ISSN 1320-0682
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ISSN 1320-0682 |
| Source: | http://www.complexity.org.au/ci/vol06/chiarella/chiarella.html | Received: | 01/07/1998 | ||
| Vol 6: | Copyright 1998 | Accepted for publication: | 15/10/1998 |
Carl Chiarella and Xue-Zhong He
School of Finance and Economics
University of Technology, Sydney
PO Box 123 Broadway
NSW 2007, Australia
Email: carl.chiarella@uts.edu.au tony.he1@uts.edu.au
WWW: http://www.bus.uts.edu.au
In this paper we consider how suppliers in a cobweb model may learn about their economic environment. Instead of assuming the one step backward-looking expectation scheme of the traditional linear cobweb model, we consider the subjective estimates of the statistical distribution of the market prices based on L-step backward time series of market clearing prices. With constant risk aversion, the cobweb model becomes nonlinear. Sufficient conditions on the local stability of the unique positive equilibrium of the nonlinear model are derived and, consequently, we show that the local stability region (of the parameters of the equation) is proportional to the lag length L. When the equilibrium loses its local stability, we show that, for L=2, the model has strong 1:3 resonance bifurcation and a family of fixed points of order 3 becomes unstable on both sides of criticality. The numerical simulations suggest that the model has a simple global structure, it has no complicated dynamics as claimed recently by Boussard. However, complicated dynamics do appear when the model is modified with constant elasticity supply and demand.
Consider the well-known cobweb model:
Here,
and
are quantities and prices, respectively, at
period t,
is the price expected at time t based
on the information at t-1, and
and
are constants.
Instead of assuming the backward-looking expectation scheme
as in Boussard [2], we rather assume that
is a random variable drawn from a normal distribution. Let
and
be the mean and variance of
,
respectively. With constant absolute risk aversion A, the
marginal revenue certainty equivalent is
1
Suppose a linear marginal cost, as in (1), so that the supply equation, under marginal revenue certainty equivalent becomes
Combining (2) and (3) and equating supply and
demand gives the market clearing price in period t as a function
of the subjective mean
and variance
We assume agents form their subjective estimates of the mean and variance by considering past market clearing prices over a window of length L, that is
and
Let
Then, it follows from (5)-(6),
Hence
Because of the dependence of the subjective mean
and
variance
as price lagged L periods equation (8)
is a difference equation of order L. It is more convenient to
reduce it to a system of L first order difference equations.
Let
and
Then equation (8) can be written as the following difference system
One can see that the system (9) has a unique positive
fixed point
satisfying
which implies
.
It can be verified that
Therefore the Jacobian matrix of the system
(9) at the steady state is given by the
matrix
Denote
and
with
. We then have
and more generally,
Thus
Using Jury's Test (see appendix A), we can derive the following local stability result. The proof of theorem 2 can be found in the appendix A.
It is interesting to notice that both the equilibrium
and the local stability condition of the
system (9) are independent from the risk aversion A. As
pointed out by Boussard [2], this is a peculiarity of the
particular expectation hypothesis chosen here. Yet, under our
assumption,
plays a key role on the local stability
of the positive equilibrium of the system (9).
Furthermore, the local stability condition
implies
that the region of the parameter
on the local stability
of the positive equilibrium of the system (9) is
proportional to the lag length L. Theorem 2 tells us that
larger time lags lead to larger region of stability (in terms of
the parameter
).
Let us consider the simplest case first, that is the case when L=2. Then we have a system
where
Let
so that the fixed point is at the origin. We then have
where
The Jacobian of the system (13) at the origin is then given by
From L=2, we have
. It follows from Theorem
2 that the equilibrium
of the system
(12) will lose its stability when
passes through
1.
Also, when
is near 1, the Jacobian matrix
has a
pair of complex eigenvalues, say
and
with
where
satisfies
Let
be the value of
when
, that is,
Obviously,
=1. For a map in
, according to
Kuznetsov [6] (page 350), there is no ''strong
resonances" if there is an eigenvalue, say
,
satisfying
for q=1, 2, 3, 4. Otherwise, we
say the map has a 1:q resonance (q=1, 2, 3, 4). Hence our
map has a 1:3 resonance. As pointed out by Hale and
Kocak [4] (page 481), the dynamics of such maps --
strong resonances -- can be exceedingly complicated and the answer is
not yet completely known. The complexity of such maps is illustrated
in one of the Example 15.34 in Hale and Kocak [4] (pages
481-482). For more detailed discussion on the bifurcations of fixed
points in discrete-time maps on
with both weak and strong
resonances, we refer the reader to Iooss [5] when the maps
involve one parameter and to Kuznetsov [6] when the maps
involve two parameters.
The rest of this section is devoted to the study of generic
bifurcations of the fixed points
of the map defined
by (12). To keep the discussion simple, we will treat
as the only parameter of the map.
To perform a standard normal form calculation (see Arrowsmith et. al. [1]) for the system (13), we write the function g in the following form
with
We now introduce complex coordinates
From which, we have
Then the linear part of the map becomes
Hence the mapping can be written in a complex form
where
The following Lemma on the normal form of the map (13) (with 1:3 resonance) can be found in Kuznetsov [6] (p.382).
It can be verified that
. Hence,
. Using a
result from Iooss [5] (p.110, Theorem 1), we have the
following bifurcation result.
Theorem 2 indicates the dynamic structure near the positive
equilibrium
and the hyperbolic periodic points
bifurcating from
near the critical value
. In the
following, two most common numerical simulation techniques, phase
diagrams and bifurcation diagrams, are used in the study of the
global dynamics of the nonlinear model.
Numerical simulations
The case L=2. In this part, we consider the case
of L=2, that is the system (12). The parameters
and
are selected to be fixed and
are
varied to characterize the changing of
. In the following
discussion, we choose
and
.
Firstly, let
so that
. In this case,
apart from the fixed positive equilibrium
with
, the system (12) has two sets of period three
fixed points, denoted
and
, where
and
It follows from Theorem 3 that, when
(so
that
),
is locally stable and the period three
point set S corresponds to the order 3 bifurcating from the
positive equilibrium
.
Fig.1 shows the phase plot of
, which is often called the pseudo-phase plot
of the system. We select four initial values:
and
.
Numerical simulations in Fig.1 indicate that,
solutions with the initial points
and
converge to the
fixed point
, while the solutions with
and
converge
to the period three point set P. One can choose other initial values
to do the simulations, but it turns out that all the solutions with
different initial values will converge to either
or P, as
indicated in Fig.1. A more detailed numerical
simulation on the basins of the attractors
and
P are plotted in Fig.2, in which, all the solutions
with initial values from the shaded area converge to
and the
rest of the solutions converge to P.
Figure 1: Pseudo-phase plot of (12) with
Figure 2: Basin plot of (12) with
In Fig.3we enlarge the central part of
Fig.1, we can then see clearly the structure of
the bifurcating point set S. We select four initial values
and (2.0, 2.8). As
suggested by Iooss [5] (pp. 127-128), S is a set of
saddle points.
Figure 3: Pseudo-phase plot of (12) with
Theorem 2 asserts the bifurcating behavior when
is
near the critical value 1. Now the question is whether the
single one-parameter family of fixed points S of order 3 exists
when
moves away from 1. In fact one can check that,
when
increases from -1.95 to -1.82207, apart from
the fixed equilibrium
, the two set of one-parameter
(
) family of fixed points S and P of order 3 continue
to exist and the distance between P and S, which is defined by
, decreases. When
,
the system has only the positive fixed equilibrium
and the
solutions with initial values (6, 8.5), (6, 8.6), (6, 8.7) and (6,
10) are plotted in Fig.4. This implies that there
exists
, or equivalently there
exists a
such that, for
, the
structure of the solutions is given by Fig.1 and
Fig.3; while for
(and near
), the structure is indicated by Fig.4.
Noting that the solutions remain near an order 3 periodic solution
before they converge to
with
. An
interesting finding is that, when we fixed the first initial
value, say
, and increase the second initial values, say
, from 10 up to near 30, the numerical steps needed for the
convergence increases, after 30, the numbers of steps decreases.
Figure 4: Solutions
Figure 5: Pseudo-phase plot of (12) with
As
decreases from
to -2 (but greater than -2),
that is
increases from
to 1, the distance
between two sets of one-parameter (
) families of fixed
points S and P of order 3 increases and, correspondingly, the
distance between S and
decreases (to zero). When
, that is
, the system has a fixed
equilibrium
with
and an order 3 periodic set P.
The phase structure in this case is indicated in Fig.5,
in which four initial values (2.2, 6), (2.62, 6), (2.3, 6) and
(3.67, 3.67) are selected. One can see that P is attracting and
is unstable and it has also the properties of the saddle
point S with both stable and unstable manifolds.
Now we choose
(so that
), then the system
(12) has a fixed equilibrium
with
and
two sets of order 3 bifurcating points P and S. In
Fig.6, we have the phase plot of the solutions
with initial values (3.43, 3.43), (1.85, 1.5), (1.8, 1.5), (2.1,
2.25) and (2.15, 2.25). It shows that P is the only attractor.
7shows the convergence of the order 3 periodic
orbit P, in which the initial value (1, 4) is selected.
Figure 6: Pseudo-phase plot of (12) with
Figure 7: Solutionof (3.1) with
General case. In general, near the critical value
, the system (9) has a periodic L orbit (fixed
points of order L) bifurcating from the positive fixed
equilibrium. The bifurcating periodic L orbit may have a similar
behaviour as the set S as in the case of L=2.
3and 3show the convergence of
the unique fixed equilibrium of the system with L=10 and
, where initial value (1.2, 1.3, 1.2, 1.3,
1.2, 1.3, 1.2, 1.3, 1.2, 1.3) is selected. When L=10 and
,
, 10 shows the
bifurcation of the positive equilibrium and the attractivity of a
family of periodic 10 orbits. Numerical simulations show that the
system (9) with L > 2 has similar dynamics to the one with
L=2.
Figure 8: Pseudo-phase plot with
Figure 9: Solution
Figure 10: Solution
Under the assumptions
and
, respectively, Boussard [2] shows that
these assumptions may result in the market generating chaotic
price and quantity series. He suggested that it would be more
rational to treat both prices and quantities as symmetrical and
this is indeed the basic assumption in this paper. Corresponding
to our case when L=2, he claimed that the main conclusions
remain approximately the same. However, our results suggest that,
under these more general symmetrical assumptions, the market
generates simpler dynamic behaviour. In order to generate more
complicated dynamics and chaotic motion, we need to replace p
and q in equations (1) by their logarithms, which is
also a natural solution to avoid negative prices and quantities
that can arise under the linear supply and demand curves. This
will be treated in the next section.
The problem of making use of linear supply and demand curves is the occurrence of negative values for prices and quantities. One solution to this problem is to replace p and q by their logarithms. That is, we replace the demand equation in (1) by
and the supply equation by
where a, b and
are positive and
are negative
constants.
One can rescale the equations by letting 2
.
We then have
and
Under constant absolute risk aversion A, the certainty
equivalent of the revenue r=pQ is
.
Thus the marginal revenue certainty equivalent is
Suppose a ``linear" (in terms of Q, not q) marginal cost so that the supply equation is
These results lead to the supply equation
that is
Assume
and
are formed as (5) and
(6) in section 1, then from (22) and (15)
the equality of supply and demand implies the market clearing
quantity
Using equation(17), we can rewrite the equation (23) in terms of the price
Let
Then the equation (24) can be written as the following L dimensional system of first order difference equations
The system (26) has a unique positive equilibrium
. One can verify
that, at the equilibrium point, the system (26) has the
Jacobian matrix J as defined in section 2. Therefore, Theorem
2 holds for system ({26) too.
Fig.11 is the phase plot of system (26)
when L=2 and
(and hence
). We select
three initial values
,
and
. The solution with
converges to the fixed equilibrium
and the solutions with
and
seem to converge to a
bounded attractor, rather than
. Fig.12
shows the case when
and the fixed equilibrium
is unstable. The corresponding attractor seems more complicated.
Figure 11: Pseudo-phase plot with
Figure 12: Pseudo-phase plot with
Fig.13 and Fig.14 show the case when
. It seems that the system has a strange
attractor when
. It may have
different shape for different
.
Figure 13: Pseudo-phase plot with
Figure 14: Pseudo-phase plot with
Figure 15: Bifurcation diagram
The above numerical simulations suggest that, for the system
(26), the market generates more complicated dynamics. In
particular, when
, the model may have chaotic
behaviour. The bifurcation plot of Q as a function of
is
shown in Fig.15, which indicates the complicated
dynamics of the system. Those simulations imply that the general
behaviour of models built along this line is very different from what
we have seen in the previous sections (certainly quite a different
picture to the one suggested by Boussard [2]).
To give the proof of Theorem 1, we need introduce concepts of the inners of a matrix and the positive innerwise matrix, which can be found from the book by Elaydi [3] (pages 180-181).
Let
be a matrix. The inners of the
matrix B are the matrix itself and all the matrices obtained by
omitting successively the first and last rows and the first and
last columns. A matrix B is said to be positive innerwise
if the determinants of all its inners are positive.
We now consider the kth order scalar equation
where the
's are real numbers. Obviously, the characteristic
equation of the equation (27) is given by
The Schue-Cohn criterion defines the conditions for the characteristic roots of equation (28) to fall inside the unit circle. More precisely, the following Jury's test will be used in our proof to Theorem 1.
Now let us prove Theorem 1. What we need to show is that
all the zeros of the characteristic polynomial
defined by (11) lie inside of the unit circle if and only
if
, that is,
satisfies the three
conditions in Theorem 3 if and only if
.
From
,it is easy to see that
and
if L is odd and
if L is even. Hence the first two conditions of Theorem
3 hold if and only if
. To show the third
condition is satisfied, it is enough to show that, for
, the matrix
with
are positive if and only if
.
Let k=2m be even. Then we have
To evaluate the determinate of
, we use (-1)
to multiply the i-th columns and add to the 2m-(i-1)-th
columns, respectively, for
. We then have
Now for
, we first add the
2m-(i-1)-the columns to the i-the columns, respectively. Then,
multiply
to the 2m-(i-1)-th column and add to the all
the first m-1 columns. as a result, the upper left block matrix
become a zero matrix and the down left block matrix has
as non-diagonal elements and
as diagonal elements.
Correspondingly,
We use -1 to time the first column and add to all the rest
columns. Then, use -1 to multiply the columns 2 to k and add
them to the first column. As as result, we have a low triangle
matrix with
. Therefore,
Similarly,
To find the
, we expand it first by the last row and
then by the first row and these lead to
. Since k=2m, it follows from the formula
that
In conclusion, we have for k=2m,
Next we assume that k=2m+1. Then
It is easy to see that
.
Using (35), we have
On the other hand,
To find the
, we multiply the i-th column by -1
and add to the 2m-(i-1)-the column,
respectively, for
.
Similarly, one can use row operations to reduce the upper left
matrix to a zero matrix and correspondingly,
Multiply the first column by -1 and add all the rest of the
columns of
and then, multiply the last column by
and add to the first column, multiply
to the
columns
and add to the first column. We then
add up with
Therefore
Then from (37) and (42), for k=2m+1,
Finally, it follows from (35) and
(43) that
are positive if and only if
and this completes the proof.
. Then maximisation of this function with
respect to
leads to the marginal revenue certainty
equivalent
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