X(P),
whereas the function of import, in contrast, is an increasing function of the
domestic price:
dM
M=M(P), 75
> 0.
A relation of the form
P*X (P*) — PuM (P*) =0
determines the condition of equilibrium of the trade balance. We assume that
this equation admits positive solutions that determine equilibrium values of
the domestic price P*.
A violation of equilibrium is accompanied by changes in the money supply
and is expressed by means of the equation
& = PX (P)— PyM(P). (2)
As follows from equation (1), a change in the money supply leads to a change
of the domestic price P(t). It is justified to assume that such a change is not
instantaneous, that is, there occurs a time delay characterized by a constant
r. In this case, equation (1) can be written as
dP(t)_ VdQ(t-7)
ds Y da (8)
Equations (2) and (3) yield a difference-differential equation that describes
the dynamics of the domestic price:
—— = —{P(t—1)X[P(t—7)] — PuM [P (t-7)}}- (4)
We will assume that the functions of export X (P) and import M (P) are
nonlinear and can be expanded into a Taylor series up to the third power,
ie.,
X(P)=Xo- MiP + XP? + X3P*+0(P*),
M (P) = My + MP + MyP? + M;P*+0(M'), (5)
where
_ dX (P*) _ dM (P*)
1 ape ap
are corresponding derivatives of the functions X (P) and M (P) at the point
of equilibrium P*. Let us introduce a new quantity P(t) = P(t) — P* that
has the meaning of a deviation of the domestic price from its equilibrium
value. In this case, equation (4), taking into account (4), is reduced to the
form
i =0,1,2,3,
dP _1V
do oY
+ (P*X2 — PyMz — X1) P? (€- 1)
+ (P*Xs — Pu Ms + X2) P*(E-1)}, (6)
{(Xo — P*X1 — PM) P (E- 1)
where ¢ = rt.
At first, we investigate the conditions of local stability, restricting our-
selves to the linear part of (6), i-e.,
“ - oP (1), @=Xy- P*X,— PyM. (7)
The characteristic equation for (7) is
TVa
Y
Applying the well-known results of the theory of the stability of difference-
differential equations [2] to equation (7), we obtain the necessary and suffi-
cient conditions of the stability:
An e*=0. (8)
0<-T- <5: (9)
As follows from the left-hand side of the double inequality (9), the quan-
tity a is negative, whereas the right-hand side implies that the absolute
3
value of a is less than zt. Condition (9) has a rather transparent economic
interpretation, which can be illustrated by a transformation of the initial
parameters.
Let
where
under the condition P*X) = PyM. We will call nx and ny elasticities of
the functions of export and import with respect to the price. As Xo > 0, the
condition a < 0 is equivalent to
1—nx — 1m <0,
or
nx +n > 1,
which corresponds to so-called Marshall-Lerner’s conditions. At the same
time, the condition (9) reduces to the form
Yu
1 1+—. 1
<tx + 1M < tow (10)
Thus, the economic interpretation of the conditions of local stability con-
sists in the fact that the sum of the elasticities should not only exceed unity,
but should also be less than another critical value. In other words, instabil-
ity in the investigated economic model may occur not only when the sum of
elasticities is sufficiently small, but also in the case of a considerably large
value of this sum.
Let us study the behavior properties of the initial dynamic system (6) in a
small neighborhood of the bounds of the inequality (10). First, we consider
the situation near the lower bound. Let us introduce a small parameter
u=1- nx — nm. In this case, with changing sign of yz, an eigenvalue of the
linearized problem passes through zero, and the stationary state P* either
may not exist or may split into several stationary states. That is, there
occurs a bifurcation of stationary solutions.
The difference-differential equation (6) can be represented as follows:
dP
7 A,P (€— 1) + AP? (f— 1) + AP? (f- 1), (11)
where
TVX TW
A= etn Ans 5 (P'X2 ~ Pula X),
TV ie
As = a (P*X3 — PuM3 + X2).
We additionally assume that the quantity v = Pxe Pui is also
small, i.e., 4g = 4*2v. Using the method of central manifold [3], one can
show that, under the condition of smallness of A;, Ay and a finite delay time
T, the difference-differential equation (11) is topologically equivalent to the
differential equation
dP =A P+ AP (D+ APO. (12)
By means of a linear change of variable P = P + aon equation (12) is given
the form
ca = 0, +02,P + AsP?, (13)
dt
where
_ 2A3 7 Aj Ag _A- A3
oa BA,’ Bay
The transformation P(t) = |B| V (¢) explicitly yields Poincaré’s normal form
for the differential equation (13):
dv
—=6,+ &V+SV', (14)
dt
where 1
ay .
B=—, fi= as, fo=a2, S=signB=+1.
Vas ry
For definitiveness, we set S = —1. Equation (14) can have three points
of equilibrium. A ”fold” bifurcation is determined by a line R in the plane
(1, (2, given by a projection of the line
Tr: By + hV-V"=0,
. By — 38V
onto the parameter plane. By eliminating V from these equations, we get
the projection
R= {(61, 5) 1463 + 278) = 0}.
The curve RF is called a semicubic parabola and has two branches R,, Rp that
meet each other tangentially at a ”cusp” point given by 6; = 0, = 0. The
separatrix divides the parameter plane into two regions (see the Figure).
In region 1, in front of the boundary line, there are three points of equi-
librium: two of them are stable and one is unstable. In region 2, beyond
the boundary line, there is a unique equilibrium point, and it is stable. A
non-degenerate ”fold” bifurcation occurs by crossing R, or R2 at any point
in the plane of the parameters {,, 62 which is different from the origin. If
the curve is crossed from region 1 to region 2, the right stable equilibrium
point merges with the unstable one and both of them vanish. Analogously,
the left stable point of equilibrium merges with the unstable one on the line
Ry. When approaching the “cusp” point in front of region 1, all the three
equilibrium states merge as a triple root of the right-hand side of the initial
equation (14). Of importance is also the fact that, by passing from a sta-
ble regime to unstable one in (14), hysteresis phenomenon is observed and a
catastrophe occurs [3].
The case S = 1 can be analyzed along the same lines.
Now we analyze the situation when the sum of elasticities 7x + qu is
close to the value at the right bound of the inequality (10). Let us introduce
a small parameter p = Xo (nx + uw — 1) — §. In this case, the difference-
differential equation (6) takes the form
dP T\ = TW aig yes
o=- (u+ 5) PO + {(P'X — Pad — Xy) P(E-1)
+ (P*X5 — Pu Ms + X2) P*(E-1)}, (15)
The characteristic equation for the linear part of (15) reads
A+ (u+ 5) eA=0. (16)
Let us find out whether this equation posses a pair of pure imaginary
roots A = tiw, i? = —1, w > 0. If A = +iw, then
T T\
(u+ >) cosw=0, w- (u+3) sinw = 0.
6
This implies that w = (2n + 1)$ (n= 0,1,...), and w= an (n= 0,2,4,...)
are critical values of yz. We will consider only the case n = 0. Thus, it has
been shown that equation (16) for 4. = 0 possesses a pair of pure imaginary
roots +4. It is not difficult to show that (16) does not posses roots with
positive real parts.
Given that \ is analytic with respect to 4, differentiation of (16) yields:
dX _ tt
= =2-_.
dH\-9 «6+
As a results, for 4s = 0, all the conditions of Hopf’s theorem are met, because
the real part of the derivative of the eigenvalue with respect to yz is not equal
to zero.
Using the above results, we will prove that the system (15) admits a family
of periodic solutions P. (#) (€ > 0), where € is a measure of the amplitude
P.O
max ;
t
and € is sufficiently small at that.
Our task is to study the bifurcation of birth (death) of a cycle in the
difference-differential equation (15). To reduce this equation to a single com-
plex equation, we will again employ the method of central manifold [4].
Equation (15) contains a great number of parameters. In order to simplify
further consideration, we make a change of variable
P= Xo (nx +n — 1)
=X Pulh —Pxy"-
For 4 = 0, equation (15) takes the form
du(t) _
dt
-+ [u(@—1) +02 (F-1) + we? F-0)]
+ (P*Xs — PuMs + Xz) PF (E-1)}, (17)
where
_ Xo (X2 + P*X3 — Pur Ms)
(Xy — P*X) + PyMoy *
By use of the theorem on central manifold, the difference-differential equation
(17) is reduced to the complex differential equation
. oT Ze - Vz V4
4 = 192 + gaz t+ 9uZ2 + Goa + 901 2
(18)
where Z = Z (2) is a complex-valued function, Z (i) is the complex conjugate
to Z(t). The coefficients of the powers of Z and Z are determined by
920 = —911 = Jon = 7D,
gn = 2x {(* mat - i917) D+ 50d +i0"}, (19)
5 4 4
14th Lng
Tm = 2
142 14
The availability of concrete values of the coefficients of the nonlinear part
of equation (18) makes it possible to employ equations of reference [4] for
the determination of the stability, the direction of birth, the period and the
asymptotic form of periodic solutions of a small amplitude that realize an
Andronov-Hopf bifurcation from the stationary state. From (19), we get an
explicit form of Liapunov’s first quantity
T 2 mill 3y7 .(m 11 390
= {22 (2 54) (2424). 2
(0) at art ra (Gt yt T)} (20)
The real parts of (20) are negative under the condition
16 — 447
= = —0.826... 21
1>%= EG 0.826 (21)
This means that a limit cycle will be stable for y > 7- and unstable under a
violation of the condition (21). As to the stable limit cycle, we have obtained
expressions for its main characteristics:
1) the amplitude is « = _—
2) the period is T. = 4 (1 + 2e);
3) the asymptotic form of the periodic solution is
at 2, 1
uc (£) = 2€ cos (=) + 2e [ sin (xt) — 3 60s (wi) - 1],
J
8
p Xo(nx +nu — 1)
PO = 54 Py; Px
In this case, the cycle is born in the direction 4 > 0, whereas the born
periodic solution is asymptotically stable. The corresponding regime of the
formation of auto-oscillations is called soft. However, under the condition
Y < Yc, we have an unstable limit cycle. The loss of stability under the
formation of auto-oscillations occurs rigidly, i-e., there exists a possibility of
a sudden transition to a new stationary or nonstationary regimes. In a real
system, this way of the loss of stability is accompanied by a catastrophe.
The initial system (17) exhibits the most interesting behavior in the sit-
uation when the parameter 7¥ is close to its critical value yc, i-e., when the
quantity € = y— 7 is small. In this case, we can observe the so-called
Bautin’s bifurcation that admits simultaneous coexistence of both the stable
and unstable cycles. An analysis of qualitative properties of this bifurcation
involves a Taylor expansion of the right-hand side of (4) up to fifth order.
Furthermore, using the method of central manifold, one has to reduce the
functional equation to a complex differential equation that contains nonlin-
ear terms up to fifth order. After that, using the corresponding bifurcation
formulas, one can evaluate Liapunov’s second quantity, whose sign uniquely
determines the main parameters and peculiarities of this bifurcation.
Thus, the results of our investigation of behavior properties of the difference-
differential equation (4) allow us to draw the following conclusions about
bifurcations of codimension two:
1) at the left boundary (7x + nw = 1), there exists a bifurcation of the
cusp” type;
2) at the right boundary (7x + = 1+ 2%), there exists a bifurcation
of Bautin’s type.
The separatrices in the parameter space for each of the above-mentioned
bifurcations, respectively, take the following form:
a) for the ”cusp” bifurcation,
Xo (1 — 9x — nu) (P*Xa — Pu Mg + Xa) = = (P*X2 — PMz — X41)’;
BLE
b) for Bautin’s bifurcation,
Xo (1 — nx — nu) (P*X3 — Pur Mz + X2).-
= 16—4hn, (P’Xe— Py Ma —Xy)’
15a?
In conclusion, we point out that an application of mathematical meth-
ods to the analysis of concrete objects is related to numerical results and
corresponding contextual interpretation. In this sense, the function of qual-
itative theory of difference-differential equations is somewhat different: It
focuses attention on characteristic features of the phenomenon as a whole,
on qualitative prediction of its behavior. Our task is a search for irreducible
topological structures that subdivide the phase picture of the system. Ap-
plied part consists in establishing a relationship between these structures of
the phase space and economic processes, complemented by implementation of
a bifurcation analysis. In this case, we would have to take into account prop-
erties of the real object that impose restrictions on both the phase variables
and the constants of equations (4)-(6).
References
[1] G. Gondolfo, Economic Dynamics (Springer, Berlin, 1996).
[2] J. Hale, Theory of Functional Differential Equations (Springer, Berlin,
1977).
[3] Yu. Kuznetsov, Elements of Applied Bifurcation Theory (Springer, Berlin,
1994).
[4] B. Hassard, N. Kazarinoff, Y.-H. Wan, Theory and Application of Hopf
Bifurcation (Cambridge University Press, London, 1981).
10
Bifurcation diagram
B,
Fig. 1
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Bifurcation diagram
B,
Fig. 1
